🔬 Model Lab

I'll solve each problem step-by-step.

## Problem 1:
**What is asked:** Find the percentage of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.

**Original mixture:**
- Yellow tint: 30% of 30 = 0.30 × 30 = 9 liters
- Total volume: 30 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total volume: 30 + 5 = 35 liters

**Percentage of yellow tint:**
$$\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%$$

ANSWER 1: C

## Problem 2:
**What is asked:** In how many ways can the ghost enter by one window and leave by a different window?

**Solution:**
- Choices for entering: 6 windows
- Choices for leaving: 5 windows (must be different from entry)
- Total ways: 6 × 5 = 30

ANSWER 2: D

## Problem 3:
**What is asked:** Calculate 1/10 + 2/20 + 3/30

**Solution:**
Simplify each fraction:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

ANSWER 3: D

## Problem 4:
**What is asked:** How much money does Susan have left?

**Solution:**
- Started with: $50
- Spent on food: $12
- Spent on rides: 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Left: $50 - $36 = $14

ANSWER 4: B

## Problem 5:
**What is asked:** Calculate 200,000 × 200,000

**Solution:**
$$200,000 × 200,000 = (2 × 10^5) × (2 × 10^5) = 4 × 10^{10} = 40,000,000,000 = 40 \text{ billion}$$

ANSWER 5: E

## Problem 6:
**What is asked:** Find the missing digit A in the 3-digit number 1A2, where 11 club members each paid an equal integer amount that totals 1A2.

**Solution:**
The number 1A2 must be divisible by 11.

Testing divisibility by 11 (alternating sum of digits):
- 1 - A + 2 = 0 (mod 11)
- 3 - A ≡ 0 (mod 11)
- A = 3

Check: 132 ÷ 11 = 12 ✓

ANSWER 6: D

## Problem 7:
**What is asked:** Which shapes can Haruki make with 24 cm of wire?

**Solution:**
Test each shape:

**Regular hexagon with side 5 cm:**
- Perimeter: 6 × 5 = 30 cm
- Need 30 cm, have 24 cm ✗

**Square with area 36 cm²:**
- Side length: √36 = 6 cm
- Perimeter: 4 × 6 = 24 cm ✓

**Right triangle with legs 6 and 8 cm:**
- Hypotenuse: √(6² + 8²) = √(36 + 64) = √100 = 10 cm
- Perimeter: 6 + 8 + 10 = 24 cm ✓

Only square and triangle can be made.

ANSWER 7: D

## Problem 8:
**What is asked:** Which number has the largest reciprocal?

**Solution:**
Find reciprocals:
- 1/(1/3) = 3
- 1/(2/5) = 5/2 = 2.5
- 1/1 = 1
- 1/5 = 0.2
- 1/1986 ≈ 0.0005

The largest reciprocal is 5 (from the number 1/5).

ANSWER 8: D

## Problem 9:
**What is asked:** Find the target heart rate for a 26-year-old athlete.

**Solution:**
- Maximum heart rate: 220 - 26 = 194 beats per minute
- Target heart rate: 80% of 194 = 0.80 × 194 = 155.2 ≈ 155

ANSWER 9: B

## Problem 10:
**What is asked:** Find H - T where H is Harry's answer and T is Terry's answer.

**Solution:**
**Harry's calculation (correct):**
$$H = 8 - (2 + 5) = 8 - 7 = 1$$

**Terry's calculation (ignoring parentheses):**
$$T = 8 - 2 + 5 = 6 + 5 = 11$$

**Difference:**
$$H - T = 1 - 11 = -10$$

ANSWER 10: A

## Problem 11:
**What is asked:** Calculate 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

**Solution:**
This is an arithmetic sequence with 10 terms. Sum = n(first + last)/2
$$81 + 83 + ... + 99 = \frac{10(81 + 99)}{2} = \frac{10 × 180}{2} = 900$$

$$2 × 900 = 1800$$

ANSWER 11: E

## Problem 12:
**What is asked:** What percentage of males surveyed listen to the station?

**Solution:**
- Total surveyed: 200
- Females: 96
- Males: 200 - 96 = 104

- Total non-listeners: 64
- Males who don't listen: 26
- Males who listen: 104 - 26 = 78

**Percentage of males who listen:**
$$\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%$$

ANSWER 12: E

Problem 1:

The original mixture is 30 liters total, with 30% yellow tint.

So the amount of yellow tint at first is:
- \(30\%\) of \(30 = 0.30 \times 30 = 9\) liters

Then 5 liters of yellow tint are added, so the new amount of yellow tint is:
- \(9 + 5 = 14\) liters

The new total mixture is:
- \(30 + 5 = 35\) liters

So the percent of yellow tint is:
- \(\frac{14}{35} = \frac{2}{5} = 0.4 = 40\%\)

Answer choice C matches 40%.

ANSWER 1: C

Problem 2:

Georgie can:
- choose 1 of the 6 windows to enter
- then choose a different window to leave

For the entry, there are 6 choices.

For the exit, there are 5 remaining choices, since it must be a different window.

So the total number of ways is:
- \(6 \times 5 = 30\)

ANSWER 2: D

Problem 3:

Compute each fraction:
- \(\frac{1}{10} = 0.1\)
- \(\frac{2}{20} = \frac{1}{10} = 0.1\)
- \(\frac{3}{30} = \frac{1}{10} = 0.1\)

Add them:
- \(0.1 + 0.1 + 0.1 = 0.3\)

So the value is \(0.3\).

ANSWER 3: D

Problem 4:

Susan had \$50.

She spent:
- \$12 on food
- twice as much on rides, so \(2 \times 12 = 24\)

Total spent:
- \(12 + 24 = 36\)

Money left:
- \(50 - 36 = 14\)

ANSWER 4: B

Problem 5:

Two hundred thousand is:
- \(200{,}000 = 2 \times 10^5\)

Multiply:
- \(200{,}000 \times 200{,}000 = (2 \times 10^5)(2 \times 10^5) = 4 \times 10^{10}\)

And \(4 \times 10^{10} = 40{,}000{,}000{,}000\), which is forty billion.

ANSWER 5: E

Problem 6:

The total is written as \(1A2\), and 11 members each paid the same integer amount.

So \(1A2\) must be divisible by 11.

Test the answer choices:
- 102 ÷ 11 is not an integer
- 112 ÷ 11 is not an integer
- 122 ÷ 11 = 11 remainder 1
- 132 ÷ 11 = 12 exactly
- 142 ÷ 11 is not an integer

So the number is 132, which means \(A = 3\).

ANSWER 6: D

Problem 7:

We check each shape.

### Hexagon
A regular hexagon has 6 equal sides.
If each side is 5 cm, perimeter is:
- \(6 \times 5 = 30\) cm

The wire is only 24 cm, so he cannot make the hexagon.

### Square
A square with area 36 has side length:
- \(\sqrt{36} = 6\) cm

Perimeter:
- \(4 \times 6 = 24\) cm

So he can make the square exactly.

### Right triangle
The legs are 6 and 8, so the hypotenuse is:
- \(\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)

Perimeter:
- \(6 + 8 + 10 = 24\) cm

So he can make the triangle exactly.

Therefore he can make the square and the triangle only.

ANSWER 7: D

Problem 8:

The reciprocal is largest when the original number is smallest, assuming positive numbers.

Compare:
- \(\frac13 \approx 0.333\)
- \(\frac25 = 0.4\)
- \(1\)
- \(5\)
- \(1986\)

The smallest number is \(\frac13\), so it has the largest reciprocal.

Its reciprocal is 3, which is larger than the reciprocals of all the others.

ANSWER 8: A

Problem 9:

Maximum heart rate:
- \(220 - 26 = 194\)

Target heart rate is 80% of that:
- \(0.8 \times 194 = 155.2\)

To the nearest whole number, that is 155.

ANSWER 9: B

Problem 10:

First, Harry uses the parentheses:
- \(8 - (2 + 5) = 8 - 7 = 1\)

So \(H = 1\).

Terry ignores parentheses:
- \(8 - 2 + 5 = 6 + 5 = 11\)

So \(T = 11\).

Difference:
- \(H - T = 1 - 11 = -10\)

ANSWER 10: A

Problem 11:

We need to evaluate:
- \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

These are 10 odd numbers from 81 to 99.

Instead of adding one by one, use the average:
- average of 81 and 99 is \( \frac{81+99}{2} = 90\)

Since there are 10 numbers, their sum is:
- \(10 \times 90 = 900\)

Now multiply by 2:
- \(2 \times 900 = 1800\)

ANSWER 11: E

Problem 12:

We want the percentage of males surveyed who listen.

From the table:
- Total surveyed = 200
- Females = 96
- So males = \(200 - 96 = 104\)

Males who don’t listen = 26, so males who do listen:
- \(104 - 26 = 78\)

Now find the percent of males who listen:
- \(\frac{78}{104} = \frac{3}{4} = 75\%\)

ANSWER 12: E

Problem 1:  
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?  

- Original yellow tint amount = 30 × 0.30 = 9 liters.  
- Add 5 liters of yellow tint → new yellow amount = 9 + 5 = 14 liters.  
- New total volume = 30 + 5 = 35 liters.  
- New percent yellow = (14 / 35) × 100% = (2/5) × 100% = 40%.  

So the correct choice is 40.  

ANSWER 1: C  

---

Problem 2:  
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?  

- Choose an entry window: 6 choices.  
- Choose a different exit window: 5 choices (cannot be the same as entry).  
- Total ways = 6 × 5 = 30.  

ANSWER 2: D  

---

Problem 3:  
Compute: \( \frac{1}{10} + \frac{2}{20} + \frac{3}{30} \).  

Convert each to a common denominator, 60:  
- \( \frac{1}{10} = \frac{6}{60} \)  
- \( \frac{2}{20} = \frac{2}{20} = \frac{6}{60} \) (since multiply top and bottom by 3)  
- \( \frac{3}{30} = \frac{3}{30} = \frac{6}{60} \) (since multiply top and bottom by 2)  

Add:  
\( \frac{6}{60} + \frac{6}{60} + \frac{6}{60} = \frac{18}{60} = \frac{3}{10} = 0.3 \)

So the value is .3.  

ANSWER 3: D  

---

Problem 4:  
Susan had $50 to spend. She spent $12 on food and twice as much on rides. How much left?  

- Food = 12  
- Rides = twice 12 = 24  
- Total spent = 12 + 24 = 36  
- Money left = 50 − 36 = 14  

ANSWER 4: B  

---

Problem 5:  
Two hundred thousand times two hundred thousand equals ?  

\(200{,}000 \times 200{,}000 = (2\times 10^5)(2\times 10^5) = 4 \times 10^{10}\).  
- \(10^{10} = 10{,}000{,}000{,}000\)  
- \(4 \times 10^{10} = 40{,}000{,}000{,}000 = 40\) billion  

ANSWER 5: E  

---

Problem 6:  
Eleven members each paid the same integer amount for a guest speaker. In all, they paid $1A2 (a 3-digit number). Find missing digit A.  

- Total = \(1A2\) means:  
  \(1A2 = 100 + 10A + 2 = 102 + 10A\).  
- This total is divided equally among 11 people:  
  So \(102 + 10A\) must be divisible by 11.  

Check \(102 + 10A \mod 11\):  
- \(102 \div 11 = 9\) remainder 3, so \(102 \equiv 3 \pmod{11}\).  
- \(10A \equiv -A \pmod{11}\) (since 10 ≡ −1 mod 11).  

So:  
\(102 + 10A \equiv 3 - A \equiv 0 \pmod{11}\)  
Thus \(A \equiv 3 \pmod{11}\).  
Since \(A\) is a digit 0–9, \(A=3\).  

ANSWER 6: D  

---

Problem 7:  
Wire length = 24 cm. Which shapes can be made (using all 24 cm of wire)?  

We need the perimeter (total boundary length) of each shape:

**(1) Regular hexagon with side length 5 cm**  
Perimeter = 6 × 5 = 30 cm → too long. Cannot.

**(2) Square with area 36 cm²**  
Side length = √36 = 6 cm  
Perimeter = 4 × 6 = 24 cm → fits. Can.

**(3) Right triangle with legs 6 and 8 cm**  
Right triangle sides are 6 and 8 legs, and hypotenuse:  
\( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)  
Perimeter = 6 + 8 + 10 = 24 cm → fits. Can.

So the shapes possible: **square and triangle only**.  

ANSWER 7: D  

---

Problem 8:  
Which number has the largest reciprocal? Recall: the reciprocal of \(x\) is \(1/x\).  
We compare \(1/3, 2/5, 1, 1/5, 1/1986\).  

Compute roughly:  
- \(1/3 \approx 0.333\)  
- \(2/5 = 0.4\)  
- \(1 = 1\)  
- \(1/5 = 0.2\)  
- \(1/1986\) is tiny  

Largest is 1 (reciprocal of 1).  

ANSWER 8: C  

---

Problem 9:  
Target heart rate = 80% of (maximum heart rate).  
Maximum heart rate = 220 − age.  
Age = 26.

- Maximum = 220 − 26 = 194  
- Target = 0.80 × 194 = 155.2  
Nearest whole number = 155  

ANSWER 9: B  

---

Problem 10:  
Expression: \(8 - (2 + 5)\)  
- Harry computes correctly:  
  \(2 + 5 = 7\), so \(8 - 7 = 1\).  
- Terry ignores parentheses:  
  \(8 - 2 + 5 = 6 + 5 = 11\).  

Difference \(H - T = 1 - 11 = -10\).  

ANSWER 10: A  

---

Problem 11:  
Compute:  
\(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

Notice these are all odd numbers from 81 to 99 stepping by 2.  
There are 10 terms.

Pair from ends:  
- 81 + 99 = 180  
- 83 + 97 = 180  
- 85 + 95 = 180  
- 87 + 93 = 180  
- 89 + 91 = 180  

There are 5 pairs, each sum 180:  
Sum = 5 × 180 = 900  

Now multiply by 2:  
2 × 900 = 1800  

ANSWER 11: E  

---

Problem 12:  
Total surveyed = 200  
Females = 96  
Females who listen = 58  
Males who don’t listen = 26  
Total listeners = 136  

We want: % of males who listen.

Step 1: Find males total.  
Males = 200 − 96 = 104

Step 2: Find males who don’t listen is given = 26.  
So males who listen = 104 − 26 = 78

Step 3: Convert to percent of males who listen:  
Percent = (78 / 104) × 100%  
Simplify 78/104 = 3/4 (divide by 26):  
So percent = (3/4) × 100% = 75%

ANSWER 12: E

Problem 1:
The original mixture is 30 liters. The amount of yellow tint is 30% of 30 liters: $0.30 \times 30 = 9$ liters.
Five liters of yellow tint are added, so the new amount of yellow tint is $9 + 5 = 14$ liters.
The new total volume of the mixture is $30 + 5 = 35$ liters.
The percentage of yellow tint is $\frac{14}{35} \times 100\%$.
Simplifying the fraction: $\frac{14}{35} = \frac{2}{5} = 0.4$.
$0.4 \times 100\% = 40\%$.
ANSWER 1: C

Problem 2:
Georgie has 6 choices for the window he enters. Since he must leave by a different window, he has $6 - 1 = 5$ choices for the window he leaves by.
The total number of ways is $6 \times 5 = 30$.
ANSWER 2: D

Problem 3:
Simplify each fraction:
$1/10 = 0.1$
$2/20 = 1/10 = 0.1$
$3/30 = 1/10 = 0.1$
Summing them: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides, which is $2 \times 12 = 24$.
Total spent: $12 + 24 = 36$.
Amount left: $50 - 36 = 14$.
ANSWER 4: B

Problem 5:
Two hundred thousand is $200,000$ (or $2 \times 10^5$).
$(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
$4 \times 10^{10} = 40,000,000,000$, which is forty billion.
ANSWER 5: E

Problem 6:
The total amount $1A2$ must be divisible by 11.
A number is divisible by 11 if the alternating sum of its digits is divisible by 11.
For $1A2$, the alternating sum is $1 - A + 2 = 3 - A$.
For this to be 0 (the only multiple of 11 possible for a single digit $A$), $3 - A = 0$, so $A = 3$.
Check: $132 / 11 = 12$. This works.
ANSWER 6: D

Problem 7:
Hexagon: Perimeter of a regular hexagon with side 5 is $6 \times 5 = 30$ cm. (Cannot make)
Square: Area is 36, so side is $\sqrt{36} = 6$. Perimeter is $4 \times 6 = 24$ cm. (Can make)
Triangle: Right triangle with legs 6 and 8. Hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. Perimeter is $6 + 8 + 10 = 24$ cm. (Can make)
ANSWER 7: D

Problem 8:
Reciprocals:
A: $1/(1/3) = 3$
B: $1/(2/5) = 2.5$
C: $1/1 = 1$
D: $1/5 = 0.2$
E: $1/1986 \approx 0.0005$
The largest is 3.
ANSWER 8: A

Problem 9:
Max heart rate = $220 - 26 = 194$.
Target heart rate = $80\%$ of $194 = 0.8 \times 194 = 155.2$.
To the nearest whole number, this is 155.
ANSWER 9: B

Problem 10:
$H = 8 - (2 + 5) = 8 - 7 = 1$.
$T = 8 - 2 + 5 = 6 + 5 = 11$.
$H - T = 1 - 11 = -10$.
ANSWER 10: A

Problem 11:
Sum inside parentheses: $81+83+85+87+89+91+93+95+97+99$.
This is an arithmetic progression with 10 terms.
Sum = $\frac{n}{2}(\text{first} + \text{last}) = \frac{10}{2}(81 + 99) = 5 \times 180 = 900$.
The expression is $2 \times 900 = 1800$.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96.
Males = $200 - 96 = 104$.
Males who don't listen = 26.
Males who listen = $104 - 26 = 78$.
Percentage of males who listen = $(78 / 104) \times 100\%$.
$\frac{78}{104} = \frac{39}{52} = \frac{3}{4} = 0.75 = 75\%$.
ANSWER 12: E

Problem 1: The original mixture has volume 30 L, of which 30% is yellow tint, so there are \(0.3 \times 30 = 9\) L of yellow tint. Adding 5 L of yellow tint produces a new total of 14 L of yellow tint and a new mixture volume of 35 L. The new percentage of yellow tint is therefore \((14/35) \times 100\% = 40\%\).

ANSWER 1: C

Problem 2: There are 6 choices for the entry window. For each of those choices there are then exactly 5 remaining choices for a different exit window, giving a total of \(6 \times 5 = 30\) ordered pairs of distinct windows.

ANSWER 2: D

Problem 3: Rewrite each term with a common denominator of 10:  
\(1/10 = 0.1\),  
\(2/20 = 1/10 = 0.1\),  
\(3/30 = 1/10 = 0.1\).  
Their sum is \(0.1 + 0.1 + 0.1 = 0.3\).

ANSWER 3: D

Problem 4: Twice the food cost is \(2 \times 12 = 24\) dollars spent on rides. The total spent is therefore \(12 + 24 = 36\) dollars, leaving \(50 - 36 = 14\) dollars.

ANSWER 4: B

Problem 5: Write the factors in scientific notation:  
\(200000 = 2 \times 10^5\), so  
\((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\),  
which is forty billion.

ANSWER 5: E

Problem 6: The three-digit total \(1A2\) must be divisible by 11. The alternating-sum divisibility rule requires that \(1 - A + 2 = 3 - A\) be a multiple of 11. The only admissible digit \(A\) (0-9) that satisfies the condition is \(A = 3\), giving the total 132, which is exactly \(11 \times 12\).

ANSWER 6: D

Problem 7: A regular hexagon with side 5 cm has perimeter 30 cm, which exceeds the 24 cm length of wire, so it is impossible. A square of area 36 cm² has side length 6 cm and perimeter 24 cm, which matches the wire length. A right triangle with legs 6 cm and 8 cm has hypotenuse 10 cm, so its perimeter is \(6 + 8 + 10 = 24\) cm, which also matches. Consequently only the square and the triangle are possible.

ANSWER 7: D

Problem 8: For positive numbers the reciprocal function is strictly decreasing, so the largest reciprocal belongs to the smallest number in the list, which is \(1/3\).

ANSWER 8: A

Problem 9: The theoretical maximum heart rate is \(220 - 26 = 194\) beats per minute. Eighty percent of that value is \(0.8 \times 194 = 155.2\), which rounds to the nearest whole number 155.

ANSWER 9: B

Problem 10: Harry evaluates the parentheses first and obtains \(8 - (2 + 5) = 1\). Terry omits the parentheses and obtains \(8 - 2 + 5 = 11\). Their difference is therefore \(1 - 11 = -10\).

ANSWER 10: A

Problem 11: The sum inside the parentheses consists of the ten consecutive odd integers from 81 to 99. Their average is \((81 + 99)/2 = 90\), so the sum equals \(10 \times 90 = 900\). Multiplying by the outer factor of 2 yields \(2 \times 900 = 1800\).

ANSWER 11: E

Problem 12: There are 200 people surveyed and 96 females, so there are \(200 - 96 = 104\) males. Of the 96 females, 58 listen, leaving \(96 - 58 = 38\) females who do not listen. Given that 26 males do not listen, the number of males who listen is \(104 - 26 = 78\). The required percentage is therefore \((78/104) \times 100\% = 75\%\).

ANSWER 12: E

Problem 1:
The problem asks us to find the percentage of yellow tint in a new mixture after adding 5 liters of yellow tint to the original 30 liters of paint mixture. The original mixture is 25% red tint, 30% yellow tint, and 45% water.

First, let's calculate the amount of yellow tint in the original mixture. 30% of 30 liters is $0.3 \times 30 = 9$ liters.

After adding 5 liters of yellow tint, the total amount of yellow tint becomes $9 + 5 = 14$ liters.

The total volume of the new mixture is $30 + 5 = 35$ liters.

To find the percentage of yellow tint in the new mixture, we divide the amount of yellow tint by the total volume and multiply by 100: $\frac{14}{35} \times 100 = 40\%$.

Let's examine the choices:
- A. 25: Too low because the percentage of yellow tint increased.
- B. 35: Not the correct calculation.
- C. 40: Matches our calculation.
- D. 45: Too high.
- E. 50: Too high.

The correct percentage is 40%.

ANSWER 1: C

Problem 2:
The problem asks us to determine the number of ways Georgie the Ghost can enter a haunted house through one of six windows and leave by a different window.

To enter, Georgie has 6 choices. After entering, there are 5 remaining windows to choose from for leaving.

The total number of ways to enter and leave is $6 \times 5 = 30$.

Let's examine the choices:
- A. 12: Too low.
- B. 15: Too low.
- C. 18: Too low.
- D. 30: Matches our calculation.
- E. 36: Too high because it would imply Georgie can leave by the same window.

The correct number of ways is 30.

ANSWER 2: D

Problem 3:
The problem requires us to simplify the expression $1⁄10 + 2⁄20 + 3⁄30$.

First, simplify each fraction:
- $1⁄10$ stays the same.
- $2⁄20 = 1⁄10$.
- $3⁄30 = 1⁄10$.

Adding them together: $1⁄10 + 1⁄10 + 1⁄10 = 3⁄10 = 0.3$.

Let's examine the choices:
- A. .1: Too low.
- B. .123: Not the correct sum.
- C. .2: Too low.
- D. .3: Matches our calculation.
- E. .6: Too high.

The correct sum is 0.3.

ANSWER 3: D

Problem 4:
The problem asks us to find out how many dollars Susan had left after spending money on food and rides at the carnival. She started with $50, spent $12 on food, and twice as much on rides.

First, calculate the amount spent on rides: $12 \times 2 = $24.

Total amount spent is $12 + $24 = $36.

The amount left is $50 - $36 = $14.

Let's examine the choices:
- A. 12: Too low.
- B. 14: Matches our calculation.
- C. 26: Too high.
- D. 38: Too high.
- E. 50: Incorrect because she spent money.

The correct amount left is $14.

ANSWER 4: B

Problem 5:
The problem asks us to calculate $200,000 \times 200,000$.

$200,000 = 2 \times 10^5$, so $200,000 \times 200,000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10} = 40,000,000,000$.

Let's examine the choices:
- A. four hundred thousand: Too low.
- B. four million: Too low.
- C. forty thousand: Too low.
- D. four hundred million: Too low.
- E. forty billion: Matches our calculation.

The correct result is forty billion.

ANSWER 5: E

Problem 6:
The problem requires us to find the missing digit A in the 3-digit number $1A2 that represents the total amount paid by 11 members of the Math Club.

The total amount paid is divisible by 11 since there are 11 members who paid the same amount.

To check divisibility by 11, we use the rule: the difference between the sum of digits at odd places and the sum of digits at even places should be divisible by 11.

For $1A2, the sum of digits at odd places is $1 + 2 = 3$, and the sum at even places is $A$. The difference is $3 - A$.

For $3 - A$ to be divisible by 11 (or 0, since $A$ is a single digit), $A = 3$ makes $3 - A = 0$, which is divisible by 11.

Let's examine the choices:
- A. 0: Not correct because $3 - 0 = 3$, not divisible by 11.
- B. 1: Not correct because $3 - 1 = 2$, not divisible by 11.
- C. 2: Not correct because $3 - 2 = 1$, not divisible by 11.
- D. 3: Correct because $3 - 3 = 0$, which is divisible by 11.
- E. 4: Not correct because $3 - 4 = -1$, not divisible by 11.

The missing digit A is 3, but we should verify if 132 is indeed divisible by 11. $132 / 11 = 12$, so it is.

ANSWER 6: D

Problem 7:
The problem asks us to determine which shapes Haruki can form using a 24 cm long wire.

1. A regular hexagon with side length 5 cm: The perimeter is $6 \times 5 = 30$ cm, which is more than 24 cm. So, Haruki cannot form this hexagon.

2. A square of area 36 cm$^2$: The side length is $\sqrt{36} = 6$ cm. The perimeter is $4 \times 6 = 24$ cm, which matches the wire length. So, Haruki can form this square.

3. A right triangle with legs 6 and 8 cm: Using the Pythagorean theorem, the hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm. The perimeter is $6 + 8 + 10 = 24$ cm, which matches the wire length. So, Haruki can form this triangle.

Let's examine the choices:
- A. Triangle only: Incorrect because Haruki can also form the square.
- B. Hexagon and square only: Incorrect because Haruki cannot form the hexagon.
- C. Hexagon and triangle only: Incorrect because Haruki cannot form the hexagon.
- D. Square and triangle only: Matches our findings.
- E. Hexagon, triangle, and square: Incorrect because Haruki cannot form the hexagon.

The correct shapes Haruki can form are the square and the triangle.

ANSWER 7: D

Problem 8:
The problem asks us to find which of the given numbers has the largest reciprocal.

The reciprocal of a number $x$ is $1/x$. The larger $x$ is, the smaller its reciprocal is, and vice versa.

Let's examine the choices:
- A. $1⁄3$: The reciprocal is 3.
- B. $2⁄5$: The reciprocal is $5/2 = 2.5$.
- C. 1: The reciprocal is 1.
- D. 5: The reciprocal is $1/5 = 0.2$.
- E. 1986: The reciprocal is $1/1986$, very small.

Comparing the reciprocals: $3 > 2.5 > 1 > 0.2 > 1/1986$. So, $1/3$ has the largest reciprocal.

Let's examine the choices again:
- A. $1⁄3$: Correct.
- B. $2⁄5$: Not correct.
- C. 1: Not correct.
- D. 5: Not correct.
- E. 1986: Not correct.

The number with the largest reciprocal is $1/3$.

ANSWER 8: A

Problem 9:
The proble

Problem 1:
Original mixture: 30 L. Yellow tint = 30% of 30 = 0.30 × 30 = 9 L.
Add 5 L yellow: new yellow = 9 + 5 = 14 L, new total volume = 30 + 5 = 35 L.
Percent yellow = (14/35) × 100% = 40%.
ANSWER 1: C

Problem 2:
6 choices for entry window, 5 remaining choices for a different exit window.
Total ways = 6 × 5 = 30.
ANSWER 2: D

Problem 3:
1/10 + 2/20 + 3/30 = 1/10 + 1/10 + 1/10 = 3/10 = 0.3.
ANSWER 3: D

Problem 4:
Spent on food: $12. Spent on rides: 2 × $12 = $24.
Total spent = $12 + $24 = $36. Left = $50 − $36 = $14.
ANSWER 4: B

Problem 5:
200,000 × 200,000 = (2×10⁵) × (2×10⁵) = 4×10¹⁰ = 40,000,000,000, which is forty billion.
ANSWER 5: E

Problem 6:
Total paid is $1A2, a multiple of 11. Divisibility by 11: (1 − A + 2) = 3 − A must be a multiple of 11.
With A a digit 0–9, 3 − A = 0 ⇒ A = 3. (Check: 132 ÷ 11 = 12.)
ANSWER 6: D

Problem 7:
Wire length = 24 cm.
- Regular hexagon, side 5 cm: perimeter = 6×5 = 30 cm > 24 cm → cannot make.
- Square, area 36 cm²: side = 6 cm, perimeter = 4×6 = 24 cm → can make.
- Right triangle, legs 6 and 8 cm: hypotenuse = 10 cm, perimeter = 6+8+10 = 24 cm → can make.
So square and triangle only.
ANSWER 7: D

Problem 8:
Reciprocals: 1/(1/3)=3, 1/(2/5)=2.5, 1/1=1, 1/5=0.2, 1/1986≈0.0005. Largest is 3, from 1/3.
ANSWER 8: A

Problem 9:
Maximum heart rate = 220 − 26 = 194 bpm.
Target = 80% of 194 = 0.8 × 194 = 155.2 → 155 to nearest whole number.
ANSWER 9: B

Problem 10:
Harry: H = 8 − (2+5) = 8 − 7 = 1.
Terry: T = 8 − 2 + 5 = 6 + 5 = 11.
H − T = 1 − 11 = −10.
ANSWER 10: A

Problem 11:
Sum inside parentheses: 10 terms, average = (81+99)/2 = 90, sum = 10×90 = 900.
2 × 900 = 1800.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96 ⇒ Males = 200 − 96 = 104.
Total listeners = 136, female listeners = 58 ⇒ male listeners = 136 − 58 = 78.
Percent of males who listen = (78/104) × 100% = 75%.
ANSWER 12: E

Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint.

**Original mixture (30 liters):**
- Red tint: 25% of 30 = 7.5 liters
- Yellow tint: 30% of 30 = 9 liters
- Water: 45% of 30 = 13.5 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total mixture: 30 + 5 = 35 liters

**New percentage of yellow tint:**
14 / 35 = 2/5 = 0.40 = 40%

Tempting wrong answers: Choice B (35) might come from just adding 5 to the original 30%, but we must recalculate based on the new total.

ANSWER 1: C

---

Problem 2:
Georgie enters through one window and leaves through a *different* window.

- Number of choices for the entry window: 6
- Number of choices for the exit window (must be different): 5
- Total ways: 6 × 5 = 30

Tempting wrong answers: Choice E (36) would be 6 × 6, which allows entering and leaving through the same window. Choice A (12) would be 6 + 6, which doesn't account for all combinations.

ANSWER 2: D

---

Problem 3:
Simplify each fraction first:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

Tempting wrong answers: Choice B (.123) might come from concatenating the digits 1, 2, 3 instead of actually computing. Choice C (.2) might come from miscounting the terms.

ANSWER 3: D

---

Problem 4:
Susan starts with $50.
- Spent on food: $12
- Spent on rides: twice the food amount = 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Amount left: $50 − $36 = $14

Tempting wrong answers: Choice C (26) might come from subtracting only the food ($50 − $24 = $26, forgetting food). Choice D (38) comes from subtracting only $12.

ANSWER 4: B

---

Problem 5:
200,000 × 200,000

Write in scientific notation:
(2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰

4 × 10¹⁰ = 40,000,000,000 = forty billion

Tempting wrong answers: Choice A (four hundred thousand) comes from adding instead of multiplying. Choice D (four hundred million) comes from miscounting zeros.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, so the total $1A2 must be divisible by 11.

**Divisibility rule for 11:** The alternating sum of digits must be a multiple of 11.
Alternating sum: 1 − A + 2 = 3 − A

For 3 − A to be divisible by 11 (and A to be a single digit 0–9):
3 − A = 0 → A = 3

**Check:** $132 ÷ 11 = $12 per member. This is a valid integer amount.

Tempting wrong answers: Other values of A don't make 3 − A a multiple of 11 within the digit range.

ANSWER 6: D

---

Problem 7:
Wire length = 24 cm. We check if each shape's perimeter ≤ 24 cm.

**Regular hexagon (side 5 cm):**
Perimeter = 6 × 5 = 30 cm. Since 30 > 24, he **cannot** make this.

**Square (area 36 cm²):**
Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Since 24 = 24, he **can** make this.

**Right triangle (legs 6 and 8 cm):**
Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Perimeter = 6 + 8 + 10 = 24 cm. Since 24 = 24, he **can** make this.

Haruki can make the square and triangle only.

Tempting wrong answers: Choice E includes the hexagon, but its perimeter exceeds the wire length.

ANSWER 7: D

---

Problem 8:
The reciprocal of a number x is 1/x. The *smaller* the positive number, the *larger* its reciprocal.

| Choice | Number | Reciprocal |
|--------|--------|------------|
| A | 1/3 | 3 |
| B | 2/5 | 5/2 = 2.5 |
| C | 1 | 1 |
| D | 5 | 1/5 = 0.2 |
| E | 1986 | 1/1986 ≈ 0.0005 |

The largest reciprocal is 3, which comes from 1/3.

Tempting wrong answers: Choice E (1986) is the largest number, so it has the *smallest* reciprocal.

ANSWER 8: A

---

Problem 9:
Athlete's age = 26 years.

**Step 1:** Maximum heart rate = 220 − 26 = 194 beats per minute.

**Step 2:** Target heart rate = 80% of 194 = 0.80 × 194 = 155.2

**Step 3:** Round to nearest whole number: 155

Tempting wrong answers: Choice C (176) might come from taking 80% of 220 without subtracting the age. Choice D (194) is the maximum heart rate itself.

ANSWER 9: B

---

Problem 10:
**Harry (correct calculation):**
H = 8 − (2 + 5) = 8 − 7 = 1

**Terry (ignores parentheses):**
T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:**
H − T = 1 − 11 = −10

Tempting wrong answers: Choice E (10) is |H − T|, but the question asks for H − T specifically, which is negative since Terry's answer is larger.

ANSWER 10: A

---

Problem 11:
We need: 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

The sum inside is an arithmetic sequence:
- First term: 81, Last term: 99
- Number of terms: 10
- Sum = (number of terms) × (first + last) / 2 = 10 × (81 + 99) / 2 = 10 × 180 / 2 = 10 × 90 = 900

Multiply by 2: 2 × 900 = 1800

Tempting wrong answers: Choice C (1700) might come from miscounting terms or miscalculating the average.

ANSWER 11: E

---

Problem 12:
Let me organize the data from the survey:

- Total surveyed: 200
- Females: 96 → Males: 200 − 96 = 104
- Total listeners: 136
- Females who listen: 58 → Males who listen: 136 − 58 = 78
- Males who don't listen: 26 (Check: 78 + 26 = 104 ✓)

**Percentage of males who listen:**
(78 / 104) × 100% = (3/4) × 100% = 75%

Tempting wrong answers: Choice C (52) might come from computing 104/200 (percentage of total who are male). Choice D (55) might come from 78/136 (percentage of listeners who are male).

ANSWER 12: E

 Problem 1:
We start with 30 liters of paint. The original amount of yellow tint is 30% of 30 liters:
\[
0.30 \times 30 = 9 \text{ liters.}
\]
Five liters of yellow tint are added, so the new amount of yellow tint is:
\[
9 + 5 = 14 \text{ liters.}
\]
The new total volume of the mixture is:
\[
30 + 5 = 35 \text{ liters.}
\]
The percent of yellow tint in the new mixture is:
\[
\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%.
\]
(Choice A is the original red percentage, D is the original water percentage, and E would require 17.5 liters of yellow, which we do not have.)
ANSWER 1: C

Problem 2:
Georgie chooses one of 6 windows to enter. Since he must leave through a different window, there are 5 remaining choices for his exit. By the multiplication principle:
\[
6 \times 5 = 30 \text{ ways.}
\]
(Choice E, 36, would count entering and leaving through the same window; the other choices are too small.)
ANSWER 2: D

Problem 3:
Simplify each fraction:
\[
\frac{2}{20} = \frac{1}{10}, \qquad \frac{3}{30} = \frac{1}{10}.
\]
So the sum is:
\[
\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3.
\]
(Choice A is just one term; C is 0.2; E is 0.6.)
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides:
\[
2 \times 12 = 24 \text{ dollars.}
\]
Total spent:
\[
12 + 24 = 36 \text{ dollars.}
\]
Amount left:
\[
50 - 36 = 14 \text{ dollars.}
\]
(Choice C is the total spent; D is the amount left if she had spent only $12.)
ANSWER 4: B

Problem 5:
\[
200{,}000 \times 200{,}000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}.
\]
\[
4 \times 10^{10} = 40{,}000{,}000{,}000,
\]
which is read as forty billion.
(Choice D is $4 \times 10^8$; B is $4 \times 10^6$.)
ANSWER 5: E

Problem 6:
The total amount paid is the three-digit number $1A2$, which equals:
\[
100 + 10A + 2 = 102 + 10A.
\]
Since 11 members each paid the same integer amount, $102 + 10A$ must be divisible by 11. Testing the choices:
- $A=0$: $102 \div 11 \approx 9.27$ (not an integer)
- $A=1$: $112 \div 11 \approx 10.18$ (not an integer)
- $A=2$: $122 \div 11 \approx 11.09$ (not an integer)
- $A=3$: $132 \div 11 = 12$ (integer!)
- $A=4$: $142 \div 11 \approx 12.9$ (not an integer)
Thus the missing digit is 3.
ANSWER 6: D

Problem 7:
The wire is 24 cm long, so a shape is possible only if its perimeter is at most 24 cm.
- **Regular hexagon, side 5 cm:** Perimeter $= 6 \times 5 = 30$ cm. $30 > 24$, so impossible.
- **Square of area 36 cm²:** Side $= \sqrt{36} = 6$ cm. Perimeter $= 4 \times 6 = 24$ cm. Exactly fits, so possible.
- **Right triangle with legs 6 and 8 cm:** Hypotenuse $= \sqrt{6^2 + 8^2} = \sqrt{100} = 10$ cm. Perimeter $= 6 + 8 + 10 = 24$ cm. Exactly fits, so possible.
Only the square and the triangle can be formed.
ANSWER 7: D

Problem 8:
For positive numbers, the reciprocal function $f(x) = \frac{1}{x}$ is decreasing, so the smallest number has the largest reciprocal. Compare the two smallest choices:
\[
\frac{1}{3} \approx 0.333\ldots, \qquad \frac{2}{5} = 0.4.
\]
Since $\frac{1}{3} < \frac{2}{5} < 1 < 5 < 1986$, the smallest number is $\frac{1}{3}$. Its reciprocal is $3$, which is larger than the reciprocals of all the other choices.
ANSWER 8: A

Problem 9:
First find the theoretical maximum heart rate:
\[
220 - 26 = 194 \text{ beats per minute.}
\]
The target rate is 80% of this:
\[
0.80 \times 194 = 155.2.
\]
Rounded to the nearest whole number, this is 155.
(Choice C is the maximum heart rate itself.)
ANSWER 9: B

Problem 10:
Harry computes correctly:
\[
H = 8 - (2 + 5) = 8 - 7 = 1.
\]
Terry ignores the parentheses:
\[
T = 8 - 2 + 5 = 6 + 5 = 11.
\]
The difference is:
\[
H - T = 1 - 11 = -10.
\]
ANSWER 10: A

Problem 11:
The ten numbers inside the parentheses form an arithmetic sequence: 81, 83, …, 99. The sum of an arithmetic sequence is:
\[
\frac{\text{number of terms}}{2} \times (\text{first} + \text{last}) = \frac{10}{2} \times (81 + 99) = 5 \times 180 = 900.
\]
Now multiply by the factor of 2 outside:
\[
2 \times 900 = 1800.
\]
(One can also pair them: $(81+99)=180$, $(83+97)=180$, etc., giving five pairs that each sum to 180.)
ANSWER 11: E

Problem 12:
Total surveyed = 200, and females surveyed = 96, so:
\[
\text{Males surveyed} = 200 - 96 = 104.
\]
Total listeners = 136, and females who listen = 58, so:
\[
\text{Males who listen} = 136 - 58 = 78.
\]
(We can check: total non-listeners = 64, males who don't listen = 26, so females who don't listen = 38; $58+38=96$ and $78+26=104$, which checks out.)

The percentage of males who listen is:
\[
\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%.
\]
ANSWER 12: E

Problem 1:
A mixture of 30 liters is 25% red, 30% yellow, and 45% water. Five liters of yellow tint are added. What is the percent of yellow tint in the new mixture?

**Setup:** Find the amount of yellow tint originally, add 5 liters, then find the new percentage.

**Original yellow tint:** 30% of 30 = 0.30 × 30 = 9 liters

**After adding 5 liters of yellow tint:**
- New yellow tint: 9 + 5 = 14 liters
- New total volume: 30 + 5 = 35 liters
- New percentage: 14/35 = 2/5 = 40%

**Checking wrong choices:** 25% would be if we just added the original percentages; 35% would be if we just added 5% to 30%; 45% and 50% are too high.

ANSWER 1: C

---

Problem 2:
Six windows. Georgie enters by one and leaves by a different one. How many ways?

**Setup:** Choose an entry window (6 choices), then choose a different exit window (5 choices).

**Calculation:** 6 × 5 = 30 ways

**Checking wrong choices:** 15 would be if we divided by 2 (treating order as irrelevant); 12 would be 6×2; 18 would be 6×3; 36 would be 6×6 (allowing same window).

ANSWER 2: D

---

Problem 3:
1/10 + 2/20 + 3/30 = ?

**Simplify each fraction:**
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

**Sum:** 1/10 + 1/10 + 1/10 = 3/10 = 0.3

**Checking wrong choices:** 0.1 would be just one term; 0.2 would be two terms; 0.6 would be adding numerators and denominators incorrectly.

ANSWER 3: D

---

Problem 4:
Susan had $50. She spent $12 on food and twice as much on rides. How much left?

**Setup:** Food = $12, Rides = 2 × $12 = $24

**Total spent:** $12 + $24 = $36

**Amount left:** $50 - $36 = $14

**Checking wrong choices:** 12 is just the food amount; 26 is 50-24 (forgetting food); 38 is 50-12 (forgetting rides).

ANSWER 4: B

---

Problem 5:
200,000 × 200,000 = ?

**Setup:** Express in scientific notation: 200,000 = 2 × 10⁵

**Calculation:** (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion

**Checking wrong choices:** Four hundred thousand is 4×10⁵; four million is 4×10⁶; forty thousand is 4×10⁴; four hundred million is 4×10⁸.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, totaling $1A2. Find the missing digit A.

**Setup:** The total $1A2 must be divisible by 11.

**Divisibility test for 11:** A number is divisible by 11 if the alternating sum of its digits is divisible by 11.

For 1A2: (1 + 2) - A = 3 - A must be divisible by 11.

Since A is a single digit (0-9), 3 - A can only equal 0 (the only multiple of 11 in range -6 to 3).

So A = 3, giving $132 = 11 × $12 ✓

**Checking:** With A=0: $102 ÷ 11 ≈ 9.27 (not integer); A=1: $112 ÷ 11 ≈ 10.18 (not integer); A=3: $132 ÷ 11 = 12 ✓

ANSWER 6: D

---

Problem 7:
Wire is 24 cm. Which shapes can be made?

**Regular hexagon with side 5 cm:** Perimeter = 6 × 5 = 30 cm. Need 30 cm, only have 24 cm. ❌

**Square with area 36 cm²:** Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Need 24 cm, have 24 cm. ✓

**Right triangle with legs 6 and 8 cm:** Hypotenuse = √(36 + 64) = √100 = 10 cm. Perimeter = 6 + 8 + 10 = 24 cm. Need 24 cm, have 24 cm. ✓

**Conclusion:** Can make the square and triangle, but not the hexagon.

ANSWER 7: D

---

Problem 8:
Which number has the largest reciprocal?

**Find each reciprocal:**
- A. 1/3 → reciprocal = 3
- B. 2/5 → reciprocal = 5/2 = 2.5
- C. 1 → reciprocal = 1
- D. 5 → reciprocal = 1/5 = 0.2
- E. 1986 → reciprocal ≈ 0.0005

**Key insight:** Smaller positive numbers have larger reciprocals. The smallest number listed is 1/3, so it has the largest reciprocal (3).

ANSWER 8: A

---

Problem 9:
Target heart rate = 80% of (220 - age). Find target for 26-year-old.

**Maximum heart rate:** 220 - 26 = 194 bpm

**Target heart rate:** 80% of 194 = 0.80 × 194 = 155.2 ≈ 155 bpm

**Checking wrong choices:** 134 would be about 69%; 176 would be about 91%; 194 is the max (100%); 243 is impossible.

ANSWER 9: B

---

Problem 10:
Harry calculates 8 − (2 + 5) correctly. Terry calculates 8 − 2 + 5. Find H − T.

**Harry's answer:** H = 8 − (2 + 5) = 8 − 7 = 1

**Terry's answer:** T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:** H − T = 1 − 11 = −10

**Checking wrong choices:** −6 would be if Terry got 7; 0 would be if they got the same answer; 6 or 10 would be T − H.

ANSWER 10: A

---

Problem 11:
2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99) = ?

**Setup:** This is 10 consecutive odd numbers. The average of the first and last is (81 + 99)/2 = 90.

**Sum of the 10 numbers:** 10 × 90 = 900

**Multiply by 2:** 2 × 900 = 1800

**Alternative verification:** Pair the numbers: (81+99) + (83+97) + (85+95) + (87+93) + (89+91) = 180 + 180 + 180 + 180 + 180 = 900. Then 2 × 900 = 1800.

ANSWER 11: E

---

Problem 12:
Find the percentage of males who listen to the station.

**Given information:**
- Total surveyed: 200
- Females: 96
- Females who listen: 58
- Males who don't listen: 26
- Total listeners: 136
- Total non-listeners: 64

**Find males:** Males = 200 - 96 = 104

**Find males who listen:** Males who listen = Total listeners - Females who listen = 136 - 58 = 78

**Verification:** Males who don't listen = 104 - 78 = 26 ✓ (matches given)

**Percentage of males who listen:** 78/104 × 100 = 75%

**Checking wrong choices:** 39% would be 78/200 (using total instead of males); 48% would be about 50/104; 52% would be about 54/104; 55% would be about 57/104.

ANSWER 12: E

**Problem 1**

The original mixture is 30 L with 30 % yellow tint → yellow = 0.30·30 = 9 L.  
Add 5 L of yellow → total yellow = 9 + 5 = 14 L.  
Total volume after addition = 30 + 5 = 35 L.  

Percent yellow = (14/35)·100 % = 40 %.

**Answer 1: C**

---

**Problem 2**

Choose an entry window (6 ways) and a different exit window (5 ways).  
Number of ways = 6 × 5 = 30.

**Answer 2: D**

---

**Problem 3**

\[
\frac{1}{10}=0.1,\qquad\frac{2}{20}=0.1,\qquad\frac{3}{30}=0.1
\]
Sum = 0.1 + 0.1 + 0.1 = 0.3.

**Answer 3: D**

---

**Problem 4**

Food cost = \$12.  
Rides cost = twice food = 2·12 = \$24.  
Total spent = 12 + 24 = \$36.  
Money left = 50 − 36 = \$14.

**Answer 4: B**

---

**Problem 5**

\(200{,}000 \times 200{,}000 = (2\times10^{5})^{2}=4\times10^{10}=40{,}000{,}000{,}000\).

That is 40 billion.

**Answer 5: E**

---

**Problem 6**

Total paid = \$1A2 = 100 + 10A + 2 = 102 + 10A.  
Since the 11 members each paid an integer amount, 102 + 10A must be divisible by 11.

\(102\equiv3\pmod{11}\).  
We need \(10A\equiv-3\equiv8\pmod{11}\).  
Because \(10\equiv-1\pmod{11}\), this is \(-A\equiv8\pmod{11}\) → \(A\equiv3\pmod{11}\).  
A digit, so \(A=3\).

**Answer 6: D**

---

**Problem 7**

- Hexagon: perimeter = 6·5 = 30 cm > 24 cm → impossible.  
- Square: side = √36 = 6 cm, perimeter = 4·6 = 24 cm → possible.  
- Right triangle: legs 6 cm, 8 cm, hypotenuse = √(6²+8²)=10 cm, perimeter = 6+8+10 = 24 cm → possible.

Only the square and triangle can be made.

**Answer 7: D**

---

**Problem 8**

Reciprocals:
- \(1/3 → 3\)  
- \(2/5 → 5/2 = 2.5\)  
- \(1 → 1\)  
- \(5 → 1/5 = 0.2\)  
- \(1986 → 1/1986 ≈ 0.0005\)

Largest reciprocal is 3, from \(1/3\).

**Answer 8: A**

---

**Problem 9**

Maximum heart rate = 220 − 26 = 194 bpm.  
Target = 80 % of 194 = 0.8·194 = 155.2 → rounded to 155 bpm.

**Answer 9: B**

---

**Problem 10**

Harry: \(8-(2+5)=8-7=1\).  
Terry (ignoring parentheses): \(8-2+5 = 6+5 = 11\).  
Difference \(H-T = 1-11 = -10\).

**Answer 10: A**

---

**Problem 11**

Sum of the ten terms: average = (81+99)/2 = 90, so sum = 90·10 = 900.  
Multiply by 2: \(2·900 = 1800\).

**Answer 11: E**

---

**Problem 12**

Total surveyed = 200.  
Females = 96 → Males = 200 − 96 = 104.  

Female listeners = 58 → Male listeners = 136 − 58 = 78.  

Percent of males who listen = (78/104)·100 % = (39/52)·100 % = 75 %.

**Answer 12: E**

---

**Final Answers**

ANSWER 1: C  
ANSWER 2: D  
ANSWER 3: D  
ANSWER 4: B  
ANSWER 5: E  
ANSWER 6: D  
ANSWER 7: D  
ANSWER 8: A  
ANSWER 9: B  
ANSWER 10: A  
ANSWER 11: E  
ANSWER 12: E

Problem 1:  
A 30-liter paint mixture is 25% red tint, 30% yellow tint, and 45% water. Five liters of yellow tint are added.  
Original yellow tint: $0.30 \times 30 = 9$ liters.  
After adding 5 liters: $9 + 5 = 14$ liters yellow.  
New total volume: $30 + 5 = 35$ liters.  
Percent yellow: $\frac{14}{35} = 0.40 = 40\%$.  
Tempting wrong choices: 35% (original yellow percent) or 45% (water percent).  
ANSWER 1: D  

Problem 2:  
Georgie enters through one of six windows and exits through a different one.  
Entry choices: 6. Exit choices: 5 (excluding entry window).  
Total ways: $6 \times 5 = 30$.  
Tempting wrong choices: 15 (if order didn’t matter) or 36 (if same window allowed).  
ANSWER 2: D  

Problem 3:  
$\frac{1}{10} + \frac{2}{20} + \frac{3}{30} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3$.  
Tempting wrong choices: 0.1 (one term) or 0.2 (two terms).  
ANSWER 3: D  

Problem 4:  
Susan starts with \$50.  
Food: \$12.  
Rides: $2 \times 12 = \$24$.  
Total spent: $12 + 24 = \$36$.  
Left: $50 - 36 = \$14$.  
Tempting wrong choices: \$26 (if spent \$24 total) or \$38 (if spent \$12 total).  
ANSWER 4: B  

Problem 5:  
$200,000 \times 200,000 = 40,000,000,000 = 40$ billion.  
Tempting wrong choices: 400,000 (added zeros) or 4 million (multiplied digits).  
ANSWER 5: E  

Problem 6:  
Total paid: \$1A2, an integer divisible by 11.  
$1A2 \mod 11 = 0$.  
$1 + 2 - A \equiv 0 \mod 11 \Rightarrow 3 - A \equiv 0 \mod 11 \Rightarrow A = 3$.  
Tempting wrong choices: 0, 1, 2, or 4 (incorrect divisibility).  
ANSWER 6: E (Wait, let's recheck: 1A2 must be divisible by 11. 132 ÷ 11 = 12, so A=3. But 132 is 11×12, so correct. However, the answer choices are A=0,B=1,C=2,D=3,E=4. So D is 3. So correct answer is D, not E. I made a mistake in the initial answer. Correcting:)  
**Correction**: $1A2 = 132$ when $A=3$, and $132 \div 11 = 12$, so $A=3$.  
ANSWER 6: D  

Problem 7:  
Wire length: 24 cm.  
- Regular hexagon with side 5 cm: Perimeter $6 \times 5 = 30$ cm (not 24).  
- Square with area 36 cm²: Side $6$ cm, perimeter $24$ cm.  
- Right triangle with legs 6 and 8 cm: Hypotenuse 10 cm, perimeter $6+8+10=24$ cm.  
Can make square and triangle.  
Tempting wrong choices: Hexagon (incorrect perimeter).  
ANSWER 7: D  

Problem 8:  
Largest reciprocal means smallest original number.  
$\frac{1}{3} \approx 0.333$, $\frac{2}{5} = 0.4$, $1$, $5$, $1986$.  
Smallest: $\frac{1}{3}$. Reciprocal: 3.  
Others: 2.5, 1, 0.2, 0.0005.  
Tempting wrong choices: $\frac{2}{5}$ (larger than $\frac{1}{3}$).  
ANSWER 8: A  

Problem 9:  
Maximum heart rate: $220 - 26 = 194$.  
Target: $0.80 \times 194 = 155.2 \approx 155$.  
Tempting wrong choices: 134 (70%), 176 (90%), or 243 (incorrect max).  
ANSWER 9: B  

Problem 10:  
Harry: $8 - (2 + 5) = 1$.  
Terry: $8 - 2 + 5 = 11$.  
Difference: $1 - 11 = -10$.  
Tempting wrong choices: 10 (absolute difference) or 0 (if parentheses ignored by both).  
ANSWER 10: A  

Problem 11:  
Sum: $81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99$.  
Average: 90, 10 terms: $90 \times 10 = 900$.  
$2 \times 900 = 1800$.  
Tempting wrong choices: 1600 (min+max=180, times 5=900, times 2=1800, but if miscount terms).  
ANSWER 11: E  

Problem 12:  
Total surveyed: 200.  
Females: 96, so males: $200 - 96 = 104$.  
Total listeners: 136.  
Female listeners: 58, so male listeners: $136 - 58 = 78$.  
Percent male listeners: $\frac{78}{104} = 0.75 = 75\%$.  
Tempting wrong choices: 39 (male non-listeners percent) or 52 (incorrect calculation).  
ANSWER 12: E

### Problem 1:
What is the percentage of yellow tint in the new 35-liter paint mixture, after adding 5 liters of yellow tint to an original 30-liter mixture that was 30% yellow?
First, calculate the original volume of yellow tint: 30% of 30 liters = \(0.3 \times 30 = 9\) liters. Add the 5 new liters of yellow tint to get \(9 + 5 = 14\) liters of yellow in the new mixture. The total new volume of the mixture is \(30 + 5 = 35\) liters. The percentage of yellow is \(\frac{14}{35} = 0.4 = 40\%\). Tempting wrong answers come from forgetting to update the total mixture volume when calculating the new percentage, or miscalculating the original yellow volume.
ANSWER 1: C

---

### Problem 2:
How many ways can Georgie enter through one window and leave through a different window, out of 6 total windows?
Georgie has 6 choices for an entry window. Since he must leave through a different window, he has 5 remaining choices for an exit window. The total number of ways is \(6 \times 5 = 30\). A common wrong answer is 15, which incorrectly treats entry/exit as unordered pairs, when order matters (entering window A and leaving B is different from entering B and leaving A).
ANSWER 2: D

---

### Problem 3:
Simplify \(\frac{1}{10} + \frac{2}{20} + \frac{3}{30}\):
Simplify each fraction first: \(\frac{2}{20} = \frac{1}{10}\) and \(\frac{3}{30} = \frac{1}{10}\). Add them together: \(\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3\). A common wrong answer is 0.1, which incorrectly adds numerators and denominators directly instead of simplifying first.
ANSWER 3: D

---

### Problem 4:
How much money does Susan have left after spending $12 on food and twice that amount on rides, starting with $50?
First calculate the amount spent on rides: twice $12 is \(2 \times 12 = 24\) dollars. Total spent is \(12 + 24 = 36\) dollars. The amount left is \(50 - 36 = 14\) dollars. A common wrong answer is 26, which incorrectly uses $12 instead of $24 for ride costs.
ANSWER 4: B

---

### Problem 5:
Calculate \(200,000 \times 200,000\):
Rewrite 200,000 as \(2 \times 10^5\). Multiplying two copies gives \((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\), which equals 40,000,000,000, or forty billion. Wrong answers come from miscounting the number of zeros in the product.
ANSWER 5: E

---

### Problem 6:
The 3-digit total cost \(1A2\) dollars is divisible by 11 (11 members paid the same integer amount). What is the missing digit A?
The divisibility rule for 11 states that alternating sum of digits must be divisible by 11. For 1A2, the alternating sum is \(1 - A + 2 = 3 - A\). The only value of A (0-9) that makes this sum divisible by 11 is A=3, since \(3-3=0\), which is divisible by 11. Verify: \(132 \div 11 = 12\), a valid integer.
ANSWER 6: D

---

### Problem 7:
Haruki has a 24cm wire. Which of the three shapes can he form? Calculate the perimeter of each shape:
1. Regular hexagon with side 5cm: perimeter = \(6 \times 5 = 30\)cm, which is longer than 24cm, so impossible.
2. Square with area 36cm²: side length = \(\sqrt{36}=6\)cm, perimeter = \(4 \times 6 = 24\)cm, which fits.
3. Right triangle with legs 6cm and 8cm: hypotenuse = 10cm (6-8-10 Pythagorean triple), perimeter = \(6+8+10=24\)cm, which fits.
Only the square and triangle can be formed.
ANSWER 7: D

---

### Problem 8:
Which number has the largest reciprocal? The reciprocal of \(x\) is \(\frac{1}{x}\), so smaller values of \(x\) produce larger reciprocals. List the values of all options:
A. \(\frac{1}{3} \approx 0.333\), B. \(\frac{2}{5}=0.4\), C.1, D.5, E.1986. The smallest value is \(\frac{1}{3}\), whose reciprocal is 3, the largest of all options.
ANSWER 8: A

---

### Problem 9:
Calculate the target heart rate for a 26-year-old athlete: target is 80% of max heart rate, where max = 220 - age.
First calculate max heart rate: \(220 - 26 = 194\) bpm. 80% of 194 is \(0.8 \times 194 = 155.2\), which rounds to 155 bpm. A common wrong answer is 176, which uses 220 instead of the correct max heart rate of 194.
ANSWER 9: B

---

### Problem 10:
Calculate \(H-T\), where H is Harry's correct answer to \(8-(2+5)\), and T is Terry's incorrect answer of \(8-2+5\).
Harry's answer: \(8 - 7 = 1 = H\). Terry's answer: \(6 + 5 = 11 = T\). The difference \(H-T = 1 - 11 = -10\). A common wrong answer is 10, which calculates \(T-H\) instead of \(H-T\).
ANSWER 10: A

---

### Problem 11:
Simplify \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\):
Pair the terms inside the parentheses to simplify addition: \((81+99)+(83+97)+(85+95)+(87+93)+(89+91) = 5 \times 180 = 900\). Multiply by 2: \(2 \times 900 = 1800\).
ANSWER 11: E

---

### Problem 12:
What percentage of males surveyed listen to the radio station? First calculate total number of males: total surveyed = 200, females = 96, so males = \(200 - 96 = 104\). Males who do not listen = 26, so males who listen = \(104 - 26 = 78\). The percentage of males who listen is \(\frac{78}{104} = 0.75 = 75\%\). A common wrong answer is 39%, which divides male listeners by the total surveyed population instead of the total male population.
ANSWER 12: E

### Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.
1.  Calculate original yellow tint volume: 25%? No wait, original is 30% yellow. So 30% of 30 L = $0.3 \times 30 = 9$ liters of yellow.
2.  Add 5 liters of yellow: total yellow = $9 + 5 = 14$ liters.
3.  Total new mixture volume: $30 + 5 = 35$ liters.
4.  Percent yellow: $\frac{14}{35} \times 100 = 40\%$.
ANSWER 1: C
---
### Problem 2:
We count the number of ways to choose an entry window and a different exit window from 6 windows.
1.  Entry window: 6 possible choices.
2.  Exit window: cannot be the entry window, so 5 remaining choices.
3.  Total ways: $6 \times 5 = 30$ (order matters because entry and exit are distinct actions).
ANSWER 2: D
---
### Problem 3:
Simplify and add the fractions:
1.  $\frac{1}{10} = 0.1$, $\frac{2}{20} = \frac{1}{10} = 0.1$, $\frac{3}{30} = \frac{1}{10} = 0.1$.
2.  Sum: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D
---
### Problem 4:
Calculate remaining money after expenses:
1.  Total initial money: $50.
2.  Money spent on rides: twice the food cost = $2 \times 12 = 24$ dollars.
3.  Total spent: $12 + 24 = 36$ dollars.
4.  Remaining: $50 - 36 = 14$ dollars.
ANSWER 4: B
---
### Problem 5:
Compute $200,000 \times 200,000$:
1.  Rewrite as $(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
2.  $10^{10} = 10,000,000,000$, so $4 \times 10^{10} = 40,000,000,000 = 40$ billion.
ANSWER 5: E
---
### Problem 6:
The total payment $1A2$ (a 3-digit number) must be divisible by 11 (since 11 people paid equal integer amounts).
1.  Use the divisibility rule for 11: the alternating sum of digits is divisible by 11. For $1A2$, this is $1 - A + 2 = 3 - A$.
2.  $A$ is a digit (0-9), so $3 - A$ must equal 0 (the only multiple of 11 in range: $3 - A = 11$ gives $A=-8$, $3-A=-11$ gives $A=14$, both invalid).
3.  $3 - A = 0 \implies A=3$. Verify: $132 \div 11 = 12$, which is an integer.
ANSWER 6: D
---
### Problem 7:
Check if the 24 cm wire can form each shape (perimeter of the shape must equal 24 cm):
1.  Regular hexagon (side 5 cm): Perimeter = $6 \times 5 = 30$ cm > 24 cm → cannot make.
2.  Square (area 36 cm²): Side length = $\sqrt{36} = 6$ cm. Perimeter = $4 \times 6 = 24$ cm → can make.
3.  Right triangle (legs 6, 8 cm): Hypotenuse = $\sqrt{6^2 + 8^2} = 10$ cm. Perimeter = $6 + 8 + 10 = 24$ cm → can make.
Only square and triangle can be made.
ANSWER 7: D
---
### Problem 8:
The reciprocal of a positive number $x$ is $\frac{1}{x}$; smaller $x$ gives larger reciprocal.
1.  List values: $\frac{1}{3} \approx 0.33$, $\frac{2}{5}=0.4$, $1$, $5$, $1986$.
2.  Smallest value is $\frac{1}{3}$, whose reciprocal is $3$ (largest among all reciprocals: $\frac{5}{2}=2.5$, $1$, $0.2$, $\frac{1}{1986} \approx 0.0005$).
ANSWER 8: A
---
### Problem 9:
Calculate target heart rate for a 26-year-old:
1.  Maximum heart rate: $220 - 26 = 194$ bpm.
2.  Target heart rate: 80% of 194 = $0.8 \times 194 = 155.2$, which rounds to 155.
ANSWER 9: B
---
### Problem 10:
Compute $H$ (Harry's correct answer) and $T$ (Terry's incorrect answer):
1.  Harry: $8 - (2 + 5) = 8 - 7 = 1$, so $H=1$.
2.  Terry (ignores parentheses, left-to-right): $8 - 2 + 5 = 6 + 5 = 11$, so $T=11$.
3.  Difference: $H - T = 1 - 11 = -10$.
ANSWER 10: A
---
### Problem 11:
Calculate $2 \times (81 + 83 + ... + 99)$:
1.  The sum inside is 10 consecutive odd numbers. Average of first and last term: $\frac{81 + 99}{2} = 90$.
2.  Sum of the 10 terms: $90 \times 10 = 900$.
3.  Multiply by 2: $2 \times 900 = 1800$.
ANSWER 11: E
---
### Problem 12:
Find the percentage of males who listen to the station:
1.  Total males surveyed: $200 - 96 = 104$.
2.  Males who do not listen: 26, so males who listen: $104 - 26 = 78$.
3.  Verify with total listeners: Total listeners = 136, female listeners = 58, so male listeners = $136 - 58 = 78$ (matches).
4.  Percentage: $\frac{78}{104} \times 100 = 75\%$.
ANSWER 12: E

Problem 1:
Original yellow = 30% of 30 = 9 L. Add 5 → 14 L yellow. New total = 35 L. Percent = 14/35 = 40%.

ANSWER 1: C

Problem 2:
Enter: 6 choices, leave: 5 different choices. 6×5 = 30.

ANSWER 2: D

Problem 3:
Each fraction = 0.1. Sum = 0.3.

ANSWER 3: D

Problem 4:
Food 12, rides twice = 24. Spent 36. Left = 50 − 36 = 14.

ANSWER 4: B

Problem 5:
200,000 × 200,000 = 40,000,000,000 = forty billion.

ANSWER 5: E

Problem 6:
1A2 divisible by 11. Alternating sum: 1 − A + 2 = 3 − A must be divisible by 11 → A = 3 (gives 0). Check 132/11 = 12. Yes.

ANSWER 6: D

Problem 7:
Hexagon: 6×5 = 30 > 24, no. Square area 36 → side 6, perimeter 24, yes. Triangle legs 6,8 → hypotenuse 10, perimeter 24, yes. Square and triangle only.

ANSWER 7: D

Problem 8:
Largest reciprocal = smallest number. Smallest is 1/3.

ANSWER 8: A

Problem 9:
Max = 220 − 26 = 194. Target = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

Problem 10:
H = 8 − 7 = 1. T = 8 − 2 + 5 = 11. H − T = 1 − 11 = −10.

ANSWER 10: A

Problem 11:
Sum of 81..99 odd numbers (10 terms): average (81+99)/2 = 90, ×10 = 900. ×2 = 1800.

ANSWER 11: E

Problem 12:
Males total = 200 − 96 = 104. Males who don't listen = 26 → males who listen = 104 − 26 = 78. Percent = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem systematically.

---

**Problem 1:**
Original mixture: 30 liters, 30% yellow = 9 liters yellow. Add 5 liters yellow → new yellow = 14 liters, new total = 35 liters.
Percent yellow = 14/35 = 40%.

ANSWER 1: C

---

**Problem 2:**
Georgie can enter through any of 6 windows, then leave through any of the remaining 5 windows.
6 × 5 = 30.

ANSWER 2: D

---

**Problem 3:**
1/10 = 0.1, 2/20 = 0.1, 3/30 = 0.1. Sum = 0.3.

ANSWER 3: D

---

**Problem 4:**
Food: $12. Rides: 2 × $12 = $24. Total spent: $12 + $24 = $36. Left: $50 − $36 = $14.

ANSWER 4: B

---

**Problem 5:**
200,000 × 200,000 = (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion.

ANSWER 5: E

---

**Problem 6:**
The number 1A2 must be divisible by 11. 1A2 ÷ 11 must be an integer.
11 × 10 = 110, 11 × 11 = 121 (no), 11 × 12 = 132 ✓ → A = 3.
Check: 132/11 = 12 ✓, each paid $12.

ANSWER 6: D

---

**Problem 7:**
Wire = 24 cm.
- Regular hexagon, side 5: perimeter = 6 × 5 = 30 cm > 24. ✗
- Square, area 36: side = 6, perimeter = 24 cm ✓
- Right triangle, legs 6 and 8: hypotenuse = 10, perimeter = 6+8+10 = 24 cm ✓

ANSWER 7: D

---

**Problem 8:**
The reciprocal of x is 1/x. The largest reciprocal comes from the smallest positive number.
Among 1/3, 2/5, 1, 5, 1986 — the smallest is 1/3, and its reciprocal is 3, which is the largest.

ANSWER 8: A

---

**Problem 9:**
Maximum heart rate = 220 − 26 = 194. Target = 80% × 194 = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

---

**Problem 10:**
Harry: 8 − (2 + 5) = 8 − 7 = 1, so H = 1.
Terry: 8 − 2 + 5 = 11, so T = 11.
H − T = 1 − 11 = −10.

ANSWER 10: A

---

**Problem 11:**
The ten odd numbers from 81 to 99: sum = (81+99)/2 × 10 = 90 × 10 = 900.
2 × 900 = 1800.

ANSWER 11: E

---

**Problem 12:**
Total surveyed = 200. Females = 96, so Males = 104.
Total non-listeners = 64. Males who don't listen = 26, so females who don't listen = 64 − 26 = 38.
Males who listen = 104 − 26 = 78.
Percentage = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem step-by-step.

## Problem 1:
**What is asked:** Find the percentage of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.

**Original mixture:**
- Yellow tint: 30% of 30 = 0.30 × 30 = 9 liters
- Total volume: 30 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total volume: 30 + 5 = 35 liters

**Percentage of yellow tint:**
$$\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%$$

ANSWER 1: C

## Problem 2:
**What is asked:** In how many ways can the ghost enter by one window and leave by a different window?

**Solution:**
- Choices for entering: 6 windows
- Choices for leaving: 5 windows (must be different from entry)
- Total ways: 6 × 5 = 30

ANSWER 2: D

## Problem 3:
**What is asked:** Calculate 1/10 + 2/20 + 3/30

**Solution:**
Simplify each fraction:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

ANSWER 3: D

## Problem 4:
**What is asked:** How much money does Susan have left?

**Solution:**
- Started with: $50
- Spent on food: $12
- Spent on rides: 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Left: $50 - $36 = $14

ANSWER 4: B

## Problem 5:
**What is asked:** Calculate 200,000 × 200,000

**Solution:**
$$200,000 × 200,000 = (2 × 10^5) × (2 × 10^5) = 4 × 10^{10} = 40,000,000,000 = 40 \text{ billion}$$

ANSWER 5: E

## Problem 6:
**What is asked:** Find the missing digit A in the 3-digit number 1A2, where 11 club members each paid an equal integer amount that totals 1A2.

**Solution:**
The number 1A2 must be divisible by 11.

Testing divisibility by 11 (alternating sum of digits):
- 1 - A + 2 = 0 (mod 11)
- 3 - A ≡ 0 (mod 11)
- A = 3

Check: 132 ÷ 11 = 12 ✓

ANSWER 6: D

## Problem 7:
**What is asked:** Which shapes can Haruki make with 24 cm of wire?

**Solution:**
Test each shape:

**Regular hexagon with side 5 cm:**
- Perimeter: 6 × 5 = 30 cm
- Need 30 cm, have 24 cm ✗

**Square with area 36 cm²:**
- Side length: √36 = 6 cm
- Perimeter: 4 × 6 = 24 cm ✓

**Right triangle with legs 6 and 8 cm:**
- Hypotenuse: √(6² + 8²) = √(36 + 64) = √100 = 10 cm
- Perimeter: 6 + 8 + 10 = 24 cm ✓

Only square and triangle can be made.

ANSWER 7: D

## Problem 8:
**What is asked:** Which number has the largest reciprocal?

**Solution:**
Find reciprocals:
- 1/(1/3) = 3
- 1/(2/5) = 5/2 = 2.5
- 1/1 = 1
- 1/5 = 0.2
- 1/1986 ≈ 0.0005

The largest reciprocal is 5 (from the number 1/5).

ANSWER 8: D

## Problem 9:
**What is asked:** Find the target heart rate for a 26-year-old athlete.

**Solution:**
- Maximum heart rate: 220 - 26 = 194 beats per minute
- Target heart rate: 80% of 194 = 0.80 × 194 = 155.2 ≈ 155

ANSWER 9: B

## Problem 10:
**What is asked:** Find H - T where H is Harry's answer and T is Terry's answer.

**Solution:**
**Harry's calculation (correct):**
$$H = 8 - (2 + 5) = 8 - 7 = 1$$

**Terry's calculation (ignoring parentheses):**
$$T = 8 - 2 + 5 = 6 + 5 = 11$$

**Difference:**
$$H - T = 1 - 11 = -10$$

ANSWER 10: A

## Problem 11:
**What is asked:** Calculate 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

**Solution:**
This is an arithmetic sequence with 10 terms. Sum = n(first + last)/2
$$81 + 83 + ... + 99 = \frac{10(81 + 99)}{2} = \frac{10 × 180}{2} = 900$$

$$2 × 900 = 1800$$

ANSWER 11: E

## Problem 12:
**What is asked:** What percentage of males surveyed listen to the station?

**Solution:**
- Total surveyed: 200
- Females: 96
- Males: 200 - 96 = 104

- Total non-listeners: 64
- Males who don't listen: 26
- Males who listen: 104 - 26 = 78

**Percentage of males who listen:**
$$\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%$$

ANSWER 12: E

Problem 1:

The original mixture is 30 liters total, with 30% yellow tint.

So the amount of yellow tint at first is:
- \(30\%\) of \(30 = 0.30 \times 30 = 9\) liters

Then 5 liters of yellow tint are added, so the new amount of yellow tint is:
- \(9 + 5 = 14\) liters

The new total mixture is:
- \(30 + 5 = 35\) liters

So the percent of yellow tint is:
- \(\frac{14}{35} = \frac{2}{5} = 0.4 = 40\%\)

Answer choice C matches 40%.

ANSWER 1: C

Problem 2:

Georgie can:
- choose 1 of the 6 windows to enter
- then choose a different window to leave

For the entry, there are 6 choices.

For the exit, there are 5 remaining choices, since it must be a different window.

So the total number of ways is:
- \(6 \times 5 = 30\)

ANSWER 2: D

Problem 3:

Compute each fraction:
- \(\frac{1}{10} = 0.1\)
- \(\frac{2}{20} = \frac{1}{10} = 0.1\)
- \(\frac{3}{30} = \frac{1}{10} = 0.1\)

Add them:
- \(0.1 + 0.1 + 0.1 = 0.3\)

So the value is \(0.3\).

ANSWER 3: D

Problem 4:

Susan had \$50.

She spent:
- \$12 on food
- twice as much on rides, so \(2 \times 12 = 24\)

Total spent:
- \(12 + 24 = 36\)

Money left:
- \(50 - 36 = 14\)

ANSWER 4: B

Problem 5:

Two hundred thousand is:
- \(200{,}000 = 2 \times 10^5\)

Multiply:
- \(200{,}000 \times 200{,}000 = (2 \times 10^5)(2 \times 10^5) = 4 \times 10^{10}\)

And \(4 \times 10^{10} = 40{,}000{,}000{,}000\), which is forty billion.

ANSWER 5: E

Problem 6:

The total is written as \(1A2\), and 11 members each paid the same integer amount.

So \(1A2\) must be divisible by 11.

Test the answer choices:
- 102 ÷ 11 is not an integer
- 112 ÷ 11 is not an integer
- 122 ÷ 11 = 11 remainder 1
- 132 ÷ 11 = 12 exactly
- 142 ÷ 11 is not an integer

So the number is 132, which means \(A = 3\).

ANSWER 6: D

Problem 7:

We check each shape.

### Hexagon
A regular hexagon has 6 equal sides.
If each side is 5 cm, perimeter is:
- \(6 \times 5 = 30\) cm

The wire is only 24 cm, so he cannot make the hexagon.

### Square
A square with area 36 has side length:
- \(\sqrt{36} = 6\) cm

Perimeter:
- \(4 \times 6 = 24\) cm

So he can make the square exactly.

### Right triangle
The legs are 6 and 8, so the hypotenuse is:
- \(\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)

Perimeter:
- \(6 + 8 + 10 = 24\) cm

So he can make the triangle exactly.

Therefore he can make the square and the triangle only.

ANSWER 7: D

Problem 8:

The reciprocal is largest when the original number is smallest, assuming positive numbers.

Compare:
- \(\frac13 \approx 0.333\)
- \(\frac25 = 0.4\)
- \(1\)
- \(5\)
- \(1986\)

The smallest number is \(\frac13\), so it has the largest reciprocal.

Its reciprocal is 3, which is larger than the reciprocals of all the others.

ANSWER 8: A

Problem 9:

Maximum heart rate:
- \(220 - 26 = 194\)

Target heart rate is 80% of that:
- \(0.8 \times 194 = 155.2\)

To the nearest whole number, that is 155.

ANSWER 9: B

Problem 10:

First, Harry uses the parentheses:
- \(8 - (2 + 5) = 8 - 7 = 1\)

So \(H = 1\).

Terry ignores parentheses:
- \(8 - 2 + 5 = 6 + 5 = 11\)

So \(T = 11\).

Difference:
- \(H - T = 1 - 11 = -10\)

ANSWER 10: A

Problem 11:

We need to evaluate:
- \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

These are 10 odd numbers from 81 to 99.

Instead of adding one by one, use the average:
- average of 81 and 99 is \( \frac{81+99}{2} = 90\)

Since there are 10 numbers, their sum is:
- \(10 \times 90 = 900\)

Now multiply by 2:
- \(2 \times 900 = 1800\)

ANSWER 11: E

Problem 12:

We want the percentage of males surveyed who listen.

From the table:
- Total surveyed = 200
- Females = 96
- So males = \(200 - 96 = 104\)

Males who don’t listen = 26, so males who do listen:
- \(104 - 26 = 78\)

Now find the percent of males who listen:
- \(\frac{78}{104} = \frac{3}{4} = 75\%\)

ANSWER 12: E

Problem 1:  
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?  

- Original yellow tint amount = 30 × 0.30 = 9 liters.  
- Add 5 liters of yellow tint → new yellow amount = 9 + 5 = 14 liters.  
- New total volume = 30 + 5 = 35 liters.  
- New percent yellow = (14 / 35) × 100% = (2/5) × 100% = 40%.  

So the correct choice is 40.  

ANSWER 1: C  

---

Problem 2:  
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?  

- Choose an entry window: 6 choices.  
- Choose a different exit window: 5 choices (cannot be the same as entry).  
- Total ways = 6 × 5 = 30.  

ANSWER 2: D  

---

Problem 3:  
Compute: \( \frac{1}{10} + \frac{2}{20} + \frac{3}{30} \).  

Convert each to a common denominator, 60:  
- \( \frac{1}{10} = \frac{6}{60} \)  
- \( \frac{2}{20} = \frac{2}{20} = \frac{6}{60} \) (since multiply top and bottom by 3)  
- \( \frac{3}{30} = \frac{3}{30} = \frac{6}{60} \) (since multiply top and bottom by 2)  

Add:  
\( \frac{6}{60} + \frac{6}{60} + \frac{6}{60} = \frac{18}{60} = \frac{3}{10} = 0.3 \)

So the value is .3.  

ANSWER 3: D  

---

Problem 4:  
Susan had $50 to spend. She spent $12 on food and twice as much on rides. How much left?  

- Food = 12  
- Rides = twice 12 = 24  
- Total spent = 12 + 24 = 36  
- Money left = 50 − 36 = 14  

ANSWER 4: B  

---

Problem 5:  
Two hundred thousand times two hundred thousand equals ?  

\(200{,}000 \times 200{,}000 = (2\times 10^5)(2\times 10^5) = 4 \times 10^{10}\).  
- \(10^{10} = 10{,}000{,}000{,}000\)  
- \(4 \times 10^{10} = 40{,}000{,}000{,}000 = 40\) billion  

ANSWER 5: E  

---

Problem 6:  
Eleven members each paid the same integer amount for a guest speaker. In all, they paid $1A2 (a 3-digit number). Find missing digit A.  

- Total = \(1A2\) means:  
  \(1A2 = 100 + 10A + 2 = 102 + 10A\).  
- This total is divided equally among 11 people:  
  So \(102 + 10A\) must be divisible by 11.  

Check \(102 + 10A \mod 11\):  
- \(102 \div 11 = 9\) remainder 3, so \(102 \equiv 3 \pmod{11}\).  
- \(10A \equiv -A \pmod{11}\) (since 10 ≡ −1 mod 11).  

So:  
\(102 + 10A \equiv 3 - A \equiv 0 \pmod{11}\)  
Thus \(A \equiv 3 \pmod{11}\).  
Since \(A\) is a digit 0–9, \(A=3\).  

ANSWER 6: D  

---

Problem 7:  
Wire length = 24 cm. Which shapes can be made (using all 24 cm of wire)?  

We need the perimeter (total boundary length) of each shape:

**(1) Regular hexagon with side length 5 cm**  
Perimeter = 6 × 5 = 30 cm → too long. Cannot.

**(2) Square with area 36 cm²**  
Side length = √36 = 6 cm  
Perimeter = 4 × 6 = 24 cm → fits. Can.

**(3) Right triangle with legs 6 and 8 cm**  
Right triangle sides are 6 and 8 legs, and hypotenuse:  
\( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)  
Perimeter = 6 + 8 + 10 = 24 cm → fits. Can.

So the shapes possible: **square and triangle only**.  

ANSWER 7: D  

---

Problem 8:  
Which number has the largest reciprocal? Recall: the reciprocal of \(x\) is \(1/x\).  
We compare \(1/3, 2/5, 1, 1/5, 1/1986\).  

Compute roughly:  
- \(1/3 \approx 0.333\)  
- \(2/5 = 0.4\)  
- \(1 = 1\)  
- \(1/5 = 0.2\)  
- \(1/1986\) is tiny  

Largest is 1 (reciprocal of 1).  

ANSWER 8: C  

---

Problem 9:  
Target heart rate = 80% of (maximum heart rate).  
Maximum heart rate = 220 − age.  
Age = 26.

- Maximum = 220 − 26 = 194  
- Target = 0.80 × 194 = 155.2  
Nearest whole number = 155  

ANSWER 9: B  

---

Problem 10:  
Expression: \(8 - (2 + 5)\)  
- Harry computes correctly:  
  \(2 + 5 = 7\), so \(8 - 7 = 1\).  
- Terry ignores parentheses:  
  \(8 - 2 + 5 = 6 + 5 = 11\).  

Difference \(H - T = 1 - 11 = -10\).  

ANSWER 10: A  

---

Problem 11:  
Compute:  
\(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

Notice these are all odd numbers from 81 to 99 stepping by 2.  
There are 10 terms.

Pair from ends:  
- 81 + 99 = 180  
- 83 + 97 = 180  
- 85 + 95 = 180  
- 87 + 93 = 180  
- 89 + 91 = 180  

There are 5 pairs, each sum 180:  
Sum = 5 × 180 = 900  

Now multiply by 2:  
2 × 900 = 1800  

ANSWER 11: E  

---

Problem 12:  
Total surveyed = 200  
Females = 96  
Females who listen = 58  
Males who don’t listen = 26  
Total listeners = 136  

We want: % of males who listen.

Step 1: Find males total.  
Males = 200 − 96 = 104

Step 2: Find males who don’t listen is given = 26.  
So males who listen = 104 − 26 = 78

Step 3: Convert to percent of males who listen:  
Percent = (78 / 104) × 100%  
Simplify 78/104 = 3/4 (divide by 26):  
So percent = (3/4) × 100% = 75%

ANSWER 12: E

Problem 1:
The original mixture is 30 liters. The amount of yellow tint is 30% of 30 liters: $0.30 \times 30 = 9$ liters.
Five liters of yellow tint are added, so the new amount of yellow tint is $9 + 5 = 14$ liters.
The new total volume of the mixture is $30 + 5 = 35$ liters.
The percentage of yellow tint is $\frac{14}{35} \times 100\%$.
Simplifying the fraction: $\frac{14}{35} = \frac{2}{5} = 0.4$.
$0.4 \times 100\% = 40\%$.
ANSWER 1: C

Problem 2:
Georgie has 6 choices for the window he enters. Since he must leave by a different window, he has $6 - 1 = 5$ choices for the window he leaves by.
The total number of ways is $6 \times 5 = 30$.
ANSWER 2: D

Problem 3:
Simplify each fraction:
$1/10 = 0.1$
$2/20 = 1/10 = 0.1$
$3/30 = 1/10 = 0.1$
Summing them: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides, which is $2 \times 12 = 24$.
Total spent: $12 + 24 = 36$.
Amount left: $50 - 36 = 14$.
ANSWER 4: B

Problem 5:
Two hundred thousand is $200,000$ (or $2 \times 10^5$).
$(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
$4 \times 10^{10} = 40,000,000,000$, which is forty billion.
ANSWER 5: E

Problem 6:
The total amount $1A2$ must be divisible by 11.
A number is divisible by 11 if the alternating sum of its digits is divisible by 11.
For $1A2$, the alternating sum is $1 - A + 2 = 3 - A$.
For this to be 0 (the only multiple of 11 possible for a single digit $A$), $3 - A = 0$, so $A = 3$.
Check: $132 / 11 = 12$. This works.
ANSWER 6: D

Problem 7:
Hexagon: Perimeter of a regular hexagon with side 5 is $6 \times 5 = 30$ cm. (Cannot make)
Square: Area is 36, so side is $\sqrt{36} = 6$. Perimeter is $4 \times 6 = 24$ cm. (Can make)
Triangle: Right triangle with legs 6 and 8. Hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. Perimeter is $6 + 8 + 10 = 24$ cm. (Can make)
ANSWER 7: D

Problem 8:
Reciprocals:
A: $1/(1/3) = 3$
B: $1/(2/5) = 2.5$
C: $1/1 = 1$
D: $1/5 = 0.2$
E: $1/1986 \approx 0.0005$
The largest is 3.
ANSWER 8: A

Problem 9:
Max heart rate = $220 - 26 = 194$.
Target heart rate = $80\%$ of $194 = 0.8 \times 194 = 155.2$.
To the nearest whole number, this is 155.
ANSWER 9: B

Problem 10:
$H = 8 - (2 + 5) = 8 - 7 = 1$.
$T = 8 - 2 + 5 = 6 + 5 = 11$.
$H - T = 1 - 11 = -10$.
ANSWER 10: A

Problem 11:
Sum inside parentheses: $81+83+85+87+89+91+93+95+97+99$.
This is an arithmetic progression with 10 terms.
Sum = $\frac{n}{2}(\text{first} + \text{last}) = \frac{10}{2}(81 + 99) = 5 \times 180 = 900$.
The expression is $2 \times 900 = 1800$.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96.
Males = $200 - 96 = 104$.
Males who don't listen = 26.
Males who listen = $104 - 26 = 78$.
Percentage of males who listen = $(78 / 104) \times 100\%$.
$\frac{78}{104} = \frac{39}{52} = \frac{3}{4} = 0.75 = 75\%$.
ANSWER 12: E

Problem 1: The original mixture has volume 30 L, of which 30% is yellow tint, so there are \(0.3 \times 30 = 9\) L of yellow tint. Adding 5 L of yellow tint produces a new total of 14 L of yellow tint and a new mixture volume of 35 L. The new percentage of yellow tint is therefore \((14/35) \times 100\% = 40\%\).

ANSWER 1: C

Problem 2: There are 6 choices for the entry window. For each of those choices there are then exactly 5 remaining choices for a different exit window, giving a total of \(6 \times 5 = 30\) ordered pairs of distinct windows.

ANSWER 2: D

Problem 3: Rewrite each term with a common denominator of 10:  
\(1/10 = 0.1\),  
\(2/20 = 1/10 = 0.1\),  
\(3/30 = 1/10 = 0.1\).  
Their sum is \(0.1 + 0.1 + 0.1 = 0.3\).

ANSWER 3: D

Problem 4: Twice the food cost is \(2 \times 12 = 24\) dollars spent on rides. The total spent is therefore \(12 + 24 = 36\) dollars, leaving \(50 - 36 = 14\) dollars.

ANSWER 4: B

Problem 5: Write the factors in scientific notation:  
\(200000 = 2 \times 10^5\), so  
\((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\),  
which is forty billion.

ANSWER 5: E

Problem 6: The three-digit total \(1A2\) must be divisible by 11. The alternating-sum divisibility rule requires that \(1 - A + 2 = 3 - A\) be a multiple of 11. The only admissible digit \(A\) (0-9) that satisfies the condition is \(A = 3\), giving the total 132, which is exactly \(11 \times 12\).

ANSWER 6: D

Problem 7: A regular hexagon with side 5 cm has perimeter 30 cm, which exceeds the 24 cm length of wire, so it is impossible. A square of area 36 cm² has side length 6 cm and perimeter 24 cm, which matches the wire length. A right triangle with legs 6 cm and 8 cm has hypotenuse 10 cm, so its perimeter is \(6 + 8 + 10 = 24\) cm, which also matches. Consequently only the square and the triangle are possible.

ANSWER 7: D

Problem 8: For positive numbers the reciprocal function is strictly decreasing, so the largest reciprocal belongs to the smallest number in the list, which is \(1/3\).

ANSWER 8: A

Problem 9: The theoretical maximum heart rate is \(220 - 26 = 194\) beats per minute. Eighty percent of that value is \(0.8 \times 194 = 155.2\), which rounds to the nearest whole number 155.

ANSWER 9: B

Problem 10: Harry evaluates the parentheses first and obtains \(8 - (2 + 5) = 1\). Terry omits the parentheses and obtains \(8 - 2 + 5 = 11\). Their difference is therefore \(1 - 11 = -10\).

ANSWER 10: A

Problem 11: The sum inside the parentheses consists of the ten consecutive odd integers from 81 to 99. Their average is \((81 + 99)/2 = 90\), so the sum equals \(10 \times 90 = 900\). Multiplying by the outer factor of 2 yields \(2 \times 900 = 1800\).

ANSWER 11: E

Problem 12: There are 200 people surveyed and 96 females, so there are \(200 - 96 = 104\) males. Of the 96 females, 58 listen, leaving \(96 - 58 = 38\) females who do not listen. Given that 26 males do not listen, the number of males who listen is \(104 - 26 = 78\). The required percentage is therefore \((78/104) \times 100\% = 75\%\).

ANSWER 12: E

Problem 1:
The problem asks us to find the percentage of yellow tint in a new mixture after adding 5 liters of yellow tint to the original 30 liters of paint mixture. The original mixture is 25% red tint, 30% yellow tint, and 45% water.

First, let's calculate the amount of yellow tint in the original mixture. 30% of 30 liters is $0.3 \times 30 = 9$ liters.

After adding 5 liters of yellow tint, the total amount of yellow tint becomes $9 + 5 = 14$ liters.

The total volume of the new mixture is $30 + 5 = 35$ liters.

To find the percentage of yellow tint in the new mixture, we divide the amount of yellow tint by the total volume and multiply by 100: $\frac{14}{35} \times 100 = 40\%$.

Let's examine the choices:
- A. 25: Too low because the percentage of yellow tint increased.
- B. 35: Not the correct calculation.
- C. 40: Matches our calculation.
- D. 45: Too high.
- E. 50: Too high.

The correct percentage is 40%.

ANSWER 1: C

Problem 2:
The problem asks us to determine the number of ways Georgie the Ghost can enter a haunted house through one of six windows and leave by a different window.

To enter, Georgie has 6 choices. After entering, there are 5 remaining windows to choose from for leaving.

The total number of ways to enter and leave is $6 \times 5 = 30$.

Let's examine the choices:
- A. 12: Too low.
- B. 15: Too low.
- C. 18: Too low.
- D. 30: Matches our calculation.
- E. 36: Too high because it would imply Georgie can leave by the same window.

The correct number of ways is 30.

ANSWER 2: D

Problem 3:
The problem requires us to simplify the expression $1⁄10 + 2⁄20 + 3⁄30$.

First, simplify each fraction:
- $1⁄10$ stays the same.
- $2⁄20 = 1⁄10$.
- $3⁄30 = 1⁄10$.

Adding them together: $1⁄10 + 1⁄10 + 1⁄10 = 3⁄10 = 0.3$.

Let's examine the choices:
- A. .1: Too low.
- B. .123: Not the correct sum.
- C. .2: Too low.
- D. .3: Matches our calculation.
- E. .6: Too high.

The correct sum is 0.3.

ANSWER 3: D

Problem 4:
The problem asks us to find out how many dollars Susan had left after spending money on food and rides at the carnival. She started with $50, spent $12 on food, and twice as much on rides.

First, calculate the amount spent on rides: $12 \times 2 = $24.

Total amount spent is $12 + $24 = $36.

The amount left is $50 - $36 = $14.

Let's examine the choices:
- A. 12: Too low.
- B. 14: Matches our calculation.
- C. 26: Too high.
- D. 38: Too high.
- E. 50: Incorrect because she spent money.

The correct amount left is $14.

ANSWER 4: B

Problem 5:
The problem asks us to calculate $200,000 \times 200,000$.

$200,000 = 2 \times 10^5$, so $200,000 \times 200,000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10} = 40,000,000,000$.

Let's examine the choices:
- A. four hundred thousand: Too low.
- B. four million: Too low.
- C. forty thousand: Too low.
- D. four hundred million: Too low.
- E. forty billion: Matches our calculation.

The correct result is forty billion.

ANSWER 5: E

Problem 6:
The problem requires us to find the missing digit A in the 3-digit number $1A2 that represents the total amount paid by 11 members of the Math Club.

The total amount paid is divisible by 11 since there are 11 members who paid the same amount.

To check divisibility by 11, we use the rule: the difference between the sum of digits at odd places and the sum of digits at even places should be divisible by 11.

For $1A2, the sum of digits at odd places is $1 + 2 = 3$, and the sum at even places is $A$. The difference is $3 - A$.

For $3 - A$ to be divisible by 11 (or 0, since $A$ is a single digit), $A = 3$ makes $3 - A = 0$, which is divisible by 11.

Let's examine the choices:
- A. 0: Not correct because $3 - 0 = 3$, not divisible by 11.
- B. 1: Not correct because $3 - 1 = 2$, not divisible by 11.
- C. 2: Not correct because $3 - 2 = 1$, not divisible by 11.
- D. 3: Correct because $3 - 3 = 0$, which is divisible by 11.
- E. 4: Not correct because $3 - 4 = -1$, not divisible by 11.

The missing digit A is 3, but we should verify if 132 is indeed divisible by 11. $132 / 11 = 12$, so it is.

ANSWER 6: D

Problem 7:
The problem asks us to determine which shapes Haruki can form using a 24 cm long wire.

1. A regular hexagon with side length 5 cm: The perimeter is $6 \times 5 = 30$ cm, which is more than 24 cm. So, Haruki cannot form this hexagon.

2. A square of area 36 cm$^2$: The side length is $\sqrt{36} = 6$ cm. The perimeter is $4 \times 6 = 24$ cm, which matches the wire length. So, Haruki can form this square.

3. A right triangle with legs 6 and 8 cm: Using the Pythagorean theorem, the hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm. The perimeter is $6 + 8 + 10 = 24$ cm, which matches the wire length. So, Haruki can form this triangle.

Let's examine the choices:
- A. Triangle only: Incorrect because Haruki can also form the square.
- B. Hexagon and square only: Incorrect because Haruki cannot form the hexagon.
- C. Hexagon and triangle only: Incorrect because Haruki cannot form the hexagon.
- D. Square and triangle only: Matches our findings.
- E. Hexagon, triangle, and square: Incorrect because Haruki cannot form the hexagon.

The correct shapes Haruki can form are the square and the triangle.

ANSWER 7: D

Problem 8:
The problem asks us to find which of the given numbers has the largest reciprocal.

The reciprocal of a number $x$ is $1/x$. The larger $x$ is, the smaller its reciprocal is, and vice versa.

Let's examine the choices:
- A. $1⁄3$: The reciprocal is 3.
- B. $2⁄5$: The reciprocal is $5/2 = 2.5$.
- C. 1: The reciprocal is 1.
- D. 5: The reciprocal is $1/5 = 0.2$.
- E. 1986: The reciprocal is $1/1986$, very small.

Comparing the reciprocals: $3 > 2.5 > 1 > 0.2 > 1/1986$. So, $1/3$ has the largest reciprocal.

Let's examine the choices again:
- A. $1⁄3$: Correct.
- B. $2⁄5$: Not correct.
- C. 1: Not correct.
- D. 5: Not correct.
- E. 1986: Not correct.

The number with the largest reciprocal is $1/3$.

ANSWER 8: A

Problem 9:
The proble

Problem 1:
Original mixture: 30 L. Yellow tint = 30% of 30 = 0.30 × 30 = 9 L.
Add 5 L yellow: new yellow = 9 + 5 = 14 L, new total volume = 30 + 5 = 35 L.
Percent yellow = (14/35) × 100% = 40%.
ANSWER 1: C

Problem 2:
6 choices for entry window, 5 remaining choices for a different exit window.
Total ways = 6 × 5 = 30.
ANSWER 2: D

Problem 3:
1/10 + 2/20 + 3/30 = 1/10 + 1/10 + 1/10 = 3/10 = 0.3.
ANSWER 3: D

Problem 4:
Spent on food: $12. Spent on rides: 2 × $12 = $24.
Total spent = $12 + $24 = $36. Left = $50 − $36 = $14.
ANSWER 4: B

Problem 5:
200,000 × 200,000 = (2×10⁵) × (2×10⁵) = 4×10¹⁰ = 40,000,000,000, which is forty billion.
ANSWER 5: E

Problem 6:
Total paid is $1A2, a multiple of 11. Divisibility by 11: (1 − A + 2) = 3 − A must be a multiple of 11.
With A a digit 0–9, 3 − A = 0 ⇒ A = 3. (Check: 132 ÷ 11 = 12.)
ANSWER 6: D

Problem 7:
Wire length = 24 cm.
- Regular hexagon, side 5 cm: perimeter = 6×5 = 30 cm > 24 cm → cannot make.
- Square, area 36 cm²: side = 6 cm, perimeter = 4×6 = 24 cm → can make.
- Right triangle, legs 6 and 8 cm: hypotenuse = 10 cm, perimeter = 6+8+10 = 24 cm → can make.
So square and triangle only.
ANSWER 7: D

Problem 8:
Reciprocals: 1/(1/3)=3, 1/(2/5)=2.5, 1/1=1, 1/5=0.2, 1/1986≈0.0005. Largest is 3, from 1/3.
ANSWER 8: A

Problem 9:
Maximum heart rate = 220 − 26 = 194 bpm.
Target = 80% of 194 = 0.8 × 194 = 155.2 → 155 to nearest whole number.
ANSWER 9: B

Problem 10:
Harry: H = 8 − (2+5) = 8 − 7 = 1.
Terry: T = 8 − 2 + 5 = 6 + 5 = 11.
H − T = 1 − 11 = −10.
ANSWER 10: A

Problem 11:
Sum inside parentheses: 10 terms, average = (81+99)/2 = 90, sum = 10×90 = 900.
2 × 900 = 1800.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96 ⇒ Males = 200 − 96 = 104.
Total listeners = 136, female listeners = 58 ⇒ male listeners = 136 − 58 = 78.
Percent of males who listen = (78/104) × 100% = 75%.
ANSWER 12: E

Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint.

**Original mixture (30 liters):**
- Red tint: 25% of 30 = 7.5 liters
- Yellow tint: 30% of 30 = 9 liters
- Water: 45% of 30 = 13.5 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total mixture: 30 + 5 = 35 liters

**New percentage of yellow tint:**
14 / 35 = 2/5 = 0.40 = 40%

Tempting wrong answers: Choice B (35) might come from just adding 5 to the original 30%, but we must recalculate based on the new total.

ANSWER 1: C

---

Problem 2:
Georgie enters through one window and leaves through a *different* window.

- Number of choices for the entry window: 6
- Number of choices for the exit window (must be different): 5
- Total ways: 6 × 5 = 30

Tempting wrong answers: Choice E (36) would be 6 × 6, which allows entering and leaving through the same window. Choice A (12) would be 6 + 6, which doesn't account for all combinations.

ANSWER 2: D

---

Problem 3:
Simplify each fraction first:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

Tempting wrong answers: Choice B (.123) might come from concatenating the digits 1, 2, 3 instead of actually computing. Choice C (.2) might come from miscounting the terms.

ANSWER 3: D

---

Problem 4:
Susan starts with $50.
- Spent on food: $12
- Spent on rides: twice the food amount = 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Amount left: $50 − $36 = $14

Tempting wrong answers: Choice C (26) might come from subtracting only the food ($50 − $24 = $26, forgetting food). Choice D (38) comes from subtracting only $12.

ANSWER 4: B

---

Problem 5:
200,000 × 200,000

Write in scientific notation:
(2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰

4 × 10¹⁰ = 40,000,000,000 = forty billion

Tempting wrong answers: Choice A (four hundred thousand) comes from adding instead of multiplying. Choice D (four hundred million) comes from miscounting zeros.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, so the total $1A2 must be divisible by 11.

**Divisibility rule for 11:** The alternating sum of digits must be a multiple of 11.
Alternating sum: 1 − A + 2 = 3 − A

For 3 − A to be divisible by 11 (and A to be a single digit 0–9):
3 − A = 0 → A = 3

**Check:** $132 ÷ 11 = $12 per member. This is a valid integer amount.

Tempting wrong answers: Other values of A don't make 3 − A a multiple of 11 within the digit range.

ANSWER 6: D

---

Problem 7:
Wire length = 24 cm. We check if each shape's perimeter ≤ 24 cm.

**Regular hexagon (side 5 cm):**
Perimeter = 6 × 5 = 30 cm. Since 30 > 24, he **cannot** make this.

**Square (area 36 cm²):**
Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Since 24 = 24, he **can** make this.

**Right triangle (legs 6 and 8 cm):**
Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Perimeter = 6 + 8 + 10 = 24 cm. Since 24 = 24, he **can** make this.

Haruki can make the square and triangle only.

Tempting wrong answers: Choice E includes the hexagon, but its perimeter exceeds the wire length.

ANSWER 7: D

---

Problem 8:
The reciprocal of a number x is 1/x. The *smaller* the positive number, the *larger* its reciprocal.

| Choice | Number | Reciprocal |
|--------|--------|------------|
| A | 1/3 | 3 |
| B | 2/5 | 5/2 = 2.5 |
| C | 1 | 1 |
| D | 5 | 1/5 = 0.2 |
| E | 1986 | 1/1986 ≈ 0.0005 |

The largest reciprocal is 3, which comes from 1/3.

Tempting wrong answers: Choice E (1986) is the largest number, so it has the *smallest* reciprocal.

ANSWER 8: A

---

Problem 9:
Athlete's age = 26 years.

**Step 1:** Maximum heart rate = 220 − 26 = 194 beats per minute.

**Step 2:** Target heart rate = 80% of 194 = 0.80 × 194 = 155.2

**Step 3:** Round to nearest whole number: 155

Tempting wrong answers: Choice C (176) might come from taking 80% of 220 without subtracting the age. Choice D (194) is the maximum heart rate itself.

ANSWER 9: B

---

Problem 10:
**Harry (correct calculation):**
H = 8 − (2 + 5) = 8 − 7 = 1

**Terry (ignores parentheses):**
T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:**
H − T = 1 − 11 = −10

Tempting wrong answers: Choice E (10) is |H − T|, but the question asks for H − T specifically, which is negative since Terry's answer is larger.

ANSWER 10: A

---

Problem 11:
We need: 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

The sum inside is an arithmetic sequence:
- First term: 81, Last term: 99
- Number of terms: 10
- Sum = (number of terms) × (first + last) / 2 = 10 × (81 + 99) / 2 = 10 × 180 / 2 = 10 × 90 = 900

Multiply by 2: 2 × 900 = 1800

Tempting wrong answers: Choice C (1700) might come from miscounting terms or miscalculating the average.

ANSWER 11: E

---

Problem 12:
Let me organize the data from the survey:

- Total surveyed: 200
- Females: 96 → Males: 200 − 96 = 104
- Total listeners: 136
- Females who listen: 58 → Males who listen: 136 − 58 = 78
- Males who don't listen: 26 (Check: 78 + 26 = 104 ✓)

**Percentage of males who listen:**
(78 / 104) × 100% = (3/4) × 100% = 75%

Tempting wrong answers: Choice C (52) might come from computing 104/200 (percentage of total who are male). Choice D (55) might come from 78/136 (percentage of listeners who are male).

ANSWER 12: E

 Problem 1:
We start with 30 liters of paint. The original amount of yellow tint is 30% of 30 liters:
\[
0.30 \times 30 = 9 \text{ liters.}
\]
Five liters of yellow tint are added, so the new amount of yellow tint is:
\[
9 + 5 = 14 \text{ liters.}
\]
The new total volume of the mixture is:
\[
30 + 5 = 35 \text{ liters.}
\]
The percent of yellow tint in the new mixture is:
\[
\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%.
\]
(Choice A is the original red percentage, D is the original water percentage, and E would require 17.5 liters of yellow, which we do not have.)
ANSWER 1: C

Problem 2:
Georgie chooses one of 6 windows to enter. Since he must leave through a different window, there are 5 remaining choices for his exit. By the multiplication principle:
\[
6 \times 5 = 30 \text{ ways.}
\]
(Choice E, 36, would count entering and leaving through the same window; the other choices are too small.)
ANSWER 2: D

Problem 3:
Simplify each fraction:
\[
\frac{2}{20} = \frac{1}{10}, \qquad \frac{3}{30} = \frac{1}{10}.
\]
So the sum is:
\[
\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3.
\]
(Choice A is just one term; C is 0.2; E is 0.6.)
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides:
\[
2 \times 12 = 24 \text{ dollars.}
\]
Total spent:
\[
12 + 24 = 36 \text{ dollars.}
\]
Amount left:
\[
50 - 36 = 14 \text{ dollars.}
\]
(Choice C is the total spent; D is the amount left if she had spent only $12.)
ANSWER 4: B

Problem 5:
\[
200{,}000 \times 200{,}000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}.
\]
\[
4 \times 10^{10} = 40{,}000{,}000{,}000,
\]
which is read as forty billion.
(Choice D is $4 \times 10^8$; B is $4 \times 10^6$.)
ANSWER 5: E

Problem 6:
The total amount paid is the three-digit number $1A2$, which equals:
\[
100 + 10A + 2 = 102 + 10A.
\]
Since 11 members each paid the same integer amount, $102 + 10A$ must be divisible by 11. Testing the choices:
- $A=0$: $102 \div 11 \approx 9.27$ (not an integer)
- $A=1$: $112 \div 11 \approx 10.18$ (not an integer)
- $A=2$: $122 \div 11 \approx 11.09$ (not an integer)
- $A=3$: $132 \div 11 = 12$ (integer!)
- $A=4$: $142 \div 11 \approx 12.9$ (not an integer)
Thus the missing digit is 3.
ANSWER 6: D

Problem 7:
The wire is 24 cm long, so a shape is possible only if its perimeter is at most 24 cm.
- **Regular hexagon, side 5 cm:** Perimeter $= 6 \times 5 = 30$ cm. $30 > 24$, so impossible.
- **Square of area 36 cm²:** Side $= \sqrt{36} = 6$ cm. Perimeter $= 4 \times 6 = 24$ cm. Exactly fits, so possible.
- **Right triangle with legs 6 and 8 cm:** Hypotenuse $= \sqrt{6^2 + 8^2} = \sqrt{100} = 10$ cm. Perimeter $= 6 + 8 + 10 = 24$ cm. Exactly fits, so possible.
Only the square and the triangle can be formed.
ANSWER 7: D

Problem 8:
For positive numbers, the reciprocal function $f(x) = \frac{1}{x}$ is decreasing, so the smallest number has the largest reciprocal. Compare the two smallest choices:
\[
\frac{1}{3} \approx 0.333\ldots, \qquad \frac{2}{5} = 0.4.
\]
Since $\frac{1}{3} < \frac{2}{5} < 1 < 5 < 1986$, the smallest number is $\frac{1}{3}$. Its reciprocal is $3$, which is larger than the reciprocals of all the other choices.
ANSWER 8: A

Problem 9:
First find the theoretical maximum heart rate:
\[
220 - 26 = 194 \text{ beats per minute.}
\]
The target rate is 80% of this:
\[
0.80 \times 194 = 155.2.
\]
Rounded to the nearest whole number, this is 155.
(Choice C is the maximum heart rate itself.)
ANSWER 9: B

Problem 10:
Harry computes correctly:
\[
H = 8 - (2 + 5) = 8 - 7 = 1.
\]
Terry ignores the parentheses:
\[
T = 8 - 2 + 5 = 6 + 5 = 11.
\]
The difference is:
\[
H - T = 1 - 11 = -10.
\]
ANSWER 10: A

Problem 11:
The ten numbers inside the parentheses form an arithmetic sequence: 81, 83, …, 99. The sum of an arithmetic sequence is:
\[
\frac{\text{number of terms}}{2} \times (\text{first} + \text{last}) = \frac{10}{2} \times (81 + 99) = 5 \times 180 = 900.
\]
Now multiply by the factor of 2 outside:
\[
2 \times 900 = 1800.
\]
(One can also pair them: $(81+99)=180$, $(83+97)=180$, etc., giving five pairs that each sum to 180.)
ANSWER 11: E

Problem 12:
Total surveyed = 200, and females surveyed = 96, so:
\[
\text{Males surveyed} = 200 - 96 = 104.
\]
Total listeners = 136, and females who listen = 58, so:
\[
\text{Males who listen} = 136 - 58 = 78.
\]
(We can check: total non-listeners = 64, males who don't listen = 26, so females who don't listen = 38; $58+38=96$ and $78+26=104$, which checks out.)

The percentage of males who listen is:
\[
\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%.
\]
ANSWER 12: E

Problem 1:
A mixture of 30 liters is 25% red, 30% yellow, and 45% water. Five liters of yellow tint are added. What is the percent of yellow tint in the new mixture?

**Setup:** Find the amount of yellow tint originally, add 5 liters, then find the new percentage.

**Original yellow tint:** 30% of 30 = 0.30 × 30 = 9 liters

**After adding 5 liters of yellow tint:**
- New yellow tint: 9 + 5 = 14 liters
- New total volume: 30 + 5 = 35 liters
- New percentage: 14/35 = 2/5 = 40%

**Checking wrong choices:** 25% would be if we just added the original percentages; 35% would be if we just added 5% to 30%; 45% and 50% are too high.

ANSWER 1: C

---

Problem 2:
Six windows. Georgie enters by one and leaves by a different one. How many ways?

**Setup:** Choose an entry window (6 choices), then choose a different exit window (5 choices).

**Calculation:** 6 × 5 = 30 ways

**Checking wrong choices:** 15 would be if we divided by 2 (treating order as irrelevant); 12 would be 6×2; 18 would be 6×3; 36 would be 6×6 (allowing same window).

ANSWER 2: D

---

Problem 3:
1/10 + 2/20 + 3/30 = ?

**Simplify each fraction:**
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

**Sum:** 1/10 + 1/10 + 1/10 = 3/10 = 0.3

**Checking wrong choices:** 0.1 would be just one term; 0.2 would be two terms; 0.6 would be adding numerators and denominators incorrectly.

ANSWER 3: D

---

Problem 4:
Susan had $50. She spent $12 on food and twice as much on rides. How much left?

**Setup:** Food = $12, Rides = 2 × $12 = $24

**Total spent:** $12 + $24 = $36

**Amount left:** $50 - $36 = $14

**Checking wrong choices:** 12 is just the food amount; 26 is 50-24 (forgetting food); 38 is 50-12 (forgetting rides).

ANSWER 4: B

---

Problem 5:
200,000 × 200,000 = ?

**Setup:** Express in scientific notation: 200,000 = 2 × 10⁵

**Calculation:** (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion

**Checking wrong choices:** Four hundred thousand is 4×10⁵; four million is 4×10⁶; forty thousand is 4×10⁴; four hundred million is 4×10⁸.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, totaling $1A2. Find the missing digit A.

**Setup:** The total $1A2 must be divisible by 11.

**Divisibility test for 11:** A number is divisible by 11 if the alternating sum of its digits is divisible by 11.

For 1A2: (1 + 2) - A = 3 - A must be divisible by 11.

Since A is a single digit (0-9), 3 - A can only equal 0 (the only multiple of 11 in range -6 to 3).

So A = 3, giving $132 = 11 × $12 ✓

**Checking:** With A=0: $102 ÷ 11 ≈ 9.27 (not integer); A=1: $112 ÷ 11 ≈ 10.18 (not integer); A=3: $132 ÷ 11 = 12 ✓

ANSWER 6: D

---

Problem 7:
Wire is 24 cm. Which shapes can be made?

**Regular hexagon with side 5 cm:** Perimeter = 6 × 5 = 30 cm. Need 30 cm, only have 24 cm. ❌

**Square with area 36 cm²:** Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Need 24 cm, have 24 cm. ✓

**Right triangle with legs 6 and 8 cm:** Hypotenuse = √(36 + 64) = √100 = 10 cm. Perimeter = 6 + 8 + 10 = 24 cm. Need 24 cm, have 24 cm. ✓

**Conclusion:** Can make the square and triangle, but not the hexagon.

ANSWER 7: D

---

Problem 8:
Which number has the largest reciprocal?

**Find each reciprocal:**
- A. 1/3 → reciprocal = 3
- B. 2/5 → reciprocal = 5/2 = 2.5
- C. 1 → reciprocal = 1
- D. 5 → reciprocal = 1/5 = 0.2
- E. 1986 → reciprocal ≈ 0.0005

**Key insight:** Smaller positive numbers have larger reciprocals. The smallest number listed is 1/3, so it has the largest reciprocal (3).

ANSWER 8: A

---

Problem 9:
Target heart rate = 80% of (220 - age). Find target for 26-year-old.

**Maximum heart rate:** 220 - 26 = 194 bpm

**Target heart rate:** 80% of 194 = 0.80 × 194 = 155.2 ≈ 155 bpm

**Checking wrong choices:** 134 would be about 69%; 176 would be about 91%; 194 is the max (100%); 243 is impossible.

ANSWER 9: B

---

Problem 10:
Harry calculates 8 − (2 + 5) correctly. Terry calculates 8 − 2 + 5. Find H − T.

**Harry's answer:** H = 8 − (2 + 5) = 8 − 7 = 1

**Terry's answer:** T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:** H − T = 1 − 11 = −10

**Checking wrong choices:** −6 would be if Terry got 7; 0 would be if they got the same answer; 6 or 10 would be T − H.

ANSWER 10: A

---

Problem 11:
2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99) = ?

**Setup:** This is 10 consecutive odd numbers. The average of the first and last is (81 + 99)/2 = 90.

**Sum of the 10 numbers:** 10 × 90 = 900

**Multiply by 2:** 2 × 900 = 1800

**Alternative verification:** Pair the numbers: (81+99) + (83+97) + (85+95) + (87+93) + (89+91) = 180 + 180 + 180 + 180 + 180 = 900. Then 2 × 900 = 1800.

ANSWER 11: E

---

Problem 12:
Find the percentage of males who listen to the station.

**Given information:**
- Total surveyed: 200
- Females: 96
- Females who listen: 58
- Males who don't listen: 26
- Total listeners: 136
- Total non-listeners: 64

**Find males:** Males = 200 - 96 = 104

**Find males who listen:** Males who listen = Total listeners - Females who listen = 136 - 58 = 78

**Verification:** Males who don't listen = 104 - 78 = 26 ✓ (matches given)

**Percentage of males who listen:** 78/104 × 100 = 75%

**Checking wrong choices:** 39% would be 78/200 (using total instead of males); 48% would be about 50/104; 52% would be about 54/104; 55% would be about 57/104.

ANSWER 12: E

**Problem 1**

The original mixture is 30 L with 30 % yellow tint → yellow = 0.30·30 = 9 L.  
Add 5 L of yellow → total yellow = 9 + 5 = 14 L.  
Total volume after addition = 30 + 5 = 35 L.  

Percent yellow = (14/35)·100 % = 40 %.

**Answer 1: C**

---

**Problem 2**

Choose an entry window (6 ways) and a different exit window (5 ways).  
Number of ways = 6 × 5 = 30.

**Answer 2: D**

---

**Problem 3**

\[
\frac{1}{10}=0.1,\qquad\frac{2}{20}=0.1,\qquad\frac{3}{30}=0.1
\]
Sum = 0.1 + 0.1 + 0.1 = 0.3.

**Answer 3: D**

---

**Problem 4**

Food cost = \$12.  
Rides cost = twice food = 2·12 = \$24.  
Total spent = 12 + 24 = \$36.  
Money left = 50 − 36 = \$14.

**Answer 4: B**

---

**Problem 5**

\(200{,}000 \times 200{,}000 = (2\times10^{5})^{2}=4\times10^{10}=40{,}000{,}000{,}000\).

That is 40 billion.

**Answer 5: E**

---

**Problem 6**

Total paid = \$1A2 = 100 + 10A + 2 = 102 + 10A.  
Since the 11 members each paid an integer amount, 102 + 10A must be divisible by 11.

\(102\equiv3\pmod{11}\).  
We need \(10A\equiv-3\equiv8\pmod{11}\).  
Because \(10\equiv-1\pmod{11}\), this is \(-A\equiv8\pmod{11}\) → \(A\equiv3\pmod{11}\).  
A digit, so \(A=3\).

**Answer 6: D**

---

**Problem 7**

- Hexagon: perimeter = 6·5 = 30 cm > 24 cm → impossible.  
- Square: side = √36 = 6 cm, perimeter = 4·6 = 24 cm → possible.  
- Right triangle: legs 6 cm, 8 cm, hypotenuse = √(6²+8²)=10 cm, perimeter = 6+8+10 = 24 cm → possible.

Only the square and triangle can be made.

**Answer 7: D**

---

**Problem 8**

Reciprocals:
- \(1/3 → 3\)  
- \(2/5 → 5/2 = 2.5\)  
- \(1 → 1\)  
- \(5 → 1/5 = 0.2\)  
- \(1986 → 1/1986 ≈ 0.0005\)

Largest reciprocal is 3, from \(1/3\).

**Answer 8: A**

---

**Problem 9**

Maximum heart rate = 220 − 26 = 194 bpm.  
Target = 80 % of 194 = 0.8·194 = 155.2 → rounded to 155 bpm.

**Answer 9: B**

---

**Problem 10**

Harry: \(8-(2+5)=8-7=1\).  
Terry (ignoring parentheses): \(8-2+5 = 6+5 = 11\).  
Difference \(H-T = 1-11 = -10\).

**Answer 10: A**

---

**Problem 11**

Sum of the ten terms: average = (81+99)/2 = 90, so sum = 90·10 = 900.  
Multiply by 2: \(2·900 = 1800\).

**Answer 11: E**

---

**Problem 12**

Total surveyed = 200.  
Females = 96 → Males = 200 − 96 = 104.  

Female listeners = 58 → Male listeners = 136 − 58 = 78.  

Percent of males who listen = (78/104)·100 % = (39/52)·100 % = 75 %.

**Answer 12: E**

---

**Final Answers**

ANSWER 1: C  
ANSWER 2: D  
ANSWER 3: D  
ANSWER 4: B  
ANSWER 5: E  
ANSWER 6: D  
ANSWER 7: D  
ANSWER 8: A  
ANSWER 9: B  
ANSWER 10: A  
ANSWER 11: E  
ANSWER 12: E

Problem 1:  
A 30-liter paint mixture is 25% red tint, 30% yellow tint, and 45% water. Five liters of yellow tint are added.  
Original yellow tint: $0.30 \times 30 = 9$ liters.  
After adding 5 liters: $9 + 5 = 14$ liters yellow.  
New total volume: $30 + 5 = 35$ liters.  
Percent yellow: $\frac{14}{35} = 0.40 = 40\%$.  
Tempting wrong choices: 35% (original yellow percent) or 45% (water percent).  
ANSWER 1: D  

Problem 2:  
Georgie enters through one of six windows and exits through a different one.  
Entry choices: 6. Exit choices: 5 (excluding entry window).  
Total ways: $6 \times 5 = 30$.  
Tempting wrong choices: 15 (if order didn’t matter) or 36 (if same window allowed).  
ANSWER 2: D  

Problem 3:  
$\frac{1}{10} + \frac{2}{20} + \frac{3}{30} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3$.  
Tempting wrong choices: 0.1 (one term) or 0.2 (two terms).  
ANSWER 3: D  

Problem 4:  
Susan starts with \$50.  
Food: \$12.  
Rides: $2 \times 12 = \$24$.  
Total spent: $12 + 24 = \$36$.  
Left: $50 - 36 = \$14$.  
Tempting wrong choices: \$26 (if spent \$24 total) or \$38 (if spent \$12 total).  
ANSWER 4: B  

Problem 5:  
$200,000 \times 200,000 = 40,000,000,000 = 40$ billion.  
Tempting wrong choices: 400,000 (added zeros) or 4 million (multiplied digits).  
ANSWER 5: E  

Problem 6:  
Total paid: \$1A2, an integer divisible by 11.  
$1A2 \mod 11 = 0$.  
$1 + 2 - A \equiv 0 \mod 11 \Rightarrow 3 - A \equiv 0 \mod 11 \Rightarrow A = 3$.  
Tempting wrong choices: 0, 1, 2, or 4 (incorrect divisibility).  
ANSWER 6: E (Wait, let's recheck: 1A2 must be divisible by 11. 132 ÷ 11 = 12, so A=3. But 132 is 11×12, so correct. However, the answer choices are A=0,B=1,C=2,D=3,E=4. So D is 3. So correct answer is D, not E. I made a mistake in the initial answer. Correcting:)  
**Correction**: $1A2 = 132$ when $A=3$, and $132 \div 11 = 12$, so $A=3$.  
ANSWER 6: D  

Problem 7:  
Wire length: 24 cm.  
- Regular hexagon with side 5 cm: Perimeter $6 \times 5 = 30$ cm (not 24).  
- Square with area 36 cm²: Side $6$ cm, perimeter $24$ cm.  
- Right triangle with legs 6 and 8 cm: Hypotenuse 10 cm, perimeter $6+8+10=24$ cm.  
Can make square and triangle.  
Tempting wrong choices: Hexagon (incorrect perimeter).  
ANSWER 7: D  

Problem 8:  
Largest reciprocal means smallest original number.  
$\frac{1}{3} \approx 0.333$, $\frac{2}{5} = 0.4$, $1$, $5$, $1986$.  
Smallest: $\frac{1}{3}$. Reciprocal: 3.  
Others: 2.5, 1, 0.2, 0.0005.  
Tempting wrong choices: $\frac{2}{5}$ (larger than $\frac{1}{3}$).  
ANSWER 8: A  

Problem 9:  
Maximum heart rate: $220 - 26 = 194$.  
Target: $0.80 \times 194 = 155.2 \approx 155$.  
Tempting wrong choices: 134 (70%), 176 (90%), or 243 (incorrect max).  
ANSWER 9: B  

Problem 10:  
Harry: $8 - (2 + 5) = 1$.  
Terry: $8 - 2 + 5 = 11$.  
Difference: $1 - 11 = -10$.  
Tempting wrong choices: 10 (absolute difference) or 0 (if parentheses ignored by both).  
ANSWER 10: A  

Problem 11:  
Sum: $81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99$.  
Average: 90, 10 terms: $90 \times 10 = 900$.  
$2 \times 900 = 1800$.  
Tempting wrong choices: 1600 (min+max=180, times 5=900, times 2=1800, but if miscount terms).  
ANSWER 11: E  

Problem 12:  
Total surveyed: 200.  
Females: 96, so males: $200 - 96 = 104$.  
Total listeners: 136.  
Female listeners: 58, so male listeners: $136 - 58 = 78$.  
Percent male listeners: $\frac{78}{104} = 0.75 = 75\%$.  
Tempting wrong choices: 39 (male non-listeners percent) or 52 (incorrect calculation).  
ANSWER 12: E

### Problem 1:
What is the percentage of yellow tint in the new 35-liter paint mixture, after adding 5 liters of yellow tint to an original 30-liter mixture that was 30% yellow?
First, calculate the original volume of yellow tint: 30% of 30 liters = \(0.3 \times 30 = 9\) liters. Add the 5 new liters of yellow tint to get \(9 + 5 = 14\) liters of yellow in the new mixture. The total new volume of the mixture is \(30 + 5 = 35\) liters. The percentage of yellow is \(\frac{14}{35} = 0.4 = 40\%\). Tempting wrong answers come from forgetting to update the total mixture volume when calculating the new percentage, or miscalculating the original yellow volume.
ANSWER 1: C

---

### Problem 2:
How many ways can Georgie enter through one window and leave through a different window, out of 6 total windows?
Georgie has 6 choices for an entry window. Since he must leave through a different window, he has 5 remaining choices for an exit window. The total number of ways is \(6 \times 5 = 30\). A common wrong answer is 15, which incorrectly treats entry/exit as unordered pairs, when order matters (entering window A and leaving B is different from entering B and leaving A).
ANSWER 2: D

---

### Problem 3:
Simplify \(\frac{1}{10} + \frac{2}{20} + \frac{3}{30}\):
Simplify each fraction first: \(\frac{2}{20} = \frac{1}{10}\) and \(\frac{3}{30} = \frac{1}{10}\). Add them together: \(\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3\). A common wrong answer is 0.1, which incorrectly adds numerators and denominators directly instead of simplifying first.
ANSWER 3: D

---

### Problem 4:
How much money does Susan have left after spending $12 on food and twice that amount on rides, starting with $50?
First calculate the amount spent on rides: twice $12 is \(2 \times 12 = 24\) dollars. Total spent is \(12 + 24 = 36\) dollars. The amount left is \(50 - 36 = 14\) dollars. A common wrong answer is 26, which incorrectly uses $12 instead of $24 for ride costs.
ANSWER 4: B

---

### Problem 5:
Calculate \(200,000 \times 200,000\):
Rewrite 200,000 as \(2 \times 10^5\). Multiplying two copies gives \((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\), which equals 40,000,000,000, or forty billion. Wrong answers come from miscounting the number of zeros in the product.
ANSWER 5: E

---

### Problem 6:
The 3-digit total cost \(1A2\) dollars is divisible by 11 (11 members paid the same integer amount). What is the missing digit A?
The divisibility rule for 11 states that alternating sum of digits must be divisible by 11. For 1A2, the alternating sum is \(1 - A + 2 = 3 - A\). The only value of A (0-9) that makes this sum divisible by 11 is A=3, since \(3-3=0\), which is divisible by 11. Verify: \(132 \div 11 = 12\), a valid integer.
ANSWER 6: D

---

### Problem 7:
Haruki has a 24cm wire. Which of the three shapes can he form? Calculate the perimeter of each shape:
1. Regular hexagon with side 5cm: perimeter = \(6 \times 5 = 30\)cm, which is longer than 24cm, so impossible.
2. Square with area 36cm²: side length = \(\sqrt{36}=6\)cm, perimeter = \(4 \times 6 = 24\)cm, which fits.
3. Right triangle with legs 6cm and 8cm: hypotenuse = 10cm (6-8-10 Pythagorean triple), perimeter = \(6+8+10=24\)cm, which fits.
Only the square and triangle can be formed.
ANSWER 7: D

---

### Problem 8:
Which number has the largest reciprocal? The reciprocal of \(x\) is \(\frac{1}{x}\), so smaller values of \(x\) produce larger reciprocals. List the values of all options:
A. \(\frac{1}{3} \approx 0.333\), B. \(\frac{2}{5}=0.4\), C.1, D.5, E.1986. The smallest value is \(\frac{1}{3}\), whose reciprocal is 3, the largest of all options.
ANSWER 8: A

---

### Problem 9:
Calculate the target heart rate for a 26-year-old athlete: target is 80% of max heart rate, where max = 220 - age.
First calculate max heart rate: \(220 - 26 = 194\) bpm. 80% of 194 is \(0.8 \times 194 = 155.2\), which rounds to 155 bpm. A common wrong answer is 176, which uses 220 instead of the correct max heart rate of 194.
ANSWER 9: B

---

### Problem 10:
Calculate \(H-T\), where H is Harry's correct answer to \(8-(2+5)\), and T is Terry's incorrect answer of \(8-2+5\).
Harry's answer: \(8 - 7 = 1 = H\). Terry's answer: \(6 + 5 = 11 = T\). The difference \(H-T = 1 - 11 = -10\). A common wrong answer is 10, which calculates \(T-H\) instead of \(H-T\).
ANSWER 10: A

---

### Problem 11:
Simplify \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\):
Pair the terms inside the parentheses to simplify addition: \((81+99)+(83+97)+(85+95)+(87+93)+(89+91) = 5 \times 180 = 900\). Multiply by 2: \(2 \times 900 = 1800\).
ANSWER 11: E

---

### Problem 12:
What percentage of males surveyed listen to the radio station? First calculate total number of males: total surveyed = 200, females = 96, so males = \(200 - 96 = 104\). Males who do not listen = 26, so males who listen = \(104 - 26 = 78\). The percentage of males who listen is \(\frac{78}{104} = 0.75 = 75\%\). A common wrong answer is 39%, which divides male listeners by the total surveyed population instead of the total male population.
ANSWER 12: E

### Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.
1.  Calculate original yellow tint volume: 25%? No wait, original is 30% yellow. So 30% of 30 L = $0.3 \times 30 = 9$ liters of yellow.
2.  Add 5 liters of yellow: total yellow = $9 + 5 = 14$ liters.
3.  Total new mixture volume: $30 + 5 = 35$ liters.
4.  Percent yellow: $\frac{14}{35} \times 100 = 40\%$.
ANSWER 1: C
---
### Problem 2:
We count the number of ways to choose an entry window and a different exit window from 6 windows.
1.  Entry window: 6 possible choices.
2.  Exit window: cannot be the entry window, so 5 remaining choices.
3.  Total ways: $6 \times 5 = 30$ (order matters because entry and exit are distinct actions).
ANSWER 2: D
---
### Problem 3:
Simplify and add the fractions:
1.  $\frac{1}{10} = 0.1$, $\frac{2}{20} = \frac{1}{10} = 0.1$, $\frac{3}{30} = \frac{1}{10} = 0.1$.
2.  Sum: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D
---
### Problem 4:
Calculate remaining money after expenses:
1.  Total initial money: $50.
2.  Money spent on rides: twice the food cost = $2 \times 12 = 24$ dollars.
3.  Total spent: $12 + 24 = 36$ dollars.
4.  Remaining: $50 - 36 = 14$ dollars.
ANSWER 4: B
---
### Problem 5:
Compute $200,000 \times 200,000$:
1.  Rewrite as $(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
2.  $10^{10} = 10,000,000,000$, so $4 \times 10^{10} = 40,000,000,000 = 40$ billion.
ANSWER 5: E
---
### Problem 6:
The total payment $1A2$ (a 3-digit number) must be divisible by 11 (since 11 people paid equal integer amounts).
1.  Use the divisibility rule for 11: the alternating sum of digits is divisible by 11. For $1A2$, this is $1 - A + 2 = 3 - A$.
2.  $A$ is a digit (0-9), so $3 - A$ must equal 0 (the only multiple of 11 in range: $3 - A = 11$ gives $A=-8$, $3-A=-11$ gives $A=14$, both invalid).
3.  $3 - A = 0 \implies A=3$. Verify: $132 \div 11 = 12$, which is an integer.
ANSWER 6: D
---
### Problem 7:
Check if the 24 cm wire can form each shape (perimeter of the shape must equal 24 cm):
1.  Regular hexagon (side 5 cm): Perimeter = $6 \times 5 = 30$ cm > 24 cm → cannot make.
2.  Square (area 36 cm²): Side length = $\sqrt{36} = 6$ cm. Perimeter = $4 \times 6 = 24$ cm → can make.
3.  Right triangle (legs 6, 8 cm): Hypotenuse = $\sqrt{6^2 + 8^2} = 10$ cm. Perimeter = $6 + 8 + 10 = 24$ cm → can make.
Only square and triangle can be made.
ANSWER 7: D
---
### Problem 8:
The reciprocal of a positive number $x$ is $\frac{1}{x}$; smaller $x$ gives larger reciprocal.
1.  List values: $\frac{1}{3} \approx 0.33$, $\frac{2}{5}=0.4$, $1$, $5$, $1986$.
2.  Smallest value is $\frac{1}{3}$, whose reciprocal is $3$ (largest among all reciprocals: $\frac{5}{2}=2.5$, $1$, $0.2$, $\frac{1}{1986} \approx 0.0005$).
ANSWER 8: A
---
### Problem 9:
Calculate target heart rate for a 26-year-old:
1.  Maximum heart rate: $220 - 26 = 194$ bpm.
2.  Target heart rate: 80% of 194 = $0.8 \times 194 = 155.2$, which rounds to 155.
ANSWER 9: B
---
### Problem 10:
Compute $H$ (Harry's correct answer) and $T$ (Terry's incorrect answer):
1.  Harry: $8 - (2 + 5) = 8 - 7 = 1$, so $H=1$.
2.  Terry (ignores parentheses, left-to-right): $8 - 2 + 5 = 6 + 5 = 11$, so $T=11$.
3.  Difference: $H - T = 1 - 11 = -10$.
ANSWER 10: A
---
### Problem 11:
Calculate $2 \times (81 + 83 + ... + 99)$:
1.  The sum inside is 10 consecutive odd numbers. Average of first and last term: $\frac{81 + 99}{2} = 90$.
2.  Sum of the 10 terms: $90 \times 10 = 900$.
3.  Multiply by 2: $2 \times 900 = 1800$.
ANSWER 11: E
---
### Problem 12:
Find the percentage of males who listen to the station:
1.  Total males surveyed: $200 - 96 = 104$.
2.  Males who do not listen: 26, so males who listen: $104 - 26 = 78$.
3.  Verify with total listeners: Total listeners = 136, female listeners = 58, so male listeners = $136 - 58 = 78$ (matches).
4.  Percentage: $\frac{78}{104} \times 100 = 75\%$.
ANSWER 12: E

Problem 1:
Original yellow = 30% of 30 = 9 L. Add 5 → 14 L yellow. New total = 35 L. Percent = 14/35 = 40%.

ANSWER 1: C

Problem 2:
Enter: 6 choices, leave: 5 different choices. 6×5 = 30.

ANSWER 2: D

Problem 3:
Each fraction = 0.1. Sum = 0.3.

ANSWER 3: D

Problem 4:
Food 12, rides twice = 24. Spent 36. Left = 50 − 36 = 14.

ANSWER 4: B

Problem 5:
200,000 × 200,000 = 40,000,000,000 = forty billion.

ANSWER 5: E

Problem 6:
1A2 divisible by 11. Alternating sum: 1 − A + 2 = 3 − A must be divisible by 11 → A = 3 (gives 0). Check 132/11 = 12. Yes.

ANSWER 6: D

Problem 7:
Hexagon: 6×5 = 30 > 24, no. Square area 36 → side 6, perimeter 24, yes. Triangle legs 6,8 → hypotenuse 10, perimeter 24, yes. Square and triangle only.

ANSWER 7: D

Problem 8:
Largest reciprocal = smallest number. Smallest is 1/3.

ANSWER 8: A

Problem 9:
Max = 220 − 26 = 194. Target = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

Problem 10:
H = 8 − 7 = 1. T = 8 − 2 + 5 = 11. H − T = 1 − 11 = −10.

ANSWER 10: A

Problem 11:
Sum of 81..99 odd numbers (10 terms): average (81+99)/2 = 90, ×10 = 900. ×2 = 1800.

ANSWER 11: E

Problem 12:
Males total = 200 − 96 = 104. Males who don't listen = 26 → males who listen = 104 − 26 = 78. Percent = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem systematically.

---

**Problem 1:**
Original mixture: 30 liters, 30% yellow = 9 liters yellow. Add 5 liters yellow → new yellow = 14 liters, new total = 35 liters.
Percent yellow = 14/35 = 40%.

ANSWER 1: C

---

**Problem 2:**
Georgie can enter through any of 6 windows, then leave through any of the remaining 5 windows.
6 × 5 = 30.

ANSWER 2: D

---

**Problem 3:**
1/10 = 0.1, 2/20 = 0.1, 3/30 = 0.1. Sum = 0.3.

ANSWER 3: D

---

**Problem 4:**
Food: $12. Rides: 2 × $12 = $24. Total spent: $12 + $24 = $36. Left: $50 − $36 = $14.

ANSWER 4: B

---

**Problem 5:**
200,000 × 200,000 = (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion.

ANSWER 5: E

---

**Problem 6:**
The number 1A2 must be divisible by 11. 1A2 ÷ 11 must be an integer.
11 × 10 = 110, 11 × 11 = 121 (no), 11 × 12 = 132 ✓ → A = 3.
Check: 132/11 = 12 ✓, each paid $12.

ANSWER 6: D

---

**Problem 7:**
Wire = 24 cm.
- Regular hexagon, side 5: perimeter = 6 × 5 = 30 cm > 24. ✗
- Square, area 36: side = 6, perimeter = 24 cm ✓
- Right triangle, legs 6 and 8: hypotenuse = 10, perimeter = 6+8+10 = 24 cm ✓

ANSWER 7: D

---

**Problem 8:**
The reciprocal of x is 1/x. The largest reciprocal comes from the smallest positive number.
Among 1/3, 2/5, 1, 5, 1986 — the smallest is 1/3, and its reciprocal is 3, which is the largest.

ANSWER 8: A

---

**Problem 9:**
Maximum heart rate = 220 − 26 = 194. Target = 80% × 194 = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

---

**Problem 10:**
Harry: 8 − (2 + 5) = 8 − 7 = 1, so H = 1.
Terry: 8 − 2 + 5 = 11, so T = 11.
H − T = 1 − 11 = −10.

ANSWER 10: A

---

**Problem 11:**
The ten odd numbers from 81 to 99: sum = (81+99)/2 × 10 = 90 × 10 = 900.
2 × 900 = 1800.

ANSWER 11: E

---

**Problem 12:**
Total surveyed = 200. Females = 96, so Males = 104.
Total non-listeners = 64. Males who don't listen = 26, so females who don't listen = 64 − 26 = 38.
Males who listen = 104 − 26 = 78.
Percentage = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem step-by-step.

## Problem 1:
**What is asked:** Find the percentage of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.

**Original mixture:**
- Yellow tint: 30% of 30 = 0.30 × 30 = 9 liters
- Total volume: 30 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total volume: 30 + 5 = 35 liters

**Percentage of yellow tint:**
$$\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%$$

ANSWER 1: C

## Problem 2:
**What is asked:** In how many ways can the ghost enter by one window and leave by a different window?

**Solution:**
- Choices for entering: 6 windows
- Choices for leaving: 5 windows (must be different from entry)
- Total ways: 6 × 5 = 30

ANSWER 2: D

## Problem 3:
**What is asked:** Calculate 1/10 + 2/20 + 3/30

**Solution:**
Simplify each fraction:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

ANSWER 3: D

## Problem 4:
**What is asked:** How much money does Susan have left?

**Solution:**
- Started with: $50
- Spent on food: $12
- Spent on rides: 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Left: $50 - $36 = $14

ANSWER 4: B

## Problem 5:
**What is asked:** Calculate 200,000 × 200,000

**Solution:**
$$200,000 × 200,000 = (2 × 10^5) × (2 × 10^5) = 4 × 10^{10} = 40,000,000,000 = 40 \text{ billion}$$

ANSWER 5: E

## Problem 6:
**What is asked:** Find the missing digit A in the 3-digit number 1A2, where 11 club members each paid an equal integer amount that totals 1A2.

**Solution:**
The number 1A2 must be divisible by 11.

Testing divisibility by 11 (alternating sum of digits):
- 1 - A + 2 = 0 (mod 11)
- 3 - A ≡ 0 (mod 11)
- A = 3

Check: 132 ÷ 11 = 12 ✓

ANSWER 6: D

## Problem 7:
**What is asked:** Which shapes can Haruki make with 24 cm of wire?

**Solution:**
Test each shape:

**Regular hexagon with side 5 cm:**
- Perimeter: 6 × 5 = 30 cm
- Need 30 cm, have 24 cm ✗

**Square with area 36 cm²:**
- Side length: √36 = 6 cm
- Perimeter: 4 × 6 = 24 cm ✓

**Right triangle with legs 6 and 8 cm:**
- Hypotenuse: √(6² + 8²) = √(36 + 64) = √100 = 10 cm
- Perimeter: 6 + 8 + 10 = 24 cm ✓

Only square and triangle can be made.

ANSWER 7: D

## Problem 8:
**What is asked:** Which number has the largest reciprocal?

**Solution:**
Find reciprocals:
- 1/(1/3) = 3
- 1/(2/5) = 5/2 = 2.5
- 1/1 = 1
- 1/5 = 0.2
- 1/1986 ≈ 0.0005

The largest reciprocal is 5 (from the number 1/5).

ANSWER 8: D

## Problem 9:
**What is asked:** Find the target heart rate for a 26-year-old athlete.

**Solution:**
- Maximum heart rate: 220 - 26 = 194 beats per minute
- Target heart rate: 80% of 194 = 0.80 × 194 = 155.2 ≈ 155

ANSWER 9: B

## Problem 10:
**What is asked:** Find H - T where H is Harry's answer and T is Terry's answer.

**Solution:**
**Harry's calculation (correct):**
$$H = 8 - (2 + 5) = 8 - 7 = 1$$

**Terry's calculation (ignoring parentheses):**
$$T = 8 - 2 + 5 = 6 + 5 = 11$$

**Difference:**
$$H - T = 1 - 11 = -10$$

ANSWER 10: A

## Problem 11:
**What is asked:** Calculate 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

**Solution:**
This is an arithmetic sequence with 10 terms. Sum = n(first + last)/2
$$81 + 83 + ... + 99 = \frac{10(81 + 99)}{2} = \frac{10 × 180}{2} = 900$$

$$2 × 900 = 1800$$

ANSWER 11: E

## Problem 12:
**What is asked:** What percentage of males surveyed listen to the station?

**Solution:**
- Total surveyed: 200
- Females: 96
- Males: 200 - 96 = 104

- Total non-listeners: 64
- Males who don't listen: 26
- Males who listen: 104 - 26 = 78

**Percentage of males who listen:**
$$\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%$$

ANSWER 12: E

Problem 1:

The original mixture is 30 liters total, with 30% yellow tint.

So the amount of yellow tint at first is:
- \(30\%\) of \(30 = 0.30 \times 30 = 9\) liters

Then 5 liters of yellow tint are added, so the new amount of yellow tint is:
- \(9 + 5 = 14\) liters

The new total mixture is:
- \(30 + 5 = 35\) liters

So the percent of yellow tint is:
- \(\frac{14}{35} = \frac{2}{5} = 0.4 = 40\%\)

Answer choice C matches 40%.

ANSWER 1: C

Problem 2:

Georgie can:
- choose 1 of the 6 windows to enter
- then choose a different window to leave

For the entry, there are 6 choices.

For the exit, there are 5 remaining choices, since it must be a different window.

So the total number of ways is:
- \(6 \times 5 = 30\)

ANSWER 2: D

Problem 3:

Compute each fraction:
- \(\frac{1}{10} = 0.1\)
- \(\frac{2}{20} = \frac{1}{10} = 0.1\)
- \(\frac{3}{30} = \frac{1}{10} = 0.1\)

Add them:
- \(0.1 + 0.1 + 0.1 = 0.3\)

So the value is \(0.3\).

ANSWER 3: D

Problem 4:

Susan had \$50.

She spent:
- \$12 on food
- twice as much on rides, so \(2 \times 12 = 24\)

Total spent:
- \(12 + 24 = 36\)

Money left:
- \(50 - 36 = 14\)

ANSWER 4: B

Problem 5:

Two hundred thousand is:
- \(200{,}000 = 2 \times 10^5\)

Multiply:
- \(200{,}000 \times 200{,}000 = (2 \times 10^5)(2 \times 10^5) = 4 \times 10^{10}\)

And \(4 \times 10^{10} = 40{,}000{,}000{,}000\), which is forty billion.

ANSWER 5: E

Problem 6:

The total is written as \(1A2\), and 11 members each paid the same integer amount.

So \(1A2\) must be divisible by 11.

Test the answer choices:
- 102 ÷ 11 is not an integer
- 112 ÷ 11 is not an integer
- 122 ÷ 11 = 11 remainder 1
- 132 ÷ 11 = 12 exactly
- 142 ÷ 11 is not an integer

So the number is 132, which means \(A = 3\).

ANSWER 6: D

Problem 7:

We check each shape.

### Hexagon
A regular hexagon has 6 equal sides.
If each side is 5 cm, perimeter is:
- \(6 \times 5 = 30\) cm

The wire is only 24 cm, so he cannot make the hexagon.

### Square
A square with area 36 has side length:
- \(\sqrt{36} = 6\) cm

Perimeter:
- \(4 \times 6 = 24\) cm

So he can make the square exactly.

### Right triangle
The legs are 6 and 8, so the hypotenuse is:
- \(\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)

Perimeter:
- \(6 + 8 + 10 = 24\) cm

So he can make the triangle exactly.

Therefore he can make the square and the triangle only.

ANSWER 7: D

Problem 8:

The reciprocal is largest when the original number is smallest, assuming positive numbers.

Compare:
- \(\frac13 \approx 0.333\)
- \(\frac25 = 0.4\)
- \(1\)
- \(5\)
- \(1986\)

The smallest number is \(\frac13\), so it has the largest reciprocal.

Its reciprocal is 3, which is larger than the reciprocals of all the others.

ANSWER 8: A

Problem 9:

Maximum heart rate:
- \(220 - 26 = 194\)

Target heart rate is 80% of that:
- \(0.8 \times 194 = 155.2\)

To the nearest whole number, that is 155.

ANSWER 9: B

Problem 10:

First, Harry uses the parentheses:
- \(8 - (2 + 5) = 8 - 7 = 1\)

So \(H = 1\).

Terry ignores parentheses:
- \(8 - 2 + 5 = 6 + 5 = 11\)

So \(T = 11\).

Difference:
- \(H - T = 1 - 11 = -10\)

ANSWER 10: A

Problem 11:

We need to evaluate:
- \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

These are 10 odd numbers from 81 to 99.

Instead of adding one by one, use the average:
- average of 81 and 99 is \( \frac{81+99}{2} = 90\)

Since there are 10 numbers, their sum is:
- \(10 \times 90 = 900\)

Now multiply by 2:
- \(2 \times 900 = 1800\)

ANSWER 11: E

Problem 12:

We want the percentage of males surveyed who listen.

From the table:
- Total surveyed = 200
- Females = 96
- So males = \(200 - 96 = 104\)

Males who don’t listen = 26, so males who do listen:
- \(104 - 26 = 78\)

Now find the percent of males who listen:
- \(\frac{78}{104} = \frac{3}{4} = 75\%\)

ANSWER 12: E

Problem 1:  
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?  

- Original yellow tint amount = 30 × 0.30 = 9 liters.  
- Add 5 liters of yellow tint → new yellow amount = 9 + 5 = 14 liters.  
- New total volume = 30 + 5 = 35 liters.  
- New percent yellow = (14 / 35) × 100% = (2/5) × 100% = 40%.  

So the correct choice is 40.  

ANSWER 1: C  

---

Problem 2:  
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?  

- Choose an entry window: 6 choices.  
- Choose a different exit window: 5 choices (cannot be the same as entry).  
- Total ways = 6 × 5 = 30.  

ANSWER 2: D  

---

Problem 3:  
Compute: \( \frac{1}{10} + \frac{2}{20} + \frac{3}{30} \).  

Convert each to a common denominator, 60:  
- \( \frac{1}{10} = \frac{6}{60} \)  
- \( \frac{2}{20} = \frac{2}{20} = \frac{6}{60} \) (since multiply top and bottom by 3)  
- \( \frac{3}{30} = \frac{3}{30} = \frac{6}{60} \) (since multiply top and bottom by 2)  

Add:  
\( \frac{6}{60} + \frac{6}{60} + \frac{6}{60} = \frac{18}{60} = \frac{3}{10} = 0.3 \)

So the value is .3.  

ANSWER 3: D  

---

Problem 4:  
Susan had $50 to spend. She spent $12 on food and twice as much on rides. How much left?  

- Food = 12  
- Rides = twice 12 = 24  
- Total spent = 12 + 24 = 36  
- Money left = 50 − 36 = 14  

ANSWER 4: B  

---

Problem 5:  
Two hundred thousand times two hundred thousand equals ?  

\(200{,}000 \times 200{,}000 = (2\times 10^5)(2\times 10^5) = 4 \times 10^{10}\).  
- \(10^{10} = 10{,}000{,}000{,}000\)  
- \(4 \times 10^{10} = 40{,}000{,}000{,}000 = 40\) billion  

ANSWER 5: E  

---

Problem 6:  
Eleven members each paid the same integer amount for a guest speaker. In all, they paid $1A2 (a 3-digit number). Find missing digit A.  

- Total = \(1A2\) means:  
  \(1A2 = 100 + 10A + 2 = 102 + 10A\).  
- This total is divided equally among 11 people:  
  So \(102 + 10A\) must be divisible by 11.  

Check \(102 + 10A \mod 11\):  
- \(102 \div 11 = 9\) remainder 3, so \(102 \equiv 3 \pmod{11}\).  
- \(10A \equiv -A \pmod{11}\) (since 10 ≡ −1 mod 11).  

So:  
\(102 + 10A \equiv 3 - A \equiv 0 \pmod{11}\)  
Thus \(A \equiv 3 \pmod{11}\).  
Since \(A\) is a digit 0–9, \(A=3\).  

ANSWER 6: D  

---

Problem 7:  
Wire length = 24 cm. Which shapes can be made (using all 24 cm of wire)?  

We need the perimeter (total boundary length) of each shape:

**(1) Regular hexagon with side length 5 cm**  
Perimeter = 6 × 5 = 30 cm → too long. Cannot.

**(2) Square with area 36 cm²**  
Side length = √36 = 6 cm  
Perimeter = 4 × 6 = 24 cm → fits. Can.

**(3) Right triangle with legs 6 and 8 cm**  
Right triangle sides are 6 and 8 legs, and hypotenuse:  
\( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)  
Perimeter = 6 + 8 + 10 = 24 cm → fits. Can.

So the shapes possible: **square and triangle only**.  

ANSWER 7: D  

---

Problem 8:  
Which number has the largest reciprocal? Recall: the reciprocal of \(x\) is \(1/x\).  
We compare \(1/3, 2/5, 1, 1/5, 1/1986\).  

Compute roughly:  
- \(1/3 \approx 0.333\)  
- \(2/5 = 0.4\)  
- \(1 = 1\)  
- \(1/5 = 0.2\)  
- \(1/1986\) is tiny  

Largest is 1 (reciprocal of 1).  

ANSWER 8: C  

---

Problem 9:  
Target heart rate = 80% of (maximum heart rate).  
Maximum heart rate = 220 − age.  
Age = 26.

- Maximum = 220 − 26 = 194  
- Target = 0.80 × 194 = 155.2  
Nearest whole number = 155  

ANSWER 9: B  

---

Problem 10:  
Expression: \(8 - (2 + 5)\)  
- Harry computes correctly:  
  \(2 + 5 = 7\), so \(8 - 7 = 1\).  
- Terry ignores parentheses:  
  \(8 - 2 + 5 = 6 + 5 = 11\).  

Difference \(H - T = 1 - 11 = -10\).  

ANSWER 10: A  

---

Problem 11:  
Compute:  
\(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

Notice these are all odd numbers from 81 to 99 stepping by 2.  
There are 10 terms.

Pair from ends:  
- 81 + 99 = 180  
- 83 + 97 = 180  
- 85 + 95 = 180  
- 87 + 93 = 180  
- 89 + 91 = 180  

There are 5 pairs, each sum 180:  
Sum = 5 × 180 = 900  

Now multiply by 2:  
2 × 900 = 1800  

ANSWER 11: E  

---

Problem 12:  
Total surveyed = 200  
Females = 96  
Females who listen = 58  
Males who don’t listen = 26  
Total listeners = 136  

We want: % of males who listen.

Step 1: Find males total.  
Males = 200 − 96 = 104

Step 2: Find males who don’t listen is given = 26.  
So males who listen = 104 − 26 = 78

Step 3: Convert to percent of males who listen:  
Percent = (78 / 104) × 100%  
Simplify 78/104 = 3/4 (divide by 26):  
So percent = (3/4) × 100% = 75%

ANSWER 12: E

Problem 1:
The original mixture is 30 liters. The amount of yellow tint is 30% of 30 liters: $0.30 \times 30 = 9$ liters.
Five liters of yellow tint are added, so the new amount of yellow tint is $9 + 5 = 14$ liters.
The new total volume of the mixture is $30 + 5 = 35$ liters.
The percentage of yellow tint is $\frac{14}{35} \times 100\%$.
Simplifying the fraction: $\frac{14}{35} = \frac{2}{5} = 0.4$.
$0.4 \times 100\% = 40\%$.
ANSWER 1: C

Problem 2:
Georgie has 6 choices for the window he enters. Since he must leave by a different window, he has $6 - 1 = 5$ choices for the window he leaves by.
The total number of ways is $6 \times 5 = 30$.
ANSWER 2: D

Problem 3:
Simplify each fraction:
$1/10 = 0.1$
$2/20 = 1/10 = 0.1$
$3/30 = 1/10 = 0.1$
Summing them: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides, which is $2 \times 12 = 24$.
Total spent: $12 + 24 = 36$.
Amount left: $50 - 36 = 14$.
ANSWER 4: B

Problem 5:
Two hundred thousand is $200,000$ (or $2 \times 10^5$).
$(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
$4 \times 10^{10} = 40,000,000,000$, which is forty billion.
ANSWER 5: E

Problem 6:
The total amount $1A2$ must be divisible by 11.
A number is divisible by 11 if the alternating sum of its digits is divisible by 11.
For $1A2$, the alternating sum is $1 - A + 2 = 3 - A$.
For this to be 0 (the only multiple of 11 possible for a single digit $A$), $3 - A = 0$, so $A = 3$.
Check: $132 / 11 = 12$. This works.
ANSWER 6: D

Problem 7:
Hexagon: Perimeter of a regular hexagon with side 5 is $6 \times 5 = 30$ cm. (Cannot make)
Square: Area is 36, so side is $\sqrt{36} = 6$. Perimeter is $4 \times 6 = 24$ cm. (Can make)
Triangle: Right triangle with legs 6 and 8. Hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. Perimeter is $6 + 8 + 10 = 24$ cm. (Can make)
ANSWER 7: D

Problem 8:
Reciprocals:
A: $1/(1/3) = 3$
B: $1/(2/5) = 2.5$
C: $1/1 = 1$
D: $1/5 = 0.2$
E: $1/1986 \approx 0.0005$
The largest is 3.
ANSWER 8: A

Problem 9:
Max heart rate = $220 - 26 = 194$.
Target heart rate = $80\%$ of $194 = 0.8 \times 194 = 155.2$.
To the nearest whole number, this is 155.
ANSWER 9: B

Problem 10:
$H = 8 - (2 + 5) = 8 - 7 = 1$.
$T = 8 - 2 + 5 = 6 + 5 = 11$.
$H - T = 1 - 11 = -10$.
ANSWER 10: A

Problem 11:
Sum inside parentheses: $81+83+85+87+89+91+93+95+97+99$.
This is an arithmetic progression with 10 terms.
Sum = $\frac{n}{2}(\text{first} + \text{last}) = \frac{10}{2}(81 + 99) = 5 \times 180 = 900$.
The expression is $2 \times 900 = 1800$.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96.
Males = $200 - 96 = 104$.
Males who don't listen = 26.
Males who listen = $104 - 26 = 78$.
Percentage of males who listen = $(78 / 104) \times 100\%$.
$\frac{78}{104} = \frac{39}{52} = \frac{3}{4} = 0.75 = 75\%$.
ANSWER 12: E

Problem 1: The original mixture has volume 30 L, of which 30% is yellow tint, so there are \(0.3 \times 30 = 9\) L of yellow tint. Adding 5 L of yellow tint produces a new total of 14 L of yellow tint and a new mixture volume of 35 L. The new percentage of yellow tint is therefore \((14/35) \times 100\% = 40\%\).

ANSWER 1: C

Problem 2: There are 6 choices for the entry window. For each of those choices there are then exactly 5 remaining choices for a different exit window, giving a total of \(6 \times 5 = 30\) ordered pairs of distinct windows.

ANSWER 2: D

Problem 3: Rewrite each term with a common denominator of 10:  
\(1/10 = 0.1\),  
\(2/20 = 1/10 = 0.1\),  
\(3/30 = 1/10 = 0.1\).  
Their sum is \(0.1 + 0.1 + 0.1 = 0.3\).

ANSWER 3: D

Problem 4: Twice the food cost is \(2 \times 12 = 24\) dollars spent on rides. The total spent is therefore \(12 + 24 = 36\) dollars, leaving \(50 - 36 = 14\) dollars.

ANSWER 4: B

Problem 5: Write the factors in scientific notation:  
\(200000 = 2 \times 10^5\), so  
\((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\),  
which is forty billion.

ANSWER 5: E

Problem 6: The three-digit total \(1A2\) must be divisible by 11. The alternating-sum divisibility rule requires that \(1 - A + 2 = 3 - A\) be a multiple of 11. The only admissible digit \(A\) (0-9) that satisfies the condition is \(A = 3\), giving the total 132, which is exactly \(11 \times 12\).

ANSWER 6: D

Problem 7: A regular hexagon with side 5 cm has perimeter 30 cm, which exceeds the 24 cm length of wire, so it is impossible. A square of area 36 cm² has side length 6 cm and perimeter 24 cm, which matches the wire length. A right triangle with legs 6 cm and 8 cm has hypotenuse 10 cm, so its perimeter is \(6 + 8 + 10 = 24\) cm, which also matches. Consequently only the square and the triangle are possible.

ANSWER 7: D

Problem 8: For positive numbers the reciprocal function is strictly decreasing, so the largest reciprocal belongs to the smallest number in the list, which is \(1/3\).

ANSWER 8: A

Problem 9: The theoretical maximum heart rate is \(220 - 26 = 194\) beats per minute. Eighty percent of that value is \(0.8 \times 194 = 155.2\), which rounds to the nearest whole number 155.

ANSWER 9: B

Problem 10: Harry evaluates the parentheses first and obtains \(8 - (2 + 5) = 1\). Terry omits the parentheses and obtains \(8 - 2 + 5 = 11\). Their difference is therefore \(1 - 11 = -10\).

ANSWER 10: A

Problem 11: The sum inside the parentheses consists of the ten consecutive odd integers from 81 to 99. Their average is \((81 + 99)/2 = 90\), so the sum equals \(10 \times 90 = 900\). Multiplying by the outer factor of 2 yields \(2 \times 900 = 1800\).

ANSWER 11: E

Problem 12: There are 200 people surveyed and 96 females, so there are \(200 - 96 = 104\) males. Of the 96 females, 58 listen, leaving \(96 - 58 = 38\) females who do not listen. Given that 26 males do not listen, the number of males who listen is \(104 - 26 = 78\). The required percentage is therefore \((78/104) \times 100\% = 75\%\).

ANSWER 12: E

Problem 1:
The problem asks us to find the percentage of yellow tint in a new mixture after adding 5 liters of yellow tint to the original 30 liters of paint mixture. The original mixture is 25% red tint, 30% yellow tint, and 45% water.

First, let's calculate the amount of yellow tint in the original mixture. 30% of 30 liters is $0.3 \times 30 = 9$ liters.

After adding 5 liters of yellow tint, the total amount of yellow tint becomes $9 + 5 = 14$ liters.

The total volume of the new mixture is $30 + 5 = 35$ liters.

To find the percentage of yellow tint in the new mixture, we divide the amount of yellow tint by the total volume and multiply by 100: $\frac{14}{35} \times 100 = 40\%$.

Let's examine the choices:
- A. 25: Too low because the percentage of yellow tint increased.
- B. 35: Not the correct calculation.
- C. 40: Matches our calculation.
- D. 45: Too high.
- E. 50: Too high.

The correct percentage is 40%.

ANSWER 1: C

Problem 2:
The problem asks us to determine the number of ways Georgie the Ghost can enter a haunted house through one of six windows and leave by a different window.

To enter, Georgie has 6 choices. After entering, there are 5 remaining windows to choose from for leaving.

The total number of ways to enter and leave is $6 \times 5 = 30$.

Let's examine the choices:
- A. 12: Too low.
- B. 15: Too low.
- C. 18: Too low.
- D. 30: Matches our calculation.
- E. 36: Too high because it would imply Georgie can leave by the same window.

The correct number of ways is 30.

ANSWER 2: D

Problem 3:
The problem requires us to simplify the expression $1⁄10 + 2⁄20 + 3⁄30$.

First, simplify each fraction:
- $1⁄10$ stays the same.
- $2⁄20 = 1⁄10$.
- $3⁄30 = 1⁄10$.

Adding them together: $1⁄10 + 1⁄10 + 1⁄10 = 3⁄10 = 0.3$.

Let's examine the choices:
- A. .1: Too low.
- B. .123: Not the correct sum.
- C. .2: Too low.
- D. .3: Matches our calculation.
- E. .6: Too high.

The correct sum is 0.3.

ANSWER 3: D

Problem 4:
The problem asks us to find out how many dollars Susan had left after spending money on food and rides at the carnival. She started with $50, spent $12 on food, and twice as much on rides.

First, calculate the amount spent on rides: $12 \times 2 = $24.

Total amount spent is $12 + $24 = $36.

The amount left is $50 - $36 = $14.

Let's examine the choices:
- A. 12: Too low.
- B. 14: Matches our calculation.
- C. 26: Too high.
- D. 38: Too high.
- E. 50: Incorrect because she spent money.

The correct amount left is $14.

ANSWER 4: B

Problem 5:
The problem asks us to calculate $200,000 \times 200,000$.

$200,000 = 2 \times 10^5$, so $200,000 \times 200,000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10} = 40,000,000,000$.

Let's examine the choices:
- A. four hundred thousand: Too low.
- B. four million: Too low.
- C. forty thousand: Too low.
- D. four hundred million: Too low.
- E. forty billion: Matches our calculation.

The correct result is forty billion.

ANSWER 5: E

Problem 6:
The problem requires us to find the missing digit A in the 3-digit number $1A2 that represents the total amount paid by 11 members of the Math Club.

The total amount paid is divisible by 11 since there are 11 members who paid the same amount.

To check divisibility by 11, we use the rule: the difference between the sum of digits at odd places and the sum of digits at even places should be divisible by 11.

For $1A2, the sum of digits at odd places is $1 + 2 = 3$, and the sum at even places is $A$. The difference is $3 - A$.

For $3 - A$ to be divisible by 11 (or 0, since $A$ is a single digit), $A = 3$ makes $3 - A = 0$, which is divisible by 11.

Let's examine the choices:
- A. 0: Not correct because $3 - 0 = 3$, not divisible by 11.
- B. 1: Not correct because $3 - 1 = 2$, not divisible by 11.
- C. 2: Not correct because $3 - 2 = 1$, not divisible by 11.
- D. 3: Correct because $3 - 3 = 0$, which is divisible by 11.
- E. 4: Not correct because $3 - 4 = -1$, not divisible by 11.

The missing digit A is 3, but we should verify if 132 is indeed divisible by 11. $132 / 11 = 12$, so it is.

ANSWER 6: D

Problem 7:
The problem asks us to determine which shapes Haruki can form using a 24 cm long wire.

1. A regular hexagon with side length 5 cm: The perimeter is $6 \times 5 = 30$ cm, which is more than 24 cm. So, Haruki cannot form this hexagon.

2. A square of area 36 cm$^2$: The side length is $\sqrt{36} = 6$ cm. The perimeter is $4 \times 6 = 24$ cm, which matches the wire length. So, Haruki can form this square.

3. A right triangle with legs 6 and 8 cm: Using the Pythagorean theorem, the hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm. The perimeter is $6 + 8 + 10 = 24$ cm, which matches the wire length. So, Haruki can form this triangle.

Let's examine the choices:
- A. Triangle only: Incorrect because Haruki can also form the square.
- B. Hexagon and square only: Incorrect because Haruki cannot form the hexagon.
- C. Hexagon and triangle only: Incorrect because Haruki cannot form the hexagon.
- D. Square and triangle only: Matches our findings.
- E. Hexagon, triangle, and square: Incorrect because Haruki cannot form the hexagon.

The correct shapes Haruki can form are the square and the triangle.

ANSWER 7: D

Problem 8:
The problem asks us to find which of the given numbers has the largest reciprocal.

The reciprocal of a number $x$ is $1/x$. The larger $x$ is, the smaller its reciprocal is, and vice versa.

Let's examine the choices:
- A. $1⁄3$: The reciprocal is 3.
- B. $2⁄5$: The reciprocal is $5/2 = 2.5$.
- C. 1: The reciprocal is 1.
- D. 5: The reciprocal is $1/5 = 0.2$.
- E. 1986: The reciprocal is $1/1986$, very small.

Comparing the reciprocals: $3 > 2.5 > 1 > 0.2 > 1/1986$. So, $1/3$ has the largest reciprocal.

Let's examine the choices again:
- A. $1⁄3$: Correct.
- B. $2⁄5$: Not correct.
- C. 1: Not correct.
- D. 5: Not correct.
- E. 1986: Not correct.

The number with the largest reciprocal is $1/3$.

ANSWER 8: A

Problem 9:
The proble

Problem 1:
Original mixture: 30 L. Yellow tint = 30% of 30 = 0.30 × 30 = 9 L.
Add 5 L yellow: new yellow = 9 + 5 = 14 L, new total volume = 30 + 5 = 35 L.
Percent yellow = (14/35) × 100% = 40%.
ANSWER 1: C

Problem 2:
6 choices for entry window, 5 remaining choices for a different exit window.
Total ways = 6 × 5 = 30.
ANSWER 2: D

Problem 3:
1/10 + 2/20 + 3/30 = 1/10 + 1/10 + 1/10 = 3/10 = 0.3.
ANSWER 3: D

Problem 4:
Spent on food: $12. Spent on rides: 2 × $12 = $24.
Total spent = $12 + $24 = $36. Left = $50 − $36 = $14.
ANSWER 4: B

Problem 5:
200,000 × 200,000 = (2×10⁵) × (2×10⁵) = 4×10¹⁰ = 40,000,000,000, which is forty billion.
ANSWER 5: E

Problem 6:
Total paid is $1A2, a multiple of 11. Divisibility by 11: (1 − A + 2) = 3 − A must be a multiple of 11.
With A a digit 0–9, 3 − A = 0 ⇒ A = 3. (Check: 132 ÷ 11 = 12.)
ANSWER 6: D

Problem 7:
Wire length = 24 cm.
- Regular hexagon, side 5 cm: perimeter = 6×5 = 30 cm > 24 cm → cannot make.
- Square, area 36 cm²: side = 6 cm, perimeter = 4×6 = 24 cm → can make.
- Right triangle, legs 6 and 8 cm: hypotenuse = 10 cm, perimeter = 6+8+10 = 24 cm → can make.
So square and triangle only.
ANSWER 7: D

Problem 8:
Reciprocals: 1/(1/3)=3, 1/(2/5)=2.5, 1/1=1, 1/5=0.2, 1/1986≈0.0005. Largest is 3, from 1/3.
ANSWER 8: A

Problem 9:
Maximum heart rate = 220 − 26 = 194 bpm.
Target = 80% of 194 = 0.8 × 194 = 155.2 → 155 to nearest whole number.
ANSWER 9: B

Problem 10:
Harry: H = 8 − (2+5) = 8 − 7 = 1.
Terry: T = 8 − 2 + 5 = 6 + 5 = 11.
H − T = 1 − 11 = −10.
ANSWER 10: A

Problem 11:
Sum inside parentheses: 10 terms, average = (81+99)/2 = 90, sum = 10×90 = 900.
2 × 900 = 1800.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96 ⇒ Males = 200 − 96 = 104.
Total listeners = 136, female listeners = 58 ⇒ male listeners = 136 − 58 = 78.
Percent of males who listen = (78/104) × 100% = 75%.
ANSWER 12: E

Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint.

**Original mixture (30 liters):**
- Red tint: 25% of 30 = 7.5 liters
- Yellow tint: 30% of 30 = 9 liters
- Water: 45% of 30 = 13.5 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total mixture: 30 + 5 = 35 liters

**New percentage of yellow tint:**
14 / 35 = 2/5 = 0.40 = 40%

Tempting wrong answers: Choice B (35) might come from just adding 5 to the original 30%, but we must recalculate based on the new total.

ANSWER 1: C

---

Problem 2:
Georgie enters through one window and leaves through a *different* window.

- Number of choices for the entry window: 6
- Number of choices for the exit window (must be different): 5
- Total ways: 6 × 5 = 30

Tempting wrong answers: Choice E (36) would be 6 × 6, which allows entering and leaving through the same window. Choice A (12) would be 6 + 6, which doesn't account for all combinations.

ANSWER 2: D

---

Problem 3:
Simplify each fraction first:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

Tempting wrong answers: Choice B (.123) might come from concatenating the digits 1, 2, 3 instead of actually computing. Choice C (.2) might come from miscounting the terms.

ANSWER 3: D

---

Problem 4:
Susan starts with $50.
- Spent on food: $12
- Spent on rides: twice the food amount = 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Amount left: $50 − $36 = $14

Tempting wrong answers: Choice C (26) might come from subtracting only the food ($50 − $24 = $26, forgetting food). Choice D (38) comes from subtracting only $12.

ANSWER 4: B

---

Problem 5:
200,000 × 200,000

Write in scientific notation:
(2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰

4 × 10¹⁰ = 40,000,000,000 = forty billion

Tempting wrong answers: Choice A (four hundred thousand) comes from adding instead of multiplying. Choice D (four hundred million) comes from miscounting zeros.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, so the total $1A2 must be divisible by 11.

**Divisibility rule for 11:** The alternating sum of digits must be a multiple of 11.
Alternating sum: 1 − A + 2 = 3 − A

For 3 − A to be divisible by 11 (and A to be a single digit 0–9):
3 − A = 0 → A = 3

**Check:** $132 ÷ 11 = $12 per member. This is a valid integer amount.

Tempting wrong answers: Other values of A don't make 3 − A a multiple of 11 within the digit range.

ANSWER 6: D

---

Problem 7:
Wire length = 24 cm. We check if each shape's perimeter ≤ 24 cm.

**Regular hexagon (side 5 cm):**
Perimeter = 6 × 5 = 30 cm. Since 30 > 24, he **cannot** make this.

**Square (area 36 cm²):**
Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Since 24 = 24, he **can** make this.

**Right triangle (legs 6 and 8 cm):**
Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Perimeter = 6 + 8 + 10 = 24 cm. Since 24 = 24, he **can** make this.

Haruki can make the square and triangle only.

Tempting wrong answers: Choice E includes the hexagon, but its perimeter exceeds the wire length.

ANSWER 7: D

---

Problem 8:
The reciprocal of a number x is 1/x. The *smaller* the positive number, the *larger* its reciprocal.

| Choice | Number | Reciprocal |
|--------|--------|------------|
| A | 1/3 | 3 |
| B | 2/5 | 5/2 = 2.5 |
| C | 1 | 1 |
| D | 5 | 1/5 = 0.2 |
| E | 1986 | 1/1986 ≈ 0.0005 |

The largest reciprocal is 3, which comes from 1/3.

Tempting wrong answers: Choice E (1986) is the largest number, so it has the *smallest* reciprocal.

ANSWER 8: A

---

Problem 9:
Athlete's age = 26 years.

**Step 1:** Maximum heart rate = 220 − 26 = 194 beats per minute.

**Step 2:** Target heart rate = 80% of 194 = 0.80 × 194 = 155.2

**Step 3:** Round to nearest whole number: 155

Tempting wrong answers: Choice C (176) might come from taking 80% of 220 without subtracting the age. Choice D (194) is the maximum heart rate itself.

ANSWER 9: B

---

Problem 10:
**Harry (correct calculation):**
H = 8 − (2 + 5) = 8 − 7 = 1

**Terry (ignores parentheses):**
T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:**
H − T = 1 − 11 = −10

Tempting wrong answers: Choice E (10) is |H − T|, but the question asks for H − T specifically, which is negative since Terry's answer is larger.

ANSWER 10: A

---

Problem 11:
We need: 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

The sum inside is an arithmetic sequence:
- First term: 81, Last term: 99
- Number of terms: 10
- Sum = (number of terms) × (first + last) / 2 = 10 × (81 + 99) / 2 = 10 × 180 / 2 = 10 × 90 = 900

Multiply by 2: 2 × 900 = 1800

Tempting wrong answers: Choice C (1700) might come from miscounting terms or miscalculating the average.

ANSWER 11: E

---

Problem 12:
Let me organize the data from the survey:

- Total surveyed: 200
- Females: 96 → Males: 200 − 96 = 104
- Total listeners: 136
- Females who listen: 58 → Males who listen: 136 − 58 = 78
- Males who don't listen: 26 (Check: 78 + 26 = 104 ✓)

**Percentage of males who listen:**
(78 / 104) × 100% = (3/4) × 100% = 75%

Tempting wrong answers: Choice C (52) might come from computing 104/200 (percentage of total who are male). Choice D (55) might come from 78/136 (percentage of listeners who are male).

ANSWER 12: E

 Problem 1:
We start with 30 liters of paint. The original amount of yellow tint is 30% of 30 liters:
\[
0.30 \times 30 = 9 \text{ liters.}
\]
Five liters of yellow tint are added, so the new amount of yellow tint is:
\[
9 + 5 = 14 \text{ liters.}
\]
The new total volume of the mixture is:
\[
30 + 5 = 35 \text{ liters.}
\]
The percent of yellow tint in the new mixture is:
\[
\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%.
\]
(Choice A is the original red percentage, D is the original water percentage, and E would require 17.5 liters of yellow, which we do not have.)
ANSWER 1: C

Problem 2:
Georgie chooses one of 6 windows to enter. Since he must leave through a different window, there are 5 remaining choices for his exit. By the multiplication principle:
\[
6 \times 5 = 30 \text{ ways.}
\]
(Choice E, 36, would count entering and leaving through the same window; the other choices are too small.)
ANSWER 2: D

Problem 3:
Simplify each fraction:
\[
\frac{2}{20} = \frac{1}{10}, \qquad \frac{3}{30} = \frac{1}{10}.
\]
So the sum is:
\[
\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3.
\]
(Choice A is just one term; C is 0.2; E is 0.6.)
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides:
\[
2 \times 12 = 24 \text{ dollars.}
\]
Total spent:
\[
12 + 24 = 36 \text{ dollars.}
\]
Amount left:
\[
50 - 36 = 14 \text{ dollars.}
\]
(Choice C is the total spent; D is the amount left if she had spent only $12.)
ANSWER 4: B

Problem 5:
\[
200{,}000 \times 200{,}000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}.
\]
\[
4 \times 10^{10} = 40{,}000{,}000{,}000,
\]
which is read as forty billion.
(Choice D is $4 \times 10^8$; B is $4 \times 10^6$.)
ANSWER 5: E

Problem 6:
The total amount paid is the three-digit number $1A2$, which equals:
\[
100 + 10A + 2 = 102 + 10A.
\]
Since 11 members each paid the same integer amount, $102 + 10A$ must be divisible by 11. Testing the choices:
- $A=0$: $102 \div 11 \approx 9.27$ (not an integer)
- $A=1$: $112 \div 11 \approx 10.18$ (not an integer)
- $A=2$: $122 \div 11 \approx 11.09$ (not an integer)
- $A=3$: $132 \div 11 = 12$ (integer!)
- $A=4$: $142 \div 11 \approx 12.9$ (not an integer)
Thus the missing digit is 3.
ANSWER 6: D

Problem 7:
The wire is 24 cm long, so a shape is possible only if its perimeter is at most 24 cm.
- **Regular hexagon, side 5 cm:** Perimeter $= 6 \times 5 = 30$ cm. $30 > 24$, so impossible.
- **Square of area 36 cm²:** Side $= \sqrt{36} = 6$ cm. Perimeter $= 4 \times 6 = 24$ cm. Exactly fits, so possible.
- **Right triangle with legs 6 and 8 cm:** Hypotenuse $= \sqrt{6^2 + 8^2} = \sqrt{100} = 10$ cm. Perimeter $= 6 + 8 + 10 = 24$ cm. Exactly fits, so possible.
Only the square and the triangle can be formed.
ANSWER 7: D

Problem 8:
For positive numbers, the reciprocal function $f(x) = \frac{1}{x}$ is decreasing, so the smallest number has the largest reciprocal. Compare the two smallest choices:
\[
\frac{1}{3} \approx 0.333\ldots, \qquad \frac{2}{5} = 0.4.
\]
Since $\frac{1}{3} < \frac{2}{5} < 1 < 5 < 1986$, the smallest number is $\frac{1}{3}$. Its reciprocal is $3$, which is larger than the reciprocals of all the other choices.
ANSWER 8: A

Problem 9:
First find the theoretical maximum heart rate:
\[
220 - 26 = 194 \text{ beats per minute.}
\]
The target rate is 80% of this:
\[
0.80 \times 194 = 155.2.
\]
Rounded to the nearest whole number, this is 155.
(Choice C is the maximum heart rate itself.)
ANSWER 9: B

Problem 10:
Harry computes correctly:
\[
H = 8 - (2 + 5) = 8 - 7 = 1.
\]
Terry ignores the parentheses:
\[
T = 8 - 2 + 5 = 6 + 5 = 11.
\]
The difference is:
\[
H - T = 1 - 11 = -10.
\]
ANSWER 10: A

Problem 11:
The ten numbers inside the parentheses form an arithmetic sequence: 81, 83, …, 99. The sum of an arithmetic sequence is:
\[
\frac{\text{number of terms}}{2} \times (\text{first} + \text{last}) = \frac{10}{2} \times (81 + 99) = 5 \times 180 = 900.
\]
Now multiply by the factor of 2 outside:
\[
2 \times 900 = 1800.
\]
(One can also pair them: $(81+99)=180$, $(83+97)=180$, etc., giving five pairs that each sum to 180.)
ANSWER 11: E

Problem 12:
Total surveyed = 200, and females surveyed = 96, so:
\[
\text{Males surveyed} = 200 - 96 = 104.
\]
Total listeners = 136, and females who listen = 58, so:
\[
\text{Males who listen} = 136 - 58 = 78.
\]
(We can check: total non-listeners = 64, males who don't listen = 26, so females who don't listen = 38; $58+38=96$ and $78+26=104$, which checks out.)

The percentage of males who listen is:
\[
\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%.
\]
ANSWER 12: E

Problem 1:
A mixture of 30 liters is 25% red, 30% yellow, and 45% water. Five liters of yellow tint are added. What is the percent of yellow tint in the new mixture?

**Setup:** Find the amount of yellow tint originally, add 5 liters, then find the new percentage.

**Original yellow tint:** 30% of 30 = 0.30 × 30 = 9 liters

**After adding 5 liters of yellow tint:**
- New yellow tint: 9 + 5 = 14 liters
- New total volume: 30 + 5 = 35 liters
- New percentage: 14/35 = 2/5 = 40%

**Checking wrong choices:** 25% would be if we just added the original percentages; 35% would be if we just added 5% to 30%; 45% and 50% are too high.

ANSWER 1: C

---

Problem 2:
Six windows. Georgie enters by one and leaves by a different one. How many ways?

**Setup:** Choose an entry window (6 choices), then choose a different exit window (5 choices).

**Calculation:** 6 × 5 = 30 ways

**Checking wrong choices:** 15 would be if we divided by 2 (treating order as irrelevant); 12 would be 6×2; 18 would be 6×3; 36 would be 6×6 (allowing same window).

ANSWER 2: D

---

Problem 3:
1/10 + 2/20 + 3/30 = ?

**Simplify each fraction:**
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

**Sum:** 1/10 + 1/10 + 1/10 = 3/10 = 0.3

**Checking wrong choices:** 0.1 would be just one term; 0.2 would be two terms; 0.6 would be adding numerators and denominators incorrectly.

ANSWER 3: D

---

Problem 4:
Susan had $50. She spent $12 on food and twice as much on rides. How much left?

**Setup:** Food = $12, Rides = 2 × $12 = $24

**Total spent:** $12 + $24 = $36

**Amount left:** $50 - $36 = $14

**Checking wrong choices:** 12 is just the food amount; 26 is 50-24 (forgetting food); 38 is 50-12 (forgetting rides).

ANSWER 4: B

---

Problem 5:
200,000 × 200,000 = ?

**Setup:** Express in scientific notation: 200,000 = 2 × 10⁵

**Calculation:** (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion

**Checking wrong choices:** Four hundred thousand is 4×10⁵; four million is 4×10⁶; forty thousand is 4×10⁴; four hundred million is 4×10⁸.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, totaling $1A2. Find the missing digit A.

**Setup:** The total $1A2 must be divisible by 11.

**Divisibility test for 11:** A number is divisible by 11 if the alternating sum of its digits is divisible by 11.

For 1A2: (1 + 2) - A = 3 - A must be divisible by 11.

Since A is a single digit (0-9), 3 - A can only equal 0 (the only multiple of 11 in range -6 to 3).

So A = 3, giving $132 = 11 × $12 ✓

**Checking:** With A=0: $102 ÷ 11 ≈ 9.27 (not integer); A=1: $112 ÷ 11 ≈ 10.18 (not integer); A=3: $132 ÷ 11 = 12 ✓

ANSWER 6: D

---

Problem 7:
Wire is 24 cm. Which shapes can be made?

**Regular hexagon with side 5 cm:** Perimeter = 6 × 5 = 30 cm. Need 30 cm, only have 24 cm. ❌

**Square with area 36 cm²:** Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Need 24 cm, have 24 cm. ✓

**Right triangle with legs 6 and 8 cm:** Hypotenuse = √(36 + 64) = √100 = 10 cm. Perimeter = 6 + 8 + 10 = 24 cm. Need 24 cm, have 24 cm. ✓

**Conclusion:** Can make the square and triangle, but not the hexagon.

ANSWER 7: D

---

Problem 8:
Which number has the largest reciprocal?

**Find each reciprocal:**
- A. 1/3 → reciprocal = 3
- B. 2/5 → reciprocal = 5/2 = 2.5
- C. 1 → reciprocal = 1
- D. 5 → reciprocal = 1/5 = 0.2
- E. 1986 → reciprocal ≈ 0.0005

**Key insight:** Smaller positive numbers have larger reciprocals. The smallest number listed is 1/3, so it has the largest reciprocal (3).

ANSWER 8: A

---

Problem 9:
Target heart rate = 80% of (220 - age). Find target for 26-year-old.

**Maximum heart rate:** 220 - 26 = 194 bpm

**Target heart rate:** 80% of 194 = 0.80 × 194 = 155.2 ≈ 155 bpm

**Checking wrong choices:** 134 would be about 69%; 176 would be about 91%; 194 is the max (100%); 243 is impossible.

ANSWER 9: B

---

Problem 10:
Harry calculates 8 − (2 + 5) correctly. Terry calculates 8 − 2 + 5. Find H − T.

**Harry's answer:** H = 8 − (2 + 5) = 8 − 7 = 1

**Terry's answer:** T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:** H − T = 1 − 11 = −10

**Checking wrong choices:** −6 would be if Terry got 7; 0 would be if they got the same answer; 6 or 10 would be T − H.

ANSWER 10: A

---

Problem 11:
2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99) = ?

**Setup:** This is 10 consecutive odd numbers. The average of the first and last is (81 + 99)/2 = 90.

**Sum of the 10 numbers:** 10 × 90 = 900

**Multiply by 2:** 2 × 900 = 1800

**Alternative verification:** Pair the numbers: (81+99) + (83+97) + (85+95) + (87+93) + (89+91) = 180 + 180 + 180 + 180 + 180 = 900. Then 2 × 900 = 1800.

ANSWER 11: E

---

Problem 12:
Find the percentage of males who listen to the station.

**Given information:**
- Total surveyed: 200
- Females: 96
- Females who listen: 58
- Males who don't listen: 26
- Total listeners: 136
- Total non-listeners: 64

**Find males:** Males = 200 - 96 = 104

**Find males who listen:** Males who listen = Total listeners - Females who listen = 136 - 58 = 78

**Verification:** Males who don't listen = 104 - 78 = 26 ✓ (matches given)

**Percentage of males who listen:** 78/104 × 100 = 75%

**Checking wrong choices:** 39% would be 78/200 (using total instead of males); 48% would be about 50/104; 52% would be about 54/104; 55% would be about 57/104.

ANSWER 12: E

**Problem 1**

The original mixture is 30 L with 30 % yellow tint → yellow = 0.30·30 = 9 L.  
Add 5 L of yellow → total yellow = 9 + 5 = 14 L.  
Total volume after addition = 30 + 5 = 35 L.  

Percent yellow = (14/35)·100 % = 40 %.

**Answer 1: C**

---

**Problem 2**

Choose an entry window (6 ways) and a different exit window (5 ways).  
Number of ways = 6 × 5 = 30.

**Answer 2: D**

---

**Problem 3**

\[
\frac{1}{10}=0.1,\qquad\frac{2}{20}=0.1,\qquad\frac{3}{30}=0.1
\]
Sum = 0.1 + 0.1 + 0.1 = 0.3.

**Answer 3: D**

---

**Problem 4**

Food cost = \$12.  
Rides cost = twice food = 2·12 = \$24.  
Total spent = 12 + 24 = \$36.  
Money left = 50 − 36 = \$14.

**Answer 4: B**

---

**Problem 5**

\(200{,}000 \times 200{,}000 = (2\times10^{5})^{2}=4\times10^{10}=40{,}000{,}000{,}000\).

That is 40 billion.

**Answer 5: E**

---

**Problem 6**

Total paid = \$1A2 = 100 + 10A + 2 = 102 + 10A.  
Since the 11 members each paid an integer amount, 102 + 10A must be divisible by 11.

\(102\equiv3\pmod{11}\).  
We need \(10A\equiv-3\equiv8\pmod{11}\).  
Because \(10\equiv-1\pmod{11}\), this is \(-A\equiv8\pmod{11}\) → \(A\equiv3\pmod{11}\).  
A digit, so \(A=3\).

**Answer 6: D**

---

**Problem 7**

- Hexagon: perimeter = 6·5 = 30 cm > 24 cm → impossible.  
- Square: side = √36 = 6 cm, perimeter = 4·6 = 24 cm → possible.  
- Right triangle: legs 6 cm, 8 cm, hypotenuse = √(6²+8²)=10 cm, perimeter = 6+8+10 = 24 cm → possible.

Only the square and triangle can be made.

**Answer 7: D**

---

**Problem 8**

Reciprocals:
- \(1/3 → 3\)  
- \(2/5 → 5/2 = 2.5\)  
- \(1 → 1\)  
- \(5 → 1/5 = 0.2\)  
- \(1986 → 1/1986 ≈ 0.0005\)

Largest reciprocal is 3, from \(1/3\).

**Answer 8: A**

---

**Problem 9**

Maximum heart rate = 220 − 26 = 194 bpm.  
Target = 80 % of 194 = 0.8·194 = 155.2 → rounded to 155 bpm.

**Answer 9: B**

---

**Problem 10**

Harry: \(8-(2+5)=8-7=1\).  
Terry (ignoring parentheses): \(8-2+5 = 6+5 = 11\).  
Difference \(H-T = 1-11 = -10\).

**Answer 10: A**

---

**Problem 11**

Sum of the ten terms: average = (81+99)/2 = 90, so sum = 90·10 = 900.  
Multiply by 2: \(2·900 = 1800\).

**Answer 11: E**

---

**Problem 12**

Total surveyed = 200.  
Females = 96 → Males = 200 − 96 = 104.  

Female listeners = 58 → Male listeners = 136 − 58 = 78.  

Percent of males who listen = (78/104)·100 % = (39/52)·100 % = 75 %.

**Answer 12: E**

---

**Final Answers**

ANSWER 1: C  
ANSWER 2: D  
ANSWER 3: D  
ANSWER 4: B  
ANSWER 5: E  
ANSWER 6: D  
ANSWER 7: D  
ANSWER 8: A  
ANSWER 9: B  
ANSWER 10: A  
ANSWER 11: E  
ANSWER 12: E

Problem 1:  
A 30-liter paint mixture is 25% red tint, 30% yellow tint, and 45% water. Five liters of yellow tint are added.  
Original yellow tint: $0.30 \times 30 = 9$ liters.  
After adding 5 liters: $9 + 5 = 14$ liters yellow.  
New total volume: $30 + 5 = 35$ liters.  
Percent yellow: $\frac{14}{35} = 0.40 = 40\%$.  
Tempting wrong choices: 35% (original yellow percent) or 45% (water percent).  
ANSWER 1: D  

Problem 2:  
Georgie enters through one of six windows and exits through a different one.  
Entry choices: 6. Exit choices: 5 (excluding entry window).  
Total ways: $6 \times 5 = 30$.  
Tempting wrong choices: 15 (if order didn’t matter) or 36 (if same window allowed).  
ANSWER 2: D  

Problem 3:  
$\frac{1}{10} + \frac{2}{20} + \frac{3}{30} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3$.  
Tempting wrong choices: 0.1 (one term) or 0.2 (two terms).  
ANSWER 3: D  

Problem 4:  
Susan starts with \$50.  
Food: \$12.  
Rides: $2 \times 12 = \$24$.  
Total spent: $12 + 24 = \$36$.  
Left: $50 - 36 = \$14$.  
Tempting wrong choices: \$26 (if spent \$24 total) or \$38 (if spent \$12 total).  
ANSWER 4: B  

Problem 5:  
$200,000 \times 200,000 = 40,000,000,000 = 40$ billion.  
Tempting wrong choices: 400,000 (added zeros) or 4 million (multiplied digits).  
ANSWER 5: E  

Problem 6:  
Total paid: \$1A2, an integer divisible by 11.  
$1A2 \mod 11 = 0$.  
$1 + 2 - A \equiv 0 \mod 11 \Rightarrow 3 - A \equiv 0 \mod 11 \Rightarrow A = 3$.  
Tempting wrong choices: 0, 1, 2, or 4 (incorrect divisibility).  
ANSWER 6: E (Wait, let's recheck: 1A2 must be divisible by 11. 132 ÷ 11 = 12, so A=3. But 132 is 11×12, so correct. However, the answer choices are A=0,B=1,C=2,D=3,E=4. So D is 3. So correct answer is D, not E. I made a mistake in the initial answer. Correcting:)  
**Correction**: $1A2 = 132$ when $A=3$, and $132 \div 11 = 12$, so $A=3$.  
ANSWER 6: D  

Problem 7:  
Wire length: 24 cm.  
- Regular hexagon with side 5 cm: Perimeter $6 \times 5 = 30$ cm (not 24).  
- Square with area 36 cm²: Side $6$ cm, perimeter $24$ cm.  
- Right triangle with legs 6 and 8 cm: Hypotenuse 10 cm, perimeter $6+8+10=24$ cm.  
Can make square and triangle.  
Tempting wrong choices: Hexagon (incorrect perimeter).  
ANSWER 7: D  

Problem 8:  
Largest reciprocal means smallest original number.  
$\frac{1}{3} \approx 0.333$, $\frac{2}{5} = 0.4$, $1$, $5$, $1986$.  
Smallest: $\frac{1}{3}$. Reciprocal: 3.  
Others: 2.5, 1, 0.2, 0.0005.  
Tempting wrong choices: $\frac{2}{5}$ (larger than $\frac{1}{3}$).  
ANSWER 8: A  

Problem 9:  
Maximum heart rate: $220 - 26 = 194$.  
Target: $0.80 \times 194 = 155.2 \approx 155$.  
Tempting wrong choices: 134 (70%), 176 (90%), or 243 (incorrect max).  
ANSWER 9: B  

Problem 10:  
Harry: $8 - (2 + 5) = 1$.  
Terry: $8 - 2 + 5 = 11$.  
Difference: $1 - 11 = -10$.  
Tempting wrong choices: 10 (absolute difference) or 0 (if parentheses ignored by both).  
ANSWER 10: A  

Problem 11:  
Sum: $81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99$.  
Average: 90, 10 terms: $90 \times 10 = 900$.  
$2 \times 900 = 1800$.  
Tempting wrong choices: 1600 (min+max=180, times 5=900, times 2=1800, but if miscount terms).  
ANSWER 11: E  

Problem 12:  
Total surveyed: 200.  
Females: 96, so males: $200 - 96 = 104$.  
Total listeners: 136.  
Female listeners: 58, so male listeners: $136 - 58 = 78$.  
Percent male listeners: $\frac{78}{104} = 0.75 = 75\%$.  
Tempting wrong choices: 39 (male non-listeners percent) or 52 (incorrect calculation).  
ANSWER 12: E

### Problem 1:
What is the percentage of yellow tint in the new 35-liter paint mixture, after adding 5 liters of yellow tint to an original 30-liter mixture that was 30% yellow?
First, calculate the original volume of yellow tint: 30% of 30 liters = \(0.3 \times 30 = 9\) liters. Add the 5 new liters of yellow tint to get \(9 + 5 = 14\) liters of yellow in the new mixture. The total new volume of the mixture is \(30 + 5 = 35\) liters. The percentage of yellow is \(\frac{14}{35} = 0.4 = 40\%\). Tempting wrong answers come from forgetting to update the total mixture volume when calculating the new percentage, or miscalculating the original yellow volume.
ANSWER 1: C

---

### Problem 2:
How many ways can Georgie enter through one window and leave through a different window, out of 6 total windows?
Georgie has 6 choices for an entry window. Since he must leave through a different window, he has 5 remaining choices for an exit window. The total number of ways is \(6 \times 5 = 30\). A common wrong answer is 15, which incorrectly treats entry/exit as unordered pairs, when order matters (entering window A and leaving B is different from entering B and leaving A).
ANSWER 2: D

---

### Problem 3:
Simplify \(\frac{1}{10} + \frac{2}{20} + \frac{3}{30}\):
Simplify each fraction first: \(\frac{2}{20} = \frac{1}{10}\) and \(\frac{3}{30} = \frac{1}{10}\). Add them together: \(\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3\). A common wrong answer is 0.1, which incorrectly adds numerators and denominators directly instead of simplifying first.
ANSWER 3: D

---

### Problem 4:
How much money does Susan have left after spending $12 on food and twice that amount on rides, starting with $50?
First calculate the amount spent on rides: twice $12 is \(2 \times 12 = 24\) dollars. Total spent is \(12 + 24 = 36\) dollars. The amount left is \(50 - 36 = 14\) dollars. A common wrong answer is 26, which incorrectly uses $12 instead of $24 for ride costs.
ANSWER 4: B

---

### Problem 5:
Calculate \(200,000 \times 200,000\):
Rewrite 200,000 as \(2 \times 10^5\). Multiplying two copies gives \((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\), which equals 40,000,000,000, or forty billion. Wrong answers come from miscounting the number of zeros in the product.
ANSWER 5: E

---

### Problem 6:
The 3-digit total cost \(1A2\) dollars is divisible by 11 (11 members paid the same integer amount). What is the missing digit A?
The divisibility rule for 11 states that alternating sum of digits must be divisible by 11. For 1A2, the alternating sum is \(1 - A + 2 = 3 - A\). The only value of A (0-9) that makes this sum divisible by 11 is A=3, since \(3-3=0\), which is divisible by 11. Verify: \(132 \div 11 = 12\), a valid integer.
ANSWER 6: D

---

### Problem 7:
Haruki has a 24cm wire. Which of the three shapes can he form? Calculate the perimeter of each shape:
1. Regular hexagon with side 5cm: perimeter = \(6 \times 5 = 30\)cm, which is longer than 24cm, so impossible.
2. Square with area 36cm²: side length = \(\sqrt{36}=6\)cm, perimeter = \(4 \times 6 = 24\)cm, which fits.
3. Right triangle with legs 6cm and 8cm: hypotenuse = 10cm (6-8-10 Pythagorean triple), perimeter = \(6+8+10=24\)cm, which fits.
Only the square and triangle can be formed.
ANSWER 7: D

---

### Problem 8:
Which number has the largest reciprocal? The reciprocal of \(x\) is \(\frac{1}{x}\), so smaller values of \(x\) produce larger reciprocals. List the values of all options:
A. \(\frac{1}{3} \approx 0.333\), B. \(\frac{2}{5}=0.4\), C.1, D.5, E.1986. The smallest value is \(\frac{1}{3}\), whose reciprocal is 3, the largest of all options.
ANSWER 8: A

---

### Problem 9:
Calculate the target heart rate for a 26-year-old athlete: target is 80% of max heart rate, where max = 220 - age.
First calculate max heart rate: \(220 - 26 = 194\) bpm. 80% of 194 is \(0.8 \times 194 = 155.2\), which rounds to 155 bpm. A common wrong answer is 176, which uses 220 instead of the correct max heart rate of 194.
ANSWER 9: B

---

### Problem 10:
Calculate \(H-T\), where H is Harry's correct answer to \(8-(2+5)\), and T is Terry's incorrect answer of \(8-2+5\).
Harry's answer: \(8 - 7 = 1 = H\). Terry's answer: \(6 + 5 = 11 = T\). The difference \(H-T = 1 - 11 = -10\). A common wrong answer is 10, which calculates \(T-H\) instead of \(H-T\).
ANSWER 10: A

---

### Problem 11:
Simplify \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\):
Pair the terms inside the parentheses to simplify addition: \((81+99)+(83+97)+(85+95)+(87+93)+(89+91) = 5 \times 180 = 900\). Multiply by 2: \(2 \times 900 = 1800\).
ANSWER 11: E

---

### Problem 12:
What percentage of males surveyed listen to the radio station? First calculate total number of males: total surveyed = 200, females = 96, so males = \(200 - 96 = 104\). Males who do not listen = 26, so males who listen = \(104 - 26 = 78\). The percentage of males who listen is \(\frac{78}{104} = 0.75 = 75\%\). A common wrong answer is 39%, which divides male listeners by the total surveyed population instead of the total male population.
ANSWER 12: E

### Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.
1.  Calculate original yellow tint volume: 25%? No wait, original is 30% yellow. So 30% of 30 L = $0.3 \times 30 = 9$ liters of yellow.
2.  Add 5 liters of yellow: total yellow = $9 + 5 = 14$ liters.
3.  Total new mixture volume: $30 + 5 = 35$ liters.
4.  Percent yellow: $\frac{14}{35} \times 100 = 40\%$.
ANSWER 1: C
---
### Problem 2:
We count the number of ways to choose an entry window and a different exit window from 6 windows.
1.  Entry window: 6 possible choices.
2.  Exit window: cannot be the entry window, so 5 remaining choices.
3.  Total ways: $6 \times 5 = 30$ (order matters because entry and exit are distinct actions).
ANSWER 2: D
---
### Problem 3:
Simplify and add the fractions:
1.  $\frac{1}{10} = 0.1$, $\frac{2}{20} = \frac{1}{10} = 0.1$, $\frac{3}{30} = \frac{1}{10} = 0.1$.
2.  Sum: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D
---
### Problem 4:
Calculate remaining money after expenses:
1.  Total initial money: $50.
2.  Money spent on rides: twice the food cost = $2 \times 12 = 24$ dollars.
3.  Total spent: $12 + 24 = 36$ dollars.
4.  Remaining: $50 - 36 = 14$ dollars.
ANSWER 4: B
---
### Problem 5:
Compute $200,000 \times 200,000$:
1.  Rewrite as $(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
2.  $10^{10} = 10,000,000,000$, so $4 \times 10^{10} = 40,000,000,000 = 40$ billion.
ANSWER 5: E
---
### Problem 6:
The total payment $1A2$ (a 3-digit number) must be divisible by 11 (since 11 people paid equal integer amounts).
1.  Use the divisibility rule for 11: the alternating sum of digits is divisible by 11. For $1A2$, this is $1 - A + 2 = 3 - A$.
2.  $A$ is a digit (0-9), so $3 - A$ must equal 0 (the only multiple of 11 in range: $3 - A = 11$ gives $A=-8$, $3-A=-11$ gives $A=14$, both invalid).
3.  $3 - A = 0 \implies A=3$. Verify: $132 \div 11 = 12$, which is an integer.
ANSWER 6: D
---
### Problem 7:
Check if the 24 cm wire can form each shape (perimeter of the shape must equal 24 cm):
1.  Regular hexagon (side 5 cm): Perimeter = $6 \times 5 = 30$ cm > 24 cm → cannot make.
2.  Square (area 36 cm²): Side length = $\sqrt{36} = 6$ cm. Perimeter = $4 \times 6 = 24$ cm → can make.
3.  Right triangle (legs 6, 8 cm): Hypotenuse = $\sqrt{6^2 + 8^2} = 10$ cm. Perimeter = $6 + 8 + 10 = 24$ cm → can make.
Only square and triangle can be made.
ANSWER 7: D
---
### Problem 8:
The reciprocal of a positive number $x$ is $\frac{1}{x}$; smaller $x$ gives larger reciprocal.
1.  List values: $\frac{1}{3} \approx 0.33$, $\frac{2}{5}=0.4$, $1$, $5$, $1986$.
2.  Smallest value is $\frac{1}{3}$, whose reciprocal is $3$ (largest among all reciprocals: $\frac{5}{2}=2.5$, $1$, $0.2$, $\frac{1}{1986} \approx 0.0005$).
ANSWER 8: A
---
### Problem 9:
Calculate target heart rate for a 26-year-old:
1.  Maximum heart rate: $220 - 26 = 194$ bpm.
2.  Target heart rate: 80% of 194 = $0.8 \times 194 = 155.2$, which rounds to 155.
ANSWER 9: B
---
### Problem 10:
Compute $H$ (Harry's correct answer) and $T$ (Terry's incorrect answer):
1.  Harry: $8 - (2 + 5) = 8 - 7 = 1$, so $H=1$.
2.  Terry (ignores parentheses, left-to-right): $8 - 2 + 5 = 6 + 5 = 11$, so $T=11$.
3.  Difference: $H - T = 1 - 11 = -10$.
ANSWER 10: A
---
### Problem 11:
Calculate $2 \times (81 + 83 + ... + 99)$:
1.  The sum inside is 10 consecutive odd numbers. Average of first and last term: $\frac{81 + 99}{2} = 90$.
2.  Sum of the 10 terms: $90 \times 10 = 900$.
3.  Multiply by 2: $2 \times 900 = 1800$.
ANSWER 11: E
---
### Problem 12:
Find the percentage of males who listen to the station:
1.  Total males surveyed: $200 - 96 = 104$.
2.  Males who do not listen: 26, so males who listen: $104 - 26 = 78$.
3.  Verify with total listeners: Total listeners = 136, female listeners = 58, so male listeners = $136 - 58 = 78$ (matches).
4.  Percentage: $\frac{78}{104} \times 100 = 75\%$.
ANSWER 12: E

Problem 1:
Original yellow = 30% of 30 = 9 L. Add 5 → 14 L yellow. New total = 35 L. Percent = 14/35 = 40%.

ANSWER 1: C

Problem 2:
Enter: 6 choices, leave: 5 different choices. 6×5 = 30.

ANSWER 2: D

Problem 3:
Each fraction = 0.1. Sum = 0.3.

ANSWER 3: D

Problem 4:
Food 12, rides twice = 24. Spent 36. Left = 50 − 36 = 14.

ANSWER 4: B

Problem 5:
200,000 × 200,000 = 40,000,000,000 = forty billion.

ANSWER 5: E

Problem 6:
1A2 divisible by 11. Alternating sum: 1 − A + 2 = 3 − A must be divisible by 11 → A = 3 (gives 0). Check 132/11 = 12. Yes.

ANSWER 6: D

Problem 7:
Hexagon: 6×5 = 30 > 24, no. Square area 36 → side 6, perimeter 24, yes. Triangle legs 6,8 → hypotenuse 10, perimeter 24, yes. Square and triangle only.

ANSWER 7: D

Problem 8:
Largest reciprocal = smallest number. Smallest is 1/3.

ANSWER 8: A

Problem 9:
Max = 220 − 26 = 194. Target = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

Problem 10:
H = 8 − 7 = 1. T = 8 − 2 + 5 = 11. H − T = 1 − 11 = −10.

ANSWER 10: A

Problem 11:
Sum of 81..99 odd numbers (10 terms): average (81+99)/2 = 90, ×10 = 900. ×2 = 1800.

ANSWER 11: E

Problem 12:
Males total = 200 − 96 = 104. Males who don't listen = 26 → males who listen = 104 − 26 = 78. Percent = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem systematically.

---

**Problem 1:**
Original mixture: 30 liters, 30% yellow = 9 liters yellow. Add 5 liters yellow → new yellow = 14 liters, new total = 35 liters.
Percent yellow = 14/35 = 40%.

ANSWER 1: C

---

**Problem 2:**
Georgie can enter through any of 6 windows, then leave through any of the remaining 5 windows.
6 × 5 = 30.

ANSWER 2: D

---

**Problem 3:**
1/10 = 0.1, 2/20 = 0.1, 3/30 = 0.1. Sum = 0.3.

ANSWER 3: D

---

**Problem 4:**
Food: $12. Rides: 2 × $12 = $24. Total spent: $12 + $24 = $36. Left: $50 − $36 = $14.

ANSWER 4: B

---

**Problem 5:**
200,000 × 200,000 = (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion.

ANSWER 5: E

---

**Problem 6:**
The number 1A2 must be divisible by 11. 1A2 ÷ 11 must be an integer.
11 × 10 = 110, 11 × 11 = 121 (no), 11 × 12 = 132 ✓ → A = 3.
Check: 132/11 = 12 ✓, each paid $12.

ANSWER 6: D

---

**Problem 7:**
Wire = 24 cm.
- Regular hexagon, side 5: perimeter = 6 × 5 = 30 cm > 24. ✗
- Square, area 36: side = 6, perimeter = 24 cm ✓
- Right triangle, legs 6 and 8: hypotenuse = 10, perimeter = 6+8+10 = 24 cm ✓

ANSWER 7: D

---

**Problem 8:**
The reciprocal of x is 1/x. The largest reciprocal comes from the smallest positive number.
Among 1/3, 2/5, 1, 5, 1986 — the smallest is 1/3, and its reciprocal is 3, which is the largest.

ANSWER 8: A

---

**Problem 9:**
Maximum heart rate = 220 − 26 = 194. Target = 80% × 194 = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

---

**Problem 10:**
Harry: 8 − (2 + 5) = 8 − 7 = 1, so H = 1.
Terry: 8 − 2 + 5 = 11, so T = 11.
H − T = 1 − 11 = −10.

ANSWER 10: A

---

**Problem 11:**
The ten odd numbers from 81 to 99: sum = (81+99)/2 × 10 = 90 × 10 = 900.
2 × 900 = 1800.

ANSWER 11: E

---

**Problem 12:**
Total surveyed = 200. Females = 96, so Males = 104.
Total non-listeners = 64. Males who don't listen = 26, so females who don't listen = 64 − 26 = 38.
Males who listen = 104 − 26 = 78.
Percentage = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem step-by-step.

## Problem 1:
**What is asked:** Find the percentage of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.

**Original mixture:**
- Yellow tint: 30% of 30 = 0.30 × 30 = 9 liters
- Total volume: 30 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total volume: 30 + 5 = 35 liters

**Percentage of yellow tint:**
$$\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%$$

ANSWER 1: C

## Problem 2:
**What is asked:** In how many ways can the ghost enter by one window and leave by a different window?

**Solution:**
- Choices for entering: 6 windows
- Choices for leaving: 5 windows (must be different from entry)
- Total ways: 6 × 5 = 30

ANSWER 2: D

## Problem 3:
**What is asked:** Calculate 1/10 + 2/20 + 3/30

**Solution:**
Simplify each fraction:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

ANSWER 3: D

## Problem 4:
**What is asked:** How much money does Susan have left?

**Solution:**
- Started with: $50
- Spent on food: $12
- Spent on rides: 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Left: $50 - $36 = $14

ANSWER 4: B

## Problem 5:
**What is asked:** Calculate 200,000 × 200,000

**Solution:**
$$200,000 × 200,000 = (2 × 10^5) × (2 × 10^5) = 4 × 10^{10} = 40,000,000,000 = 40 \text{ billion}$$

ANSWER 5: E

## Problem 6:
**What is asked:** Find the missing digit A in the 3-digit number 1A2, where 11 club members each paid an equal integer amount that totals 1A2.

**Solution:**
The number 1A2 must be divisible by 11.

Testing divisibility by 11 (alternating sum of digits):
- 1 - A + 2 = 0 (mod 11)
- 3 - A ≡ 0 (mod 11)
- A = 3

Check: 132 ÷ 11 = 12 ✓

ANSWER 6: D

## Problem 7:
**What is asked:** Which shapes can Haruki make with 24 cm of wire?

**Solution:**
Test each shape:

**Regular hexagon with side 5 cm:**
- Perimeter: 6 × 5 = 30 cm
- Need 30 cm, have 24 cm ✗

**Square with area 36 cm²:**
- Side length: √36 = 6 cm
- Perimeter: 4 × 6 = 24 cm ✓

**Right triangle with legs 6 and 8 cm:**
- Hypotenuse: √(6² + 8²) = √(36 + 64) = √100 = 10 cm
- Perimeter: 6 + 8 + 10 = 24 cm ✓

Only square and triangle can be made.

ANSWER 7: D

## Problem 8:
**What is asked:** Which number has the largest reciprocal?

**Solution:**
Find reciprocals:
- 1/(1/3) = 3
- 1/(2/5) = 5/2 = 2.5
- 1/1 = 1
- 1/5 = 0.2
- 1/1986 ≈ 0.0005

The largest reciprocal is 5 (from the number 1/5).

ANSWER 8: D

## Problem 9:
**What is asked:** Find the target heart rate for a 26-year-old athlete.

**Solution:**
- Maximum heart rate: 220 - 26 = 194 beats per minute
- Target heart rate: 80% of 194 = 0.80 × 194 = 155.2 ≈ 155

ANSWER 9: B

## Problem 10:
**What is asked:** Find H - T where H is Harry's answer and T is Terry's answer.

**Solution:**
**Harry's calculation (correct):**
$$H = 8 - (2 + 5) = 8 - 7 = 1$$

**Terry's calculation (ignoring parentheses):**
$$T = 8 - 2 + 5 = 6 + 5 = 11$$

**Difference:**
$$H - T = 1 - 11 = -10$$

ANSWER 10: A

## Problem 11:
**What is asked:** Calculate 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

**Solution:**
This is an arithmetic sequence with 10 terms. Sum = n(first + last)/2
$$81 + 83 + ... + 99 = \frac{10(81 + 99)}{2} = \frac{10 × 180}{2} = 900$$

$$2 × 900 = 1800$$

ANSWER 11: E

## Problem 12:
**What is asked:** What percentage of males surveyed listen to the station?

**Solution:**
- Total surveyed: 200
- Females: 96
- Males: 200 - 96 = 104

- Total non-listeners: 64
- Males who don't listen: 26
- Males who listen: 104 - 26 = 78

**Percentage of males who listen:**
$$\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%$$

ANSWER 12: E

Problem 1:

The original mixture is 30 liters total, with 30% yellow tint.

So the amount of yellow tint at first is:
- \(30\%\) of \(30 = 0.30 \times 30 = 9\) liters

Then 5 liters of yellow tint are added, so the new amount of yellow tint is:
- \(9 + 5 = 14\) liters

The new total mixture is:
- \(30 + 5 = 35\) liters

So the percent of yellow tint is:
- \(\frac{14}{35} = \frac{2}{5} = 0.4 = 40\%\)

Answer choice C matches 40%.

ANSWER 1: C

Problem 2:

Georgie can:
- choose 1 of the 6 windows to enter
- then choose a different window to leave

For the entry, there are 6 choices.

For the exit, there are 5 remaining choices, since it must be a different window.

So the total number of ways is:
- \(6 \times 5 = 30\)

ANSWER 2: D

Problem 3:

Compute each fraction:
- \(\frac{1}{10} = 0.1\)
- \(\frac{2}{20} = \frac{1}{10} = 0.1\)
- \(\frac{3}{30} = \frac{1}{10} = 0.1\)

Add them:
- \(0.1 + 0.1 + 0.1 = 0.3\)

So the value is \(0.3\).

ANSWER 3: D

Problem 4:

Susan had \$50.

She spent:
- \$12 on food
- twice as much on rides, so \(2 \times 12 = 24\)

Total spent:
- \(12 + 24 = 36\)

Money left:
- \(50 - 36 = 14\)

ANSWER 4: B

Problem 5:

Two hundred thousand is:
- \(200{,}000 = 2 \times 10^5\)

Multiply:
- \(200{,}000 \times 200{,}000 = (2 \times 10^5)(2 \times 10^5) = 4 \times 10^{10}\)

And \(4 \times 10^{10} = 40{,}000{,}000{,}000\), which is forty billion.

ANSWER 5: E

Problem 6:

The total is written as \(1A2\), and 11 members each paid the same integer amount.

So \(1A2\) must be divisible by 11.

Test the answer choices:
- 102 ÷ 11 is not an integer
- 112 ÷ 11 is not an integer
- 122 ÷ 11 = 11 remainder 1
- 132 ÷ 11 = 12 exactly
- 142 ÷ 11 is not an integer

So the number is 132, which means \(A = 3\).

ANSWER 6: D

Problem 7:

We check each shape.

### Hexagon
A regular hexagon has 6 equal sides.
If each side is 5 cm, perimeter is:
- \(6 \times 5 = 30\) cm

The wire is only 24 cm, so he cannot make the hexagon.

### Square
A square with area 36 has side length:
- \(\sqrt{36} = 6\) cm

Perimeter:
- \(4 \times 6 = 24\) cm

So he can make the square exactly.

### Right triangle
The legs are 6 and 8, so the hypotenuse is:
- \(\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)

Perimeter:
- \(6 + 8 + 10 = 24\) cm

So he can make the triangle exactly.

Therefore he can make the square and the triangle only.

ANSWER 7: D

Problem 8:

The reciprocal is largest when the original number is smallest, assuming positive numbers.

Compare:
- \(\frac13 \approx 0.333\)
- \(\frac25 = 0.4\)
- \(1\)
- \(5\)
- \(1986\)

The smallest number is \(\frac13\), so it has the largest reciprocal.

Its reciprocal is 3, which is larger than the reciprocals of all the others.

ANSWER 8: A

Problem 9:

Maximum heart rate:
- \(220 - 26 = 194\)

Target heart rate is 80% of that:
- \(0.8 \times 194 = 155.2\)

To the nearest whole number, that is 155.

ANSWER 9: B

Problem 10:

First, Harry uses the parentheses:
- \(8 - (2 + 5) = 8 - 7 = 1\)

So \(H = 1\).

Terry ignores parentheses:
- \(8 - 2 + 5 = 6 + 5 = 11\)

So \(T = 11\).

Difference:
- \(H - T = 1 - 11 = -10\)

ANSWER 10: A

Problem 11:

We need to evaluate:
- \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

These are 10 odd numbers from 81 to 99.

Instead of adding one by one, use the average:
- average of 81 and 99 is \( \frac{81+99}{2} = 90\)

Since there are 10 numbers, their sum is:
- \(10 \times 90 = 900\)

Now multiply by 2:
- \(2 \times 900 = 1800\)

ANSWER 11: E

Problem 12:

We want the percentage of males surveyed who listen.

From the table:
- Total surveyed = 200
- Females = 96
- So males = \(200 - 96 = 104\)

Males who don’t listen = 26, so males who do listen:
- \(104 - 26 = 78\)

Now find the percent of males who listen:
- \(\frac{78}{104} = \frac{3}{4} = 75\%\)

ANSWER 12: E

Problem 1:  
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?  

- Original yellow tint amount = 30 × 0.30 = 9 liters.  
- Add 5 liters of yellow tint → new yellow amount = 9 + 5 = 14 liters.  
- New total volume = 30 + 5 = 35 liters.  
- New percent yellow = (14 / 35) × 100% = (2/5) × 100% = 40%.  

So the correct choice is 40.  

ANSWER 1: C  

---

Problem 2:  
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?  

- Choose an entry window: 6 choices.  
- Choose a different exit window: 5 choices (cannot be the same as entry).  
- Total ways = 6 × 5 = 30.  

ANSWER 2: D  

---

Problem 3:  
Compute: \( \frac{1}{10} + \frac{2}{20} + \frac{3}{30} \).  

Convert each to a common denominator, 60:  
- \( \frac{1}{10} = \frac{6}{60} \)  
- \( \frac{2}{20} = \frac{2}{20} = \frac{6}{60} \) (since multiply top and bottom by 3)  
- \( \frac{3}{30} = \frac{3}{30} = \frac{6}{60} \) (since multiply top and bottom by 2)  

Add:  
\( \frac{6}{60} + \frac{6}{60} + \frac{6}{60} = \frac{18}{60} = \frac{3}{10} = 0.3 \)

So the value is .3.  

ANSWER 3: D  

---

Problem 4:  
Susan had $50 to spend. She spent $12 on food and twice as much on rides. How much left?  

- Food = 12  
- Rides = twice 12 = 24  
- Total spent = 12 + 24 = 36  
- Money left = 50 − 36 = 14  

ANSWER 4: B  

---

Problem 5:  
Two hundred thousand times two hundred thousand equals ?  

\(200{,}000 \times 200{,}000 = (2\times 10^5)(2\times 10^5) = 4 \times 10^{10}\).  
- \(10^{10} = 10{,}000{,}000{,}000\)  
- \(4 \times 10^{10} = 40{,}000{,}000{,}000 = 40\) billion  

ANSWER 5: E  

---

Problem 6:  
Eleven members each paid the same integer amount for a guest speaker. In all, they paid $1A2 (a 3-digit number). Find missing digit A.  

- Total = \(1A2\) means:  
  \(1A2 = 100 + 10A + 2 = 102 + 10A\).  
- This total is divided equally among 11 people:  
  So \(102 + 10A\) must be divisible by 11.  

Check \(102 + 10A \mod 11\):  
- \(102 \div 11 = 9\) remainder 3, so \(102 \equiv 3 \pmod{11}\).  
- \(10A \equiv -A \pmod{11}\) (since 10 ≡ −1 mod 11).  

So:  
\(102 + 10A \equiv 3 - A \equiv 0 \pmod{11}\)  
Thus \(A \equiv 3 \pmod{11}\).  
Since \(A\) is a digit 0–9, \(A=3\).  

ANSWER 6: D  

---

Problem 7:  
Wire length = 24 cm. Which shapes can be made (using all 24 cm of wire)?  

We need the perimeter (total boundary length) of each shape:

**(1) Regular hexagon with side length 5 cm**  
Perimeter = 6 × 5 = 30 cm → too long. Cannot.

**(2) Square with area 36 cm²**  
Side length = √36 = 6 cm  
Perimeter = 4 × 6 = 24 cm → fits. Can.

**(3) Right triangle with legs 6 and 8 cm**  
Right triangle sides are 6 and 8 legs, and hypotenuse:  
\( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)  
Perimeter = 6 + 8 + 10 = 24 cm → fits. Can.

So the shapes possible: **square and triangle only**.  

ANSWER 7: D  

---

Problem 8:  
Which number has the largest reciprocal? Recall: the reciprocal of \(x\) is \(1/x\).  
We compare \(1/3, 2/5, 1, 1/5, 1/1986\).  

Compute roughly:  
- \(1/3 \approx 0.333\)  
- \(2/5 = 0.4\)  
- \(1 = 1\)  
- \(1/5 = 0.2\)  
- \(1/1986\) is tiny  

Largest is 1 (reciprocal of 1).  

ANSWER 8: C  

---

Problem 9:  
Target heart rate = 80% of (maximum heart rate).  
Maximum heart rate = 220 − age.  
Age = 26.

- Maximum = 220 − 26 = 194  
- Target = 0.80 × 194 = 155.2  
Nearest whole number = 155  

ANSWER 9: B  

---

Problem 10:  
Expression: \(8 - (2 + 5)\)  
- Harry computes correctly:  
  \(2 + 5 = 7\), so \(8 - 7 = 1\).  
- Terry ignores parentheses:  
  \(8 - 2 + 5 = 6 + 5 = 11\).  

Difference \(H - T = 1 - 11 = -10\).  

ANSWER 10: A  

---

Problem 11:  
Compute:  
\(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

Notice these are all odd numbers from 81 to 99 stepping by 2.  
There are 10 terms.

Pair from ends:  
- 81 + 99 = 180  
- 83 + 97 = 180  
- 85 + 95 = 180  
- 87 + 93 = 180  
- 89 + 91 = 180  

There are 5 pairs, each sum 180:  
Sum = 5 × 180 = 900  

Now multiply by 2:  
2 × 900 = 1800  

ANSWER 11: E  

---

Problem 12:  
Total surveyed = 200  
Females = 96  
Females who listen = 58  
Males who don’t listen = 26  
Total listeners = 136  

We want: % of males who listen.

Step 1: Find males total.  
Males = 200 − 96 = 104

Step 2: Find males who don’t listen is given = 26.  
So males who listen = 104 − 26 = 78

Step 3: Convert to percent of males who listen:  
Percent = (78 / 104) × 100%  
Simplify 78/104 = 3/4 (divide by 26):  
So percent = (3/4) × 100% = 75%

ANSWER 12: E

Problem 1:
The original mixture is 30 liters. The amount of yellow tint is 30% of 30 liters: $0.30 \times 30 = 9$ liters.
Five liters of yellow tint are added, so the new amount of yellow tint is $9 + 5 = 14$ liters.
The new total volume of the mixture is $30 + 5 = 35$ liters.
The percentage of yellow tint is $\frac{14}{35} \times 100\%$.
Simplifying the fraction: $\frac{14}{35} = \frac{2}{5} = 0.4$.
$0.4 \times 100\% = 40\%$.
ANSWER 1: C

Problem 2:
Georgie has 6 choices for the window he enters. Since he must leave by a different window, he has $6 - 1 = 5$ choices for the window he leaves by.
The total number of ways is $6 \times 5 = 30$.
ANSWER 2: D

Problem 3:
Simplify each fraction:
$1/10 = 0.1$
$2/20 = 1/10 = 0.1$
$3/30 = 1/10 = 0.1$
Summing them: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides, which is $2 \times 12 = 24$.
Total spent: $12 + 24 = 36$.
Amount left: $50 - 36 = 14$.
ANSWER 4: B

Problem 5:
Two hundred thousand is $200,000$ (or $2 \times 10^5$).
$(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
$4 \times 10^{10} = 40,000,000,000$, which is forty billion.
ANSWER 5: E

Problem 6:
The total amount $1A2$ must be divisible by 11.
A number is divisible by 11 if the alternating sum of its digits is divisible by 11.
For $1A2$, the alternating sum is $1 - A + 2 = 3 - A$.
For this to be 0 (the only multiple of 11 possible for a single digit $A$), $3 - A = 0$, so $A = 3$.
Check: $132 / 11 = 12$. This works.
ANSWER 6: D

Problem 7:
Hexagon: Perimeter of a regular hexagon with side 5 is $6 \times 5 = 30$ cm. (Cannot make)
Square: Area is 36, so side is $\sqrt{36} = 6$. Perimeter is $4 \times 6 = 24$ cm. (Can make)
Triangle: Right triangle with legs 6 and 8. Hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. Perimeter is $6 + 8 + 10 = 24$ cm. (Can make)
ANSWER 7: D

Problem 8:
Reciprocals:
A: $1/(1/3) = 3$
B: $1/(2/5) = 2.5$
C: $1/1 = 1$
D: $1/5 = 0.2$
E: $1/1986 \approx 0.0005$
The largest is 3.
ANSWER 8: A

Problem 9:
Max heart rate = $220 - 26 = 194$.
Target heart rate = $80\%$ of $194 = 0.8 \times 194 = 155.2$.
To the nearest whole number, this is 155.
ANSWER 9: B

Problem 10:
$H = 8 - (2 + 5) = 8 - 7 = 1$.
$T = 8 - 2 + 5 = 6 + 5 = 11$.
$H - T = 1 - 11 = -10$.
ANSWER 10: A

Problem 11:
Sum inside parentheses: $81+83+85+87+89+91+93+95+97+99$.
This is an arithmetic progression with 10 terms.
Sum = $\frac{n}{2}(\text{first} + \text{last}) = \frac{10}{2}(81 + 99) = 5 \times 180 = 900$.
The expression is $2 \times 900 = 1800$.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96.
Males = $200 - 96 = 104$.
Males who don't listen = 26.
Males who listen = $104 - 26 = 78$.
Percentage of males who listen = $(78 / 104) \times 100\%$.
$\frac{78}{104} = \frac{39}{52} = \frac{3}{4} = 0.75 = 75\%$.
ANSWER 12: E

Problem 1: The original mixture has volume 30 L, of which 30% is yellow tint, so there are \(0.3 \times 30 = 9\) L of yellow tint. Adding 5 L of yellow tint produces a new total of 14 L of yellow tint and a new mixture volume of 35 L. The new percentage of yellow tint is therefore \((14/35) \times 100\% = 40\%\).

ANSWER 1: C

Problem 2: There are 6 choices for the entry window. For each of those choices there are then exactly 5 remaining choices for a different exit window, giving a total of \(6 \times 5 = 30\) ordered pairs of distinct windows.

ANSWER 2: D

Problem 3: Rewrite each term with a common denominator of 10:  
\(1/10 = 0.1\),  
\(2/20 = 1/10 = 0.1\),  
\(3/30 = 1/10 = 0.1\).  
Their sum is \(0.1 + 0.1 + 0.1 = 0.3\).

ANSWER 3: D

Problem 4: Twice the food cost is \(2 \times 12 = 24\) dollars spent on rides. The total spent is therefore \(12 + 24 = 36\) dollars, leaving \(50 - 36 = 14\) dollars.

ANSWER 4: B

Problem 5: Write the factors in scientific notation:  
\(200000 = 2 \times 10^5\), so  
\((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\),  
which is forty billion.

ANSWER 5: E

Problem 6: The three-digit total \(1A2\) must be divisible by 11. The alternating-sum divisibility rule requires that \(1 - A + 2 = 3 - A\) be a multiple of 11. The only admissible digit \(A\) (0-9) that satisfies the condition is \(A = 3\), giving the total 132, which is exactly \(11 \times 12\).

ANSWER 6: D

Problem 7: A regular hexagon with side 5 cm has perimeter 30 cm, which exceeds the 24 cm length of wire, so it is impossible. A square of area 36 cm² has side length 6 cm and perimeter 24 cm, which matches the wire length. A right triangle with legs 6 cm and 8 cm has hypotenuse 10 cm, so its perimeter is \(6 + 8 + 10 = 24\) cm, which also matches. Consequently only the square and the triangle are possible.

ANSWER 7: D

Problem 8: For positive numbers the reciprocal function is strictly decreasing, so the largest reciprocal belongs to the smallest number in the list, which is \(1/3\).

ANSWER 8: A

Problem 9: The theoretical maximum heart rate is \(220 - 26 = 194\) beats per minute. Eighty percent of that value is \(0.8 \times 194 = 155.2\), which rounds to the nearest whole number 155.

ANSWER 9: B

Problem 10: Harry evaluates the parentheses first and obtains \(8 - (2 + 5) = 1\). Terry omits the parentheses and obtains \(8 - 2 + 5 = 11\). Their difference is therefore \(1 - 11 = -10\).

ANSWER 10: A

Problem 11: The sum inside the parentheses consists of the ten consecutive odd integers from 81 to 99. Their average is \((81 + 99)/2 = 90\), so the sum equals \(10 \times 90 = 900\). Multiplying by the outer factor of 2 yields \(2 \times 900 = 1800\).

ANSWER 11: E

Problem 12: There are 200 people surveyed and 96 females, so there are \(200 - 96 = 104\) males. Of the 96 females, 58 listen, leaving \(96 - 58 = 38\) females who do not listen. Given that 26 males do not listen, the number of males who listen is \(104 - 26 = 78\). The required percentage is therefore \((78/104) \times 100\% = 75\%\).

ANSWER 12: E

Problem 1:
The problem asks us to find the percentage of yellow tint in a new mixture after adding 5 liters of yellow tint to the original 30 liters of paint mixture. The original mixture is 25% red tint, 30% yellow tint, and 45% water.

First, let's calculate the amount of yellow tint in the original mixture. 30% of 30 liters is $0.3 \times 30 = 9$ liters.

After adding 5 liters of yellow tint, the total amount of yellow tint becomes $9 + 5 = 14$ liters.

The total volume of the new mixture is $30 + 5 = 35$ liters.

To find the percentage of yellow tint in the new mixture, we divide the amount of yellow tint by the total volume and multiply by 100: $\frac{14}{35} \times 100 = 40\%$.

Let's examine the choices:
- A. 25: Too low because the percentage of yellow tint increased.
- B. 35: Not the correct calculation.
- C. 40: Matches our calculation.
- D. 45: Too high.
- E. 50: Too high.

The correct percentage is 40%.

ANSWER 1: C

Problem 2:
The problem asks us to determine the number of ways Georgie the Ghost can enter a haunted house through one of six windows and leave by a different window.

To enter, Georgie has 6 choices. After entering, there are 5 remaining windows to choose from for leaving.

The total number of ways to enter and leave is $6 \times 5 = 30$.

Let's examine the choices:
- A. 12: Too low.
- B. 15: Too low.
- C. 18: Too low.
- D. 30: Matches our calculation.
- E. 36: Too high because it would imply Georgie can leave by the same window.

The correct number of ways is 30.

ANSWER 2: D

Problem 3:
The problem requires us to simplify the expression $1⁄10 + 2⁄20 + 3⁄30$.

First, simplify each fraction:
- $1⁄10$ stays the same.
- $2⁄20 = 1⁄10$.
- $3⁄30 = 1⁄10$.

Adding them together: $1⁄10 + 1⁄10 + 1⁄10 = 3⁄10 = 0.3$.

Let's examine the choices:
- A. .1: Too low.
- B. .123: Not the correct sum.
- C. .2: Too low.
- D. .3: Matches our calculation.
- E. .6: Too high.

The correct sum is 0.3.

ANSWER 3: D

Problem 4:
The problem asks us to find out how many dollars Susan had left after spending money on food and rides at the carnival. She started with $50, spent $12 on food, and twice as much on rides.

First, calculate the amount spent on rides: $12 \times 2 = $24.

Total amount spent is $12 + $24 = $36.

The amount left is $50 - $36 = $14.

Let's examine the choices:
- A. 12: Too low.
- B. 14: Matches our calculation.
- C. 26: Too high.
- D. 38: Too high.
- E. 50: Incorrect because she spent money.

The correct amount left is $14.

ANSWER 4: B

Problem 5:
The problem asks us to calculate $200,000 \times 200,000$.

$200,000 = 2 \times 10^5$, so $200,000 \times 200,000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10} = 40,000,000,000$.

Let's examine the choices:
- A. four hundred thousand: Too low.
- B. four million: Too low.
- C. forty thousand: Too low.
- D. four hundred million: Too low.
- E. forty billion: Matches our calculation.

The correct result is forty billion.

ANSWER 5: E

Problem 6:
The problem requires us to find the missing digit A in the 3-digit number $1A2 that represents the total amount paid by 11 members of the Math Club.

The total amount paid is divisible by 11 since there are 11 members who paid the same amount.

To check divisibility by 11, we use the rule: the difference between the sum of digits at odd places and the sum of digits at even places should be divisible by 11.

For $1A2, the sum of digits at odd places is $1 + 2 = 3$, and the sum at even places is $A$. The difference is $3 - A$.

For $3 - A$ to be divisible by 11 (or 0, since $A$ is a single digit), $A = 3$ makes $3 - A = 0$, which is divisible by 11.

Let's examine the choices:
- A. 0: Not correct because $3 - 0 = 3$, not divisible by 11.
- B. 1: Not correct because $3 - 1 = 2$, not divisible by 11.
- C. 2: Not correct because $3 - 2 = 1$, not divisible by 11.
- D. 3: Correct because $3 - 3 = 0$, which is divisible by 11.
- E. 4: Not correct because $3 - 4 = -1$, not divisible by 11.

The missing digit A is 3, but we should verify if 132 is indeed divisible by 11. $132 / 11 = 12$, so it is.

ANSWER 6: D

Problem 7:
The problem asks us to determine which shapes Haruki can form using a 24 cm long wire.

1. A regular hexagon with side length 5 cm: The perimeter is $6 \times 5 = 30$ cm, which is more than 24 cm. So, Haruki cannot form this hexagon.

2. A square of area 36 cm$^2$: The side length is $\sqrt{36} = 6$ cm. The perimeter is $4 \times 6 = 24$ cm, which matches the wire length. So, Haruki can form this square.

3. A right triangle with legs 6 and 8 cm: Using the Pythagorean theorem, the hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm. The perimeter is $6 + 8 + 10 = 24$ cm, which matches the wire length. So, Haruki can form this triangle.

Let's examine the choices:
- A. Triangle only: Incorrect because Haruki can also form the square.
- B. Hexagon and square only: Incorrect because Haruki cannot form the hexagon.
- C. Hexagon and triangle only: Incorrect because Haruki cannot form the hexagon.
- D. Square and triangle only: Matches our findings.
- E. Hexagon, triangle, and square: Incorrect because Haruki cannot form the hexagon.

The correct shapes Haruki can form are the square and the triangle.

ANSWER 7: D

Problem 8:
The problem asks us to find which of the given numbers has the largest reciprocal.

The reciprocal of a number $x$ is $1/x$. The larger $x$ is, the smaller its reciprocal is, and vice versa.

Let's examine the choices:
- A. $1⁄3$: The reciprocal is 3.
- B. $2⁄5$: The reciprocal is $5/2 = 2.5$.
- C. 1: The reciprocal is 1.
- D. 5: The reciprocal is $1/5 = 0.2$.
- E. 1986: The reciprocal is $1/1986$, very small.

Comparing the reciprocals: $3 > 2.5 > 1 > 0.2 > 1/1986$. So, $1/3$ has the largest reciprocal.

Let's examine the choices again:
- A. $1⁄3$: Correct.
- B. $2⁄5$: Not correct.
- C. 1: Not correct.
- D. 5: Not correct.
- E. 1986: Not correct.

The number with the largest reciprocal is $1/3$.

ANSWER 8: A

Problem 9:
The proble

Problem 1:
Original mixture: 30 L. Yellow tint = 30% of 30 = 0.30 × 30 = 9 L.
Add 5 L yellow: new yellow = 9 + 5 = 14 L, new total volume = 30 + 5 = 35 L.
Percent yellow = (14/35) × 100% = 40%.
ANSWER 1: C

Problem 2:
6 choices for entry window, 5 remaining choices for a different exit window.
Total ways = 6 × 5 = 30.
ANSWER 2: D

Problem 3:
1/10 + 2/20 + 3/30 = 1/10 + 1/10 + 1/10 = 3/10 = 0.3.
ANSWER 3: D

Problem 4:
Spent on food: $12. Spent on rides: 2 × $12 = $24.
Total spent = $12 + $24 = $36. Left = $50 − $36 = $14.
ANSWER 4: B

Problem 5:
200,000 × 200,000 = (2×10⁵) × (2×10⁵) = 4×10¹⁰ = 40,000,000,000, which is forty billion.
ANSWER 5: E

Problem 6:
Total paid is $1A2, a multiple of 11. Divisibility by 11: (1 − A + 2) = 3 − A must be a multiple of 11.
With A a digit 0–9, 3 − A = 0 ⇒ A = 3. (Check: 132 ÷ 11 = 12.)
ANSWER 6: D

Problem 7:
Wire length = 24 cm.
- Regular hexagon, side 5 cm: perimeter = 6×5 = 30 cm > 24 cm → cannot make.
- Square, area 36 cm²: side = 6 cm, perimeter = 4×6 = 24 cm → can make.
- Right triangle, legs 6 and 8 cm: hypotenuse = 10 cm, perimeter = 6+8+10 = 24 cm → can make.
So square and triangle only.
ANSWER 7: D

Problem 8:
Reciprocals: 1/(1/3)=3, 1/(2/5)=2.5, 1/1=1, 1/5=0.2, 1/1986≈0.0005. Largest is 3, from 1/3.
ANSWER 8: A

Problem 9:
Maximum heart rate = 220 − 26 = 194 bpm.
Target = 80% of 194 = 0.8 × 194 = 155.2 → 155 to nearest whole number.
ANSWER 9: B

Problem 10:
Harry: H = 8 − (2+5) = 8 − 7 = 1.
Terry: T = 8 − 2 + 5 = 6 + 5 = 11.
H − T = 1 − 11 = −10.
ANSWER 10: A

Problem 11:
Sum inside parentheses: 10 terms, average = (81+99)/2 = 90, sum = 10×90 = 900.
2 × 900 = 1800.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96 ⇒ Males = 200 − 96 = 104.
Total listeners = 136, female listeners = 58 ⇒ male listeners = 136 − 58 = 78.
Percent of males who listen = (78/104) × 100% = 75%.
ANSWER 12: E

Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint.

**Original mixture (30 liters):**
- Red tint: 25% of 30 = 7.5 liters
- Yellow tint: 30% of 30 = 9 liters
- Water: 45% of 30 = 13.5 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total mixture: 30 + 5 = 35 liters

**New percentage of yellow tint:**
14 / 35 = 2/5 = 0.40 = 40%

Tempting wrong answers: Choice B (35) might come from just adding 5 to the original 30%, but we must recalculate based on the new total.

ANSWER 1: C

---

Problem 2:
Georgie enters through one window and leaves through a *different* window.

- Number of choices for the entry window: 6
- Number of choices for the exit window (must be different): 5
- Total ways: 6 × 5 = 30

Tempting wrong answers: Choice E (36) would be 6 × 6, which allows entering and leaving through the same window. Choice A (12) would be 6 + 6, which doesn't account for all combinations.

ANSWER 2: D

---

Problem 3:
Simplify each fraction first:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

Tempting wrong answers: Choice B (.123) might come from concatenating the digits 1, 2, 3 instead of actually computing. Choice C (.2) might come from miscounting the terms.

ANSWER 3: D

---

Problem 4:
Susan starts with $50.
- Spent on food: $12
- Spent on rides: twice the food amount = 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Amount left: $50 − $36 = $14

Tempting wrong answers: Choice C (26) might come from subtracting only the food ($50 − $24 = $26, forgetting food). Choice D (38) comes from subtracting only $12.

ANSWER 4: B

---

Problem 5:
200,000 × 200,000

Write in scientific notation:
(2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰

4 × 10¹⁰ = 40,000,000,000 = forty billion

Tempting wrong answers: Choice A (four hundred thousand) comes from adding instead of multiplying. Choice D (four hundred million) comes from miscounting zeros.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, so the total $1A2 must be divisible by 11.

**Divisibility rule for 11:** The alternating sum of digits must be a multiple of 11.
Alternating sum: 1 − A + 2 = 3 − A

For 3 − A to be divisible by 11 (and A to be a single digit 0–9):
3 − A = 0 → A = 3

**Check:** $132 ÷ 11 = $12 per member. This is a valid integer amount.

Tempting wrong answers: Other values of A don't make 3 − A a multiple of 11 within the digit range.

ANSWER 6: D

---

Problem 7:
Wire length = 24 cm. We check if each shape's perimeter ≤ 24 cm.

**Regular hexagon (side 5 cm):**
Perimeter = 6 × 5 = 30 cm. Since 30 > 24, he **cannot** make this.

**Square (area 36 cm²):**
Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Since 24 = 24, he **can** make this.

**Right triangle (legs 6 and 8 cm):**
Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Perimeter = 6 + 8 + 10 = 24 cm. Since 24 = 24, he **can** make this.

Haruki can make the square and triangle only.

Tempting wrong answers: Choice E includes the hexagon, but its perimeter exceeds the wire length.

ANSWER 7: D

---

Problem 8:
The reciprocal of a number x is 1/x. The *smaller* the positive number, the *larger* its reciprocal.

| Choice | Number | Reciprocal |
|--------|--------|------------|
| A | 1/3 | 3 |
| B | 2/5 | 5/2 = 2.5 |
| C | 1 | 1 |
| D | 5 | 1/5 = 0.2 |
| E | 1986 | 1/1986 ≈ 0.0005 |

The largest reciprocal is 3, which comes from 1/3.

Tempting wrong answers: Choice E (1986) is the largest number, so it has the *smallest* reciprocal.

ANSWER 8: A

---

Problem 9:
Athlete's age = 26 years.

**Step 1:** Maximum heart rate = 220 − 26 = 194 beats per minute.

**Step 2:** Target heart rate = 80% of 194 = 0.80 × 194 = 155.2

**Step 3:** Round to nearest whole number: 155

Tempting wrong answers: Choice C (176) might come from taking 80% of 220 without subtracting the age. Choice D (194) is the maximum heart rate itself.

ANSWER 9: B

---

Problem 10:
**Harry (correct calculation):**
H = 8 − (2 + 5) = 8 − 7 = 1

**Terry (ignores parentheses):**
T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:**
H − T = 1 − 11 = −10

Tempting wrong answers: Choice E (10) is |H − T|, but the question asks for H − T specifically, which is negative since Terry's answer is larger.

ANSWER 10: A

---

Problem 11:
We need: 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

The sum inside is an arithmetic sequence:
- First term: 81, Last term: 99
- Number of terms: 10
- Sum = (number of terms) × (first + last) / 2 = 10 × (81 + 99) / 2 = 10 × 180 / 2 = 10 × 90 = 900

Multiply by 2: 2 × 900 = 1800

Tempting wrong answers: Choice C (1700) might come from miscounting terms or miscalculating the average.

ANSWER 11: E

---

Problem 12:
Let me organize the data from the survey:

- Total surveyed: 200
- Females: 96 → Males: 200 − 96 = 104
- Total listeners: 136
- Females who listen: 58 → Males who listen: 136 − 58 = 78
- Males who don't listen: 26 (Check: 78 + 26 = 104 ✓)

**Percentage of males who listen:**
(78 / 104) × 100% = (3/4) × 100% = 75%

Tempting wrong answers: Choice C (52) might come from computing 104/200 (percentage of total who are male). Choice D (55) might come from 78/136 (percentage of listeners who are male).

ANSWER 12: E

 Problem 1:
We start with 30 liters of paint. The original amount of yellow tint is 30% of 30 liters:
\[
0.30 \times 30 = 9 \text{ liters.}
\]
Five liters of yellow tint are added, so the new amount of yellow tint is:
\[
9 + 5 = 14 \text{ liters.}
\]
The new total volume of the mixture is:
\[
30 + 5 = 35 \text{ liters.}
\]
The percent of yellow tint in the new mixture is:
\[
\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%.
\]
(Choice A is the original red percentage, D is the original water percentage, and E would require 17.5 liters of yellow, which we do not have.)
ANSWER 1: C

Problem 2:
Georgie chooses one of 6 windows to enter. Since he must leave through a different window, there are 5 remaining choices for his exit. By the multiplication principle:
\[
6 \times 5 = 30 \text{ ways.}
\]
(Choice E, 36, would count entering and leaving through the same window; the other choices are too small.)
ANSWER 2: D

Problem 3:
Simplify each fraction:
\[
\frac{2}{20} = \frac{1}{10}, \qquad \frac{3}{30} = \frac{1}{10}.
\]
So the sum is:
\[
\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3.
\]
(Choice A is just one term; C is 0.2; E is 0.6.)
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides:
\[
2 \times 12 = 24 \text{ dollars.}
\]
Total spent:
\[
12 + 24 = 36 \text{ dollars.}
\]
Amount left:
\[
50 - 36 = 14 \text{ dollars.}
\]
(Choice C is the total spent; D is the amount left if she had spent only $12.)
ANSWER 4: B

Problem 5:
\[
200{,}000 \times 200{,}000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}.
\]
\[
4 \times 10^{10} = 40{,}000{,}000{,}000,
\]
which is read as forty billion.
(Choice D is $4 \times 10^8$; B is $4 \times 10^6$.)
ANSWER 5: E

Problem 6:
The total amount paid is the three-digit number $1A2$, which equals:
\[
100 + 10A + 2 = 102 + 10A.
\]
Since 11 members each paid the same integer amount, $102 + 10A$ must be divisible by 11. Testing the choices:
- $A=0$: $102 \div 11 \approx 9.27$ (not an integer)
- $A=1$: $112 \div 11 \approx 10.18$ (not an integer)
- $A=2$: $122 \div 11 \approx 11.09$ (not an integer)
- $A=3$: $132 \div 11 = 12$ (integer!)
- $A=4$: $142 \div 11 \approx 12.9$ (not an integer)
Thus the missing digit is 3.
ANSWER 6: D

Problem 7:
The wire is 24 cm long, so a shape is possible only if its perimeter is at most 24 cm.
- **Regular hexagon, side 5 cm:** Perimeter $= 6 \times 5 = 30$ cm. $30 > 24$, so impossible.
- **Square of area 36 cm²:** Side $= \sqrt{36} = 6$ cm. Perimeter $= 4 \times 6 = 24$ cm. Exactly fits, so possible.
- **Right triangle with legs 6 and 8 cm:** Hypotenuse $= \sqrt{6^2 + 8^2} = \sqrt{100} = 10$ cm. Perimeter $= 6 + 8 + 10 = 24$ cm. Exactly fits, so possible.
Only the square and the triangle can be formed.
ANSWER 7: D

Problem 8:
For positive numbers, the reciprocal function $f(x) = \frac{1}{x}$ is decreasing, so the smallest number has the largest reciprocal. Compare the two smallest choices:
\[
\frac{1}{3} \approx 0.333\ldots, \qquad \frac{2}{5} = 0.4.
\]
Since $\frac{1}{3} < \frac{2}{5} < 1 < 5 < 1986$, the smallest number is $\frac{1}{3}$. Its reciprocal is $3$, which is larger than the reciprocals of all the other choices.
ANSWER 8: A

Problem 9:
First find the theoretical maximum heart rate:
\[
220 - 26 = 194 \text{ beats per minute.}
\]
The target rate is 80% of this:
\[
0.80 \times 194 = 155.2.
\]
Rounded to the nearest whole number, this is 155.
(Choice C is the maximum heart rate itself.)
ANSWER 9: B

Problem 10:
Harry computes correctly:
\[
H = 8 - (2 + 5) = 8 - 7 = 1.
\]
Terry ignores the parentheses:
\[
T = 8 - 2 + 5 = 6 + 5 = 11.
\]
The difference is:
\[
H - T = 1 - 11 = -10.
\]
ANSWER 10: A

Problem 11:
The ten numbers inside the parentheses form an arithmetic sequence: 81, 83, …, 99. The sum of an arithmetic sequence is:
\[
\frac{\text{number of terms}}{2} \times (\text{first} + \text{last}) = \frac{10}{2} \times (81 + 99) = 5 \times 180 = 900.
\]
Now multiply by the factor of 2 outside:
\[
2 \times 900 = 1800.
\]
(One can also pair them: $(81+99)=180$, $(83+97)=180$, etc., giving five pairs that each sum to 180.)
ANSWER 11: E

Problem 12:
Total surveyed = 200, and females surveyed = 96, so:
\[
\text{Males surveyed} = 200 - 96 = 104.
\]
Total listeners = 136, and females who listen = 58, so:
\[
\text{Males who listen} = 136 - 58 = 78.
\]
(We can check: total non-listeners = 64, males who don't listen = 26, so females who don't listen = 38; $58+38=96$ and $78+26=104$, which checks out.)

The percentage of males who listen is:
\[
\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%.
\]
ANSWER 12: E

Problem 1:
A mixture of 30 liters is 25% red, 30% yellow, and 45% water. Five liters of yellow tint are added. What is the percent of yellow tint in the new mixture?

**Setup:** Find the amount of yellow tint originally, add 5 liters, then find the new percentage.

**Original yellow tint:** 30% of 30 = 0.30 × 30 = 9 liters

**After adding 5 liters of yellow tint:**
- New yellow tint: 9 + 5 = 14 liters
- New total volume: 30 + 5 = 35 liters
- New percentage: 14/35 = 2/5 = 40%

**Checking wrong choices:** 25% would be if we just added the original percentages; 35% would be if we just added 5% to 30%; 45% and 50% are too high.

ANSWER 1: C

---

Problem 2:
Six windows. Georgie enters by one and leaves by a different one. How many ways?

**Setup:** Choose an entry window (6 choices), then choose a different exit window (5 choices).

**Calculation:** 6 × 5 = 30 ways

**Checking wrong choices:** 15 would be if we divided by 2 (treating order as irrelevant); 12 would be 6×2; 18 would be 6×3; 36 would be 6×6 (allowing same window).

ANSWER 2: D

---

Problem 3:
1/10 + 2/20 + 3/30 = ?

**Simplify each fraction:**
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

**Sum:** 1/10 + 1/10 + 1/10 = 3/10 = 0.3

**Checking wrong choices:** 0.1 would be just one term; 0.2 would be two terms; 0.6 would be adding numerators and denominators incorrectly.

ANSWER 3: D

---

Problem 4:
Susan had $50. She spent $12 on food and twice as much on rides. How much left?

**Setup:** Food = $12, Rides = 2 × $12 = $24

**Total spent:** $12 + $24 = $36

**Amount left:** $50 - $36 = $14

**Checking wrong choices:** 12 is just the food amount; 26 is 50-24 (forgetting food); 38 is 50-12 (forgetting rides).

ANSWER 4: B

---

Problem 5:
200,000 × 200,000 = ?

**Setup:** Express in scientific notation: 200,000 = 2 × 10⁵

**Calculation:** (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion

**Checking wrong choices:** Four hundred thousand is 4×10⁵; four million is 4×10⁶; forty thousand is 4×10⁴; four hundred million is 4×10⁸.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, totaling $1A2. Find the missing digit A.

**Setup:** The total $1A2 must be divisible by 11.

**Divisibility test for 11:** A number is divisible by 11 if the alternating sum of its digits is divisible by 11.

For 1A2: (1 + 2) - A = 3 - A must be divisible by 11.

Since A is a single digit (0-9), 3 - A can only equal 0 (the only multiple of 11 in range -6 to 3).

So A = 3, giving $132 = 11 × $12 ✓

**Checking:** With A=0: $102 ÷ 11 ≈ 9.27 (not integer); A=1: $112 ÷ 11 ≈ 10.18 (not integer); A=3: $132 ÷ 11 = 12 ✓

ANSWER 6: D

---

Problem 7:
Wire is 24 cm. Which shapes can be made?

**Regular hexagon with side 5 cm:** Perimeter = 6 × 5 = 30 cm. Need 30 cm, only have 24 cm. ❌

**Square with area 36 cm²:** Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Need 24 cm, have 24 cm. ✓

**Right triangle with legs 6 and 8 cm:** Hypotenuse = √(36 + 64) = √100 = 10 cm. Perimeter = 6 + 8 + 10 = 24 cm. Need 24 cm, have 24 cm. ✓

**Conclusion:** Can make the square and triangle, but not the hexagon.

ANSWER 7: D

---

Problem 8:
Which number has the largest reciprocal?

**Find each reciprocal:**
- A. 1/3 → reciprocal = 3
- B. 2/5 → reciprocal = 5/2 = 2.5
- C. 1 → reciprocal = 1
- D. 5 → reciprocal = 1/5 = 0.2
- E. 1986 → reciprocal ≈ 0.0005

**Key insight:** Smaller positive numbers have larger reciprocals. The smallest number listed is 1/3, so it has the largest reciprocal (3).

ANSWER 8: A

---

Problem 9:
Target heart rate = 80% of (220 - age). Find target for 26-year-old.

**Maximum heart rate:** 220 - 26 = 194 bpm

**Target heart rate:** 80% of 194 = 0.80 × 194 = 155.2 ≈ 155 bpm

**Checking wrong choices:** 134 would be about 69%; 176 would be about 91%; 194 is the max (100%); 243 is impossible.

ANSWER 9: B

---

Problem 10:
Harry calculates 8 − (2 + 5) correctly. Terry calculates 8 − 2 + 5. Find H − T.

**Harry's answer:** H = 8 − (2 + 5) = 8 − 7 = 1

**Terry's answer:** T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:** H − T = 1 − 11 = −10

**Checking wrong choices:** −6 would be if Terry got 7; 0 would be if they got the same answer; 6 or 10 would be T − H.

ANSWER 10: A

---

Problem 11:
2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99) = ?

**Setup:** This is 10 consecutive odd numbers. The average of the first and last is (81 + 99)/2 = 90.

**Sum of the 10 numbers:** 10 × 90 = 900

**Multiply by 2:** 2 × 900 = 1800

**Alternative verification:** Pair the numbers: (81+99) + (83+97) + (85+95) + (87+93) + (89+91) = 180 + 180 + 180 + 180 + 180 = 900. Then 2 × 900 = 1800.

ANSWER 11: E

---

Problem 12:
Find the percentage of males who listen to the station.

**Given information:**
- Total surveyed: 200
- Females: 96
- Females who listen: 58
- Males who don't listen: 26
- Total listeners: 136
- Total non-listeners: 64

**Find males:** Males = 200 - 96 = 104

**Find males who listen:** Males who listen = Total listeners - Females who listen = 136 - 58 = 78

**Verification:** Males who don't listen = 104 - 78 = 26 ✓ (matches given)

**Percentage of males who listen:** 78/104 × 100 = 75%

**Checking wrong choices:** 39% would be 78/200 (using total instead of males); 48% would be about 50/104; 52% would be about 54/104; 55% would be about 57/104.

ANSWER 12: E

**Problem 1**

The original mixture is 30 L with 30 % yellow tint → yellow = 0.30·30 = 9 L.  
Add 5 L of yellow → total yellow = 9 + 5 = 14 L.  
Total volume after addition = 30 + 5 = 35 L.  

Percent yellow = (14/35)·100 % = 40 %.

**Answer 1: C**

---

**Problem 2**

Choose an entry window (6 ways) and a different exit window (5 ways).  
Number of ways = 6 × 5 = 30.

**Answer 2: D**

---

**Problem 3**

\[
\frac{1}{10}=0.1,\qquad\frac{2}{20}=0.1,\qquad\frac{3}{30}=0.1
\]
Sum = 0.1 + 0.1 + 0.1 = 0.3.

**Answer 3: D**

---

**Problem 4**

Food cost = \$12.  
Rides cost = twice food = 2·12 = \$24.  
Total spent = 12 + 24 = \$36.  
Money left = 50 − 36 = \$14.

**Answer 4: B**

---

**Problem 5**

\(200{,}000 \times 200{,}000 = (2\times10^{5})^{2}=4\times10^{10}=40{,}000{,}000{,}000\).

That is 40 billion.

**Answer 5: E**

---

**Problem 6**

Total paid = \$1A2 = 100 + 10A + 2 = 102 + 10A.  
Since the 11 members each paid an integer amount, 102 + 10A must be divisible by 11.

\(102\equiv3\pmod{11}\).  
We need \(10A\equiv-3\equiv8\pmod{11}\).  
Because \(10\equiv-1\pmod{11}\), this is \(-A\equiv8\pmod{11}\) → \(A\equiv3\pmod{11}\).  
A digit, so \(A=3\).

**Answer 6: D**

---

**Problem 7**

- Hexagon: perimeter = 6·5 = 30 cm > 24 cm → impossible.  
- Square: side = √36 = 6 cm, perimeter = 4·6 = 24 cm → possible.  
- Right triangle: legs 6 cm, 8 cm, hypotenuse = √(6²+8²)=10 cm, perimeter = 6+8+10 = 24 cm → possible.

Only the square and triangle can be made.

**Answer 7: D**

---

**Problem 8**

Reciprocals:
- \(1/3 → 3\)  
- \(2/5 → 5/2 = 2.5\)  
- \(1 → 1\)  
- \(5 → 1/5 = 0.2\)  
- \(1986 → 1/1986 ≈ 0.0005\)

Largest reciprocal is 3, from \(1/3\).

**Answer 8: A**

---

**Problem 9**

Maximum heart rate = 220 − 26 = 194 bpm.  
Target = 80 % of 194 = 0.8·194 = 155.2 → rounded to 155 bpm.

**Answer 9: B**

---

**Problem 10**

Harry: \(8-(2+5)=8-7=1\).  
Terry (ignoring parentheses): \(8-2+5 = 6+5 = 11\).  
Difference \(H-T = 1-11 = -10\).

**Answer 10: A**

---

**Problem 11**

Sum of the ten terms: average = (81+99)/2 = 90, so sum = 90·10 = 900.  
Multiply by 2: \(2·900 = 1800\).

**Answer 11: E**

---

**Problem 12**

Total surveyed = 200.  
Females = 96 → Males = 200 − 96 = 104.  

Female listeners = 58 → Male listeners = 136 − 58 = 78.  

Percent of males who listen = (78/104)·100 % = (39/52)·100 % = 75 %.

**Answer 12: E**

---

**Final Answers**

ANSWER 1: C  
ANSWER 2: D  
ANSWER 3: D  
ANSWER 4: B  
ANSWER 5: E  
ANSWER 6: D  
ANSWER 7: D  
ANSWER 8: A  
ANSWER 9: B  
ANSWER 10: A  
ANSWER 11: E  
ANSWER 12: E

Problem 1:  
A 30-liter paint mixture is 25% red tint, 30% yellow tint, and 45% water. Five liters of yellow tint are added.  
Original yellow tint: $0.30 \times 30 = 9$ liters.  
After adding 5 liters: $9 + 5 = 14$ liters yellow.  
New total volume: $30 + 5 = 35$ liters.  
Percent yellow: $\frac{14}{35} = 0.40 = 40\%$.  
Tempting wrong choices: 35% (original yellow percent) or 45% (water percent).  
ANSWER 1: D  

Problem 2:  
Georgie enters through one of six windows and exits through a different one.  
Entry choices: 6. Exit choices: 5 (excluding entry window).  
Total ways: $6 \times 5 = 30$.  
Tempting wrong choices: 15 (if order didn’t matter) or 36 (if same window allowed).  
ANSWER 2: D  

Problem 3:  
$\frac{1}{10} + \frac{2}{20} + \frac{3}{30} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3$.  
Tempting wrong choices: 0.1 (one term) or 0.2 (two terms).  
ANSWER 3: D  

Problem 4:  
Susan starts with \$50.  
Food: \$12.  
Rides: $2 \times 12 = \$24$.  
Total spent: $12 + 24 = \$36$.  
Left: $50 - 36 = \$14$.  
Tempting wrong choices: \$26 (if spent \$24 total) or \$38 (if spent \$12 total).  
ANSWER 4: B  

Problem 5:  
$200,000 \times 200,000 = 40,000,000,000 = 40$ billion.  
Tempting wrong choices: 400,000 (added zeros) or 4 million (multiplied digits).  
ANSWER 5: E  

Problem 6:  
Total paid: \$1A2, an integer divisible by 11.  
$1A2 \mod 11 = 0$.  
$1 + 2 - A \equiv 0 \mod 11 \Rightarrow 3 - A \equiv 0 \mod 11 \Rightarrow A = 3$.  
Tempting wrong choices: 0, 1, 2, or 4 (incorrect divisibility).  
ANSWER 6: E (Wait, let's recheck: 1A2 must be divisible by 11. 132 ÷ 11 = 12, so A=3. But 132 is 11×12, so correct. However, the answer choices are A=0,B=1,C=2,D=3,E=4. So D is 3. So correct answer is D, not E. I made a mistake in the initial answer. Correcting:)  
**Correction**: $1A2 = 132$ when $A=3$, and $132 \div 11 = 12$, so $A=3$.  
ANSWER 6: D  

Problem 7:  
Wire length: 24 cm.  
- Regular hexagon with side 5 cm: Perimeter $6 \times 5 = 30$ cm (not 24).  
- Square with area 36 cm²: Side $6$ cm, perimeter $24$ cm.  
- Right triangle with legs 6 and 8 cm: Hypotenuse 10 cm, perimeter $6+8+10=24$ cm.  
Can make square and triangle.  
Tempting wrong choices: Hexagon (incorrect perimeter).  
ANSWER 7: D  

Problem 8:  
Largest reciprocal means smallest original number.  
$\frac{1}{3} \approx 0.333$, $\frac{2}{5} = 0.4$, $1$, $5$, $1986$.  
Smallest: $\frac{1}{3}$. Reciprocal: 3.  
Others: 2.5, 1, 0.2, 0.0005.  
Tempting wrong choices: $\frac{2}{5}$ (larger than $\frac{1}{3}$).  
ANSWER 8: A  

Problem 9:  
Maximum heart rate: $220 - 26 = 194$.  
Target: $0.80 \times 194 = 155.2 \approx 155$.  
Tempting wrong choices: 134 (70%), 176 (90%), or 243 (incorrect max).  
ANSWER 9: B  

Problem 10:  
Harry: $8 - (2 + 5) = 1$.  
Terry: $8 - 2 + 5 = 11$.  
Difference: $1 - 11 = -10$.  
Tempting wrong choices: 10 (absolute difference) or 0 (if parentheses ignored by both).  
ANSWER 10: A  

Problem 11:  
Sum: $81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99$.  
Average: 90, 10 terms: $90 \times 10 = 900$.  
$2 \times 900 = 1800$.  
Tempting wrong choices: 1600 (min+max=180, times 5=900, times 2=1800, but if miscount terms).  
ANSWER 11: E  

Problem 12:  
Total surveyed: 200.  
Females: 96, so males: $200 - 96 = 104$.  
Total listeners: 136.  
Female listeners: 58, so male listeners: $136 - 58 = 78$.  
Percent male listeners: $\frac{78}{104} = 0.75 = 75\%$.  
Tempting wrong choices: 39 (male non-listeners percent) or 52 (incorrect calculation).  
ANSWER 12: E

### Problem 1:
What is the percentage of yellow tint in the new 35-liter paint mixture, after adding 5 liters of yellow tint to an original 30-liter mixture that was 30% yellow?
First, calculate the original volume of yellow tint: 30% of 30 liters = \(0.3 \times 30 = 9\) liters. Add the 5 new liters of yellow tint to get \(9 + 5 = 14\) liters of yellow in the new mixture. The total new volume of the mixture is \(30 + 5 = 35\) liters. The percentage of yellow is \(\frac{14}{35} = 0.4 = 40\%\). Tempting wrong answers come from forgetting to update the total mixture volume when calculating the new percentage, or miscalculating the original yellow volume.
ANSWER 1: C

---

### Problem 2:
How many ways can Georgie enter through one window and leave through a different window, out of 6 total windows?
Georgie has 6 choices for an entry window. Since he must leave through a different window, he has 5 remaining choices for an exit window. The total number of ways is \(6 \times 5 = 30\). A common wrong answer is 15, which incorrectly treats entry/exit as unordered pairs, when order matters (entering window A and leaving B is different from entering B and leaving A).
ANSWER 2: D

---

### Problem 3:
Simplify \(\frac{1}{10} + \frac{2}{20} + \frac{3}{30}\):
Simplify each fraction first: \(\frac{2}{20} = \frac{1}{10}\) and \(\frac{3}{30} = \frac{1}{10}\). Add them together: \(\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3\). A common wrong answer is 0.1, which incorrectly adds numerators and denominators directly instead of simplifying first.
ANSWER 3: D

---

### Problem 4:
How much money does Susan have left after spending $12 on food and twice that amount on rides, starting with $50?
First calculate the amount spent on rides: twice $12 is \(2 \times 12 = 24\) dollars. Total spent is \(12 + 24 = 36\) dollars. The amount left is \(50 - 36 = 14\) dollars. A common wrong answer is 26, which incorrectly uses $12 instead of $24 for ride costs.
ANSWER 4: B

---

### Problem 5:
Calculate \(200,000 \times 200,000\):
Rewrite 200,000 as \(2 \times 10^5\). Multiplying two copies gives \((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\), which equals 40,000,000,000, or forty billion. Wrong answers come from miscounting the number of zeros in the product.
ANSWER 5: E

---

### Problem 6:
The 3-digit total cost \(1A2\) dollars is divisible by 11 (11 members paid the same integer amount). What is the missing digit A?
The divisibility rule for 11 states that alternating sum of digits must be divisible by 11. For 1A2, the alternating sum is \(1 - A + 2 = 3 - A\). The only value of A (0-9) that makes this sum divisible by 11 is A=3, since \(3-3=0\), which is divisible by 11. Verify: \(132 \div 11 = 12\), a valid integer.
ANSWER 6: D

---

### Problem 7:
Haruki has a 24cm wire. Which of the three shapes can he form? Calculate the perimeter of each shape:
1. Regular hexagon with side 5cm: perimeter = \(6 \times 5 = 30\)cm, which is longer than 24cm, so impossible.
2. Square with area 36cm²: side length = \(\sqrt{36}=6\)cm, perimeter = \(4 \times 6 = 24\)cm, which fits.
3. Right triangle with legs 6cm and 8cm: hypotenuse = 10cm (6-8-10 Pythagorean triple), perimeter = \(6+8+10=24\)cm, which fits.
Only the square and triangle can be formed.
ANSWER 7: D

---

### Problem 8:
Which number has the largest reciprocal? The reciprocal of \(x\) is \(\frac{1}{x}\), so smaller values of \(x\) produce larger reciprocals. List the values of all options:
A. \(\frac{1}{3} \approx 0.333\), B. \(\frac{2}{5}=0.4\), C.1, D.5, E.1986. The smallest value is \(\frac{1}{3}\), whose reciprocal is 3, the largest of all options.
ANSWER 8: A

---

### Problem 9:
Calculate the target heart rate for a 26-year-old athlete: target is 80% of max heart rate, where max = 220 - age.
First calculate max heart rate: \(220 - 26 = 194\) bpm. 80% of 194 is \(0.8 \times 194 = 155.2\), which rounds to 155 bpm. A common wrong answer is 176, which uses 220 instead of the correct max heart rate of 194.
ANSWER 9: B

---

### Problem 10:
Calculate \(H-T\), where H is Harry's correct answer to \(8-(2+5)\), and T is Terry's incorrect answer of \(8-2+5\).
Harry's answer: \(8 - 7 = 1 = H\). Terry's answer: \(6 + 5 = 11 = T\). The difference \(H-T = 1 - 11 = -10\). A common wrong answer is 10, which calculates \(T-H\) instead of \(H-T\).
ANSWER 10: A

---

### Problem 11:
Simplify \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\):
Pair the terms inside the parentheses to simplify addition: \((81+99)+(83+97)+(85+95)+(87+93)+(89+91) = 5 \times 180 = 900\). Multiply by 2: \(2 \times 900 = 1800\).
ANSWER 11: E

---

### Problem 12:
What percentage of males surveyed listen to the radio station? First calculate total number of males: total surveyed = 200, females = 96, so males = \(200 - 96 = 104\). Males who do not listen = 26, so males who listen = \(104 - 26 = 78\). The percentage of males who listen is \(\frac{78}{104} = 0.75 = 75\%\). A common wrong answer is 39%, which divides male listeners by the total surveyed population instead of the total male population.
ANSWER 12: E

### Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.
1.  Calculate original yellow tint volume: 25%? No wait, original is 30% yellow. So 30% of 30 L = $0.3 \times 30 = 9$ liters of yellow.
2.  Add 5 liters of yellow: total yellow = $9 + 5 = 14$ liters.
3.  Total new mixture volume: $30 + 5 = 35$ liters.
4.  Percent yellow: $\frac{14}{35} \times 100 = 40\%$.
ANSWER 1: C
---
### Problem 2:
We count the number of ways to choose an entry window and a different exit window from 6 windows.
1.  Entry window: 6 possible choices.
2.  Exit window: cannot be the entry window, so 5 remaining choices.
3.  Total ways: $6 \times 5 = 30$ (order matters because entry and exit are distinct actions).
ANSWER 2: D
---
### Problem 3:
Simplify and add the fractions:
1.  $\frac{1}{10} = 0.1$, $\frac{2}{20} = \frac{1}{10} = 0.1$, $\frac{3}{30} = \frac{1}{10} = 0.1$.
2.  Sum: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D
---
### Problem 4:
Calculate remaining money after expenses:
1.  Total initial money: $50.
2.  Money spent on rides: twice the food cost = $2 \times 12 = 24$ dollars.
3.  Total spent: $12 + 24 = 36$ dollars.
4.  Remaining: $50 - 36 = 14$ dollars.
ANSWER 4: B
---
### Problem 5:
Compute $200,000 \times 200,000$:
1.  Rewrite as $(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
2.  $10^{10} = 10,000,000,000$, so $4 \times 10^{10} = 40,000,000,000 = 40$ billion.
ANSWER 5: E
---
### Problem 6:
The total payment $1A2$ (a 3-digit number) must be divisible by 11 (since 11 people paid equal integer amounts).
1.  Use the divisibility rule for 11: the alternating sum of digits is divisible by 11. For $1A2$, this is $1 - A + 2 = 3 - A$.
2.  $A$ is a digit (0-9), so $3 - A$ must equal 0 (the only multiple of 11 in range: $3 - A = 11$ gives $A=-8$, $3-A=-11$ gives $A=14$, both invalid).
3.  $3 - A = 0 \implies A=3$. Verify: $132 \div 11 = 12$, which is an integer.
ANSWER 6: D
---
### Problem 7:
Check if the 24 cm wire can form each shape (perimeter of the shape must equal 24 cm):
1.  Regular hexagon (side 5 cm): Perimeter = $6 \times 5 = 30$ cm > 24 cm → cannot make.
2.  Square (area 36 cm²): Side length = $\sqrt{36} = 6$ cm. Perimeter = $4 \times 6 = 24$ cm → can make.
3.  Right triangle (legs 6, 8 cm): Hypotenuse = $\sqrt{6^2 + 8^2} = 10$ cm. Perimeter = $6 + 8 + 10 = 24$ cm → can make.
Only square and triangle can be made.
ANSWER 7: D
---
### Problem 8:
The reciprocal of a positive number $x$ is $\frac{1}{x}$; smaller $x$ gives larger reciprocal.
1.  List values: $\frac{1}{3} \approx 0.33$, $\frac{2}{5}=0.4$, $1$, $5$, $1986$.
2.  Smallest value is $\frac{1}{3}$, whose reciprocal is $3$ (largest among all reciprocals: $\frac{5}{2}=2.5$, $1$, $0.2$, $\frac{1}{1986} \approx 0.0005$).
ANSWER 8: A
---
### Problem 9:
Calculate target heart rate for a 26-year-old:
1.  Maximum heart rate: $220 - 26 = 194$ bpm.
2.  Target heart rate: 80% of 194 = $0.8 \times 194 = 155.2$, which rounds to 155.
ANSWER 9: B
---
### Problem 10:
Compute $H$ (Harry's correct answer) and $T$ (Terry's incorrect answer):
1.  Harry: $8 - (2 + 5) = 8 - 7 = 1$, so $H=1$.
2.  Terry (ignores parentheses, left-to-right): $8 - 2 + 5 = 6 + 5 = 11$, so $T=11$.
3.  Difference: $H - T = 1 - 11 = -10$.
ANSWER 10: A
---
### Problem 11:
Calculate $2 \times (81 + 83 + ... + 99)$:
1.  The sum inside is 10 consecutive odd numbers. Average of first and last term: $\frac{81 + 99}{2} = 90$.
2.  Sum of the 10 terms: $90 \times 10 = 900$.
3.  Multiply by 2: $2 \times 900 = 1800$.
ANSWER 11: E
---
### Problem 12:
Find the percentage of males who listen to the station:
1.  Total males surveyed: $200 - 96 = 104$.
2.  Males who do not listen: 26, so males who listen: $104 - 26 = 78$.
3.  Verify with total listeners: Total listeners = 136, female listeners = 58, so male listeners = $136 - 58 = 78$ (matches).
4.  Percentage: $\frac{78}{104} \times 100 = 75\%$.
ANSWER 12: E

Problem 1:
Original yellow = 30% of 30 = 9 L. Add 5 → 14 L yellow. New total = 35 L. Percent = 14/35 = 40%.

ANSWER 1: C

Problem 2:
Enter: 6 choices, leave: 5 different choices. 6×5 = 30.

ANSWER 2: D

Problem 3:
Each fraction = 0.1. Sum = 0.3.

ANSWER 3: D

Problem 4:
Food 12, rides twice = 24. Spent 36. Left = 50 − 36 = 14.

ANSWER 4: B

Problem 5:
200,000 × 200,000 = 40,000,000,000 = forty billion.

ANSWER 5: E

Problem 6:
1A2 divisible by 11. Alternating sum: 1 − A + 2 = 3 − A must be divisible by 11 → A = 3 (gives 0). Check 132/11 = 12. Yes.

ANSWER 6: D

Problem 7:
Hexagon: 6×5 = 30 > 24, no. Square area 36 → side 6, perimeter 24, yes. Triangle legs 6,8 → hypotenuse 10, perimeter 24, yes. Square and triangle only.

ANSWER 7: D

Problem 8:
Largest reciprocal = smallest number. Smallest is 1/3.

ANSWER 8: A

Problem 9:
Max = 220 − 26 = 194. Target = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

Problem 10:
H = 8 − 7 = 1. T = 8 − 2 + 5 = 11. H − T = 1 − 11 = −10.

ANSWER 10: A

Problem 11:
Sum of 81..99 odd numbers (10 terms): average (81+99)/2 = 90, ×10 = 900. ×2 = 1800.

ANSWER 11: E

Problem 12:
Males total = 200 − 96 = 104. Males who don't listen = 26 → males who listen = 104 − 26 = 78. Percent = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem systematically.

---

**Problem 1:**
Original mixture: 30 liters, 30% yellow = 9 liters yellow. Add 5 liters yellow → new yellow = 14 liters, new total = 35 liters.
Percent yellow = 14/35 = 40%.

ANSWER 1: C

---

**Problem 2:**
Georgie can enter through any of 6 windows, then leave through any of the remaining 5 windows.
6 × 5 = 30.

ANSWER 2: D

---

**Problem 3:**
1/10 = 0.1, 2/20 = 0.1, 3/30 = 0.1. Sum = 0.3.

ANSWER 3: D

---

**Problem 4:**
Food: $12. Rides: 2 × $12 = $24. Total spent: $12 + $24 = $36. Left: $50 − $36 = $14.

ANSWER 4: B

---

**Problem 5:**
200,000 × 200,000 = (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion.

ANSWER 5: E

---

**Problem 6:**
The number 1A2 must be divisible by 11. 1A2 ÷ 11 must be an integer.
11 × 10 = 110, 11 × 11 = 121 (no), 11 × 12 = 132 ✓ → A = 3.
Check: 132/11 = 12 ✓, each paid $12.

ANSWER 6: D

---

**Problem 7:**
Wire = 24 cm.
- Regular hexagon, side 5: perimeter = 6 × 5 = 30 cm > 24. ✗
- Square, area 36: side = 6, perimeter = 24 cm ✓
- Right triangle, legs 6 and 8: hypotenuse = 10, perimeter = 6+8+10 = 24 cm ✓

ANSWER 7: D

---

**Problem 8:**
The reciprocal of x is 1/x. The largest reciprocal comes from the smallest positive number.
Among 1/3, 2/5, 1, 5, 1986 — the smallest is 1/3, and its reciprocal is 3, which is the largest.

ANSWER 8: A

---

**Problem 9:**
Maximum heart rate = 220 − 26 = 194. Target = 80% × 194 = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

---

**Problem 10:**
Harry: 8 − (2 + 5) = 8 − 7 = 1, so H = 1.
Terry: 8 − 2 + 5 = 11, so T = 11.
H − T = 1 − 11 = −10.

ANSWER 10: A

---

**Problem 11:**
The ten odd numbers from 81 to 99: sum = (81+99)/2 × 10 = 90 × 10 = 900.
2 × 900 = 1800.

ANSWER 11: E

---

**Problem 12:**
Total surveyed = 200. Females = 96, so Males = 104.
Total non-listeners = 64. Males who don't listen = 26, so females who don't listen = 64 − 26 = 38.
Males who listen = 104 − 26 = 78.
Percentage = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem step-by-step.

## Problem 1:
**What is asked:** Find the percentage of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.

**Original mixture:**
- Yellow tint: 30% of 30 = 0.30 × 30 = 9 liters
- Total volume: 30 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total volume: 30 + 5 = 35 liters

**Percentage of yellow tint:**
$$\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%$$

ANSWER 1: C

## Problem 2:
**What is asked:** In how many ways can the ghost enter by one window and leave by a different window?

**Solution:**
- Choices for entering: 6 windows
- Choices for leaving: 5 windows (must be different from entry)
- Total ways: 6 × 5 = 30

ANSWER 2: D

## Problem 3:
**What is asked:** Calculate 1/10 + 2/20 + 3/30

**Solution:**
Simplify each fraction:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

ANSWER 3: D

## Problem 4:
**What is asked:** How much money does Susan have left?

**Solution:**
- Started with: $50
- Spent on food: $12
- Spent on rides: 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Left: $50 - $36 = $14

ANSWER 4: B

## Problem 5:
**What is asked:** Calculate 200,000 × 200,000

**Solution:**
$$200,000 × 200,000 = (2 × 10^5) × (2 × 10^5) = 4 × 10^{10} = 40,000,000,000 = 40 \text{ billion}$$

ANSWER 5: E

## Problem 6:
**What is asked:** Find the missing digit A in the 3-digit number 1A2, where 11 club members each paid an equal integer amount that totals 1A2.

**Solution:**
The number 1A2 must be divisible by 11.

Testing divisibility by 11 (alternating sum of digits):
- 1 - A + 2 = 0 (mod 11)
- 3 - A ≡ 0 (mod 11)
- A = 3

Check: 132 ÷ 11 = 12 ✓

ANSWER 6: D

## Problem 7:
**What is asked:** Which shapes can Haruki make with 24 cm of wire?

**Solution:**
Test each shape:

**Regular hexagon with side 5 cm:**
- Perimeter: 6 × 5 = 30 cm
- Need 30 cm, have 24 cm ✗

**Square with area 36 cm²:**
- Side length: √36 = 6 cm
- Perimeter: 4 × 6 = 24 cm ✓

**Right triangle with legs 6 and 8 cm:**
- Hypotenuse: √(6² + 8²) = √(36 + 64) = √100 = 10 cm
- Perimeter: 6 + 8 + 10 = 24 cm ✓

Only square and triangle can be made.

ANSWER 7: D

## Problem 8:
**What is asked:** Which number has the largest reciprocal?

**Solution:**
Find reciprocals:
- 1/(1/3) = 3
- 1/(2/5) = 5/2 = 2.5
- 1/1 = 1
- 1/5 = 0.2
- 1/1986 ≈ 0.0005

The largest reciprocal is 5 (from the number 1/5).

ANSWER 8: D

## Problem 9:
**What is asked:** Find the target heart rate for a 26-year-old athlete.

**Solution:**
- Maximum heart rate: 220 - 26 = 194 beats per minute
- Target heart rate: 80% of 194 = 0.80 × 194 = 155.2 ≈ 155

ANSWER 9: B

## Problem 10:
**What is asked:** Find H - T where H is Harry's answer and T is Terry's answer.

**Solution:**
**Harry's calculation (correct):**
$$H = 8 - (2 + 5) = 8 - 7 = 1$$

**Terry's calculation (ignoring parentheses):**
$$T = 8 - 2 + 5 = 6 + 5 = 11$$

**Difference:**
$$H - T = 1 - 11 = -10$$

ANSWER 10: A

## Problem 11:
**What is asked:** Calculate 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

**Solution:**
This is an arithmetic sequence with 10 terms. Sum = n(first + last)/2
$$81 + 83 + ... + 99 = \frac{10(81 + 99)}{2} = \frac{10 × 180}{2} = 900$$

$$2 × 900 = 1800$$

ANSWER 11: E

## Problem 12:
**What is asked:** What percentage of males surveyed listen to the station?

**Solution:**
- Total surveyed: 200
- Females: 96
- Males: 200 - 96 = 104

- Total non-listeners: 64
- Males who don't listen: 26
- Males who listen: 104 - 26 = 78

**Percentage of males who listen:**
$$\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%$$

ANSWER 12: E

Problem 1:

The original mixture is 30 liters total, with 30% yellow tint.

So the amount of yellow tint at first is:
- \(30\%\) of \(30 = 0.30 \times 30 = 9\) liters

Then 5 liters of yellow tint are added, so the new amount of yellow tint is:
- \(9 + 5 = 14\) liters

The new total mixture is:
- \(30 + 5 = 35\) liters

So the percent of yellow tint is:
- \(\frac{14}{35} = \frac{2}{5} = 0.4 = 40\%\)

Answer choice C matches 40%.

ANSWER 1: C

Problem 2:

Georgie can:
- choose 1 of the 6 windows to enter
- then choose a different window to leave

For the entry, there are 6 choices.

For the exit, there are 5 remaining choices, since it must be a different window.

So the total number of ways is:
- \(6 \times 5 = 30\)

ANSWER 2: D

Problem 3:

Compute each fraction:
- \(\frac{1}{10} = 0.1\)
- \(\frac{2}{20} = \frac{1}{10} = 0.1\)
- \(\frac{3}{30} = \frac{1}{10} = 0.1\)

Add them:
- \(0.1 + 0.1 + 0.1 = 0.3\)

So the value is \(0.3\).

ANSWER 3: D

Problem 4:

Susan had \$50.

She spent:
- \$12 on food
- twice as much on rides, so \(2 \times 12 = 24\)

Total spent:
- \(12 + 24 = 36\)

Money left:
- \(50 - 36 = 14\)

ANSWER 4: B

Problem 5:

Two hundred thousand is:
- \(200{,}000 = 2 \times 10^5\)

Multiply:
- \(200{,}000 \times 200{,}000 = (2 \times 10^5)(2 \times 10^5) = 4 \times 10^{10}\)

And \(4 \times 10^{10} = 40{,}000{,}000{,}000\), which is forty billion.

ANSWER 5: E

Problem 6:

The total is written as \(1A2\), and 11 members each paid the same integer amount.

So \(1A2\) must be divisible by 11.

Test the answer choices:
- 102 ÷ 11 is not an integer
- 112 ÷ 11 is not an integer
- 122 ÷ 11 = 11 remainder 1
- 132 ÷ 11 = 12 exactly
- 142 ÷ 11 is not an integer

So the number is 132, which means \(A = 3\).

ANSWER 6: D

Problem 7:

We check each shape.

### Hexagon
A regular hexagon has 6 equal sides.
If each side is 5 cm, perimeter is:
- \(6 \times 5 = 30\) cm

The wire is only 24 cm, so he cannot make the hexagon.

### Square
A square with area 36 has side length:
- \(\sqrt{36} = 6\) cm

Perimeter:
- \(4 \times 6 = 24\) cm

So he can make the square exactly.

### Right triangle
The legs are 6 and 8, so the hypotenuse is:
- \(\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)

Perimeter:
- \(6 + 8 + 10 = 24\) cm

So he can make the triangle exactly.

Therefore he can make the square and the triangle only.

ANSWER 7: D

Problem 8:

The reciprocal is largest when the original number is smallest, assuming positive numbers.

Compare:
- \(\frac13 \approx 0.333\)
- \(\frac25 = 0.4\)
- \(1\)
- \(5\)
- \(1986\)

The smallest number is \(\frac13\), so it has the largest reciprocal.

Its reciprocal is 3, which is larger than the reciprocals of all the others.

ANSWER 8: A

Problem 9:

Maximum heart rate:
- \(220 - 26 = 194\)

Target heart rate is 80% of that:
- \(0.8 \times 194 = 155.2\)

To the nearest whole number, that is 155.

ANSWER 9: B

Problem 10:

First, Harry uses the parentheses:
- \(8 - (2 + 5) = 8 - 7 = 1\)

So \(H = 1\).

Terry ignores parentheses:
- \(8 - 2 + 5 = 6 + 5 = 11\)

So \(T = 11\).

Difference:
- \(H - T = 1 - 11 = -10\)

ANSWER 10: A

Problem 11:

We need to evaluate:
- \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

These are 10 odd numbers from 81 to 99.

Instead of adding one by one, use the average:
- average of 81 and 99 is \( \frac{81+99}{2} = 90\)

Since there are 10 numbers, their sum is:
- \(10 \times 90 = 900\)

Now multiply by 2:
- \(2 \times 900 = 1800\)

ANSWER 11: E

Problem 12:

We want the percentage of males surveyed who listen.

From the table:
- Total surveyed = 200
- Females = 96
- So males = \(200 - 96 = 104\)

Males who don’t listen = 26, so males who do listen:
- \(104 - 26 = 78\)

Now find the percent of males who listen:
- \(\frac{78}{104} = \frac{3}{4} = 75\%\)

ANSWER 12: E

Problem 1:  
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?  

- Original yellow tint amount = 30 × 0.30 = 9 liters.  
- Add 5 liters of yellow tint → new yellow amount = 9 + 5 = 14 liters.  
- New total volume = 30 + 5 = 35 liters.  
- New percent yellow = (14 / 35) × 100% = (2/5) × 100% = 40%.  

So the correct choice is 40.  

ANSWER 1: C  

---

Problem 2:  
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?  

- Choose an entry window: 6 choices.  
- Choose a different exit window: 5 choices (cannot be the same as entry).  
- Total ways = 6 × 5 = 30.  

ANSWER 2: D  

---

Problem 3:  
Compute: \( \frac{1}{10} + \frac{2}{20} + \frac{3}{30} \).  

Convert each to a common denominator, 60:  
- \( \frac{1}{10} = \frac{6}{60} \)  
- \( \frac{2}{20} = \frac{2}{20} = \frac{6}{60} \) (since multiply top and bottom by 3)  
- \( \frac{3}{30} = \frac{3}{30} = \frac{6}{60} \) (since multiply top and bottom by 2)  

Add:  
\( \frac{6}{60} + \frac{6}{60} + \frac{6}{60} = \frac{18}{60} = \frac{3}{10} = 0.3 \)

So the value is .3.  

ANSWER 3: D  

---

Problem 4:  
Susan had $50 to spend. She spent $12 on food and twice as much on rides. How much left?  

- Food = 12  
- Rides = twice 12 = 24  
- Total spent = 12 + 24 = 36  
- Money left = 50 − 36 = 14  

ANSWER 4: B  

---

Problem 5:  
Two hundred thousand times two hundred thousand equals ?  

\(200{,}000 \times 200{,}000 = (2\times 10^5)(2\times 10^5) = 4 \times 10^{10}\).  
- \(10^{10} = 10{,}000{,}000{,}000\)  
- \(4 \times 10^{10} = 40{,}000{,}000{,}000 = 40\) billion  

ANSWER 5: E  

---

Problem 6:  
Eleven members each paid the same integer amount for a guest speaker. In all, they paid $1A2 (a 3-digit number). Find missing digit A.  

- Total = \(1A2\) means:  
  \(1A2 = 100 + 10A + 2 = 102 + 10A\).  
- This total is divided equally among 11 people:  
  So \(102 + 10A\) must be divisible by 11.  

Check \(102 + 10A \mod 11\):  
- \(102 \div 11 = 9\) remainder 3, so \(102 \equiv 3 \pmod{11}\).  
- \(10A \equiv -A \pmod{11}\) (since 10 ≡ −1 mod 11).  

So:  
\(102 + 10A \equiv 3 - A \equiv 0 \pmod{11}\)  
Thus \(A \equiv 3 \pmod{11}\).  
Since \(A\) is a digit 0–9, \(A=3\).  

ANSWER 6: D  

---

Problem 7:  
Wire length = 24 cm. Which shapes can be made (using all 24 cm of wire)?  

We need the perimeter (total boundary length) of each shape:

**(1) Regular hexagon with side length 5 cm**  
Perimeter = 6 × 5 = 30 cm → too long. Cannot.

**(2) Square with area 36 cm²**  
Side length = √36 = 6 cm  
Perimeter = 4 × 6 = 24 cm → fits. Can.

**(3) Right triangle with legs 6 and 8 cm**  
Right triangle sides are 6 and 8 legs, and hypotenuse:  
\( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)  
Perimeter = 6 + 8 + 10 = 24 cm → fits. Can.

So the shapes possible: **square and triangle only**.  

ANSWER 7: D  

---

Problem 8:  
Which number has the largest reciprocal? Recall: the reciprocal of \(x\) is \(1/x\).  
We compare \(1/3, 2/5, 1, 1/5, 1/1986\).  

Compute roughly:  
- \(1/3 \approx 0.333\)  
- \(2/5 = 0.4\)  
- \(1 = 1\)  
- \(1/5 = 0.2\)  
- \(1/1986\) is tiny  

Largest is 1 (reciprocal of 1).  

ANSWER 8: C  

---

Problem 9:  
Target heart rate = 80% of (maximum heart rate).  
Maximum heart rate = 220 − age.  
Age = 26.

- Maximum = 220 − 26 = 194  
- Target = 0.80 × 194 = 155.2  
Nearest whole number = 155  

ANSWER 9: B  

---

Problem 10:  
Expression: \(8 - (2 + 5)\)  
- Harry computes correctly:  
  \(2 + 5 = 7\), so \(8 - 7 = 1\).  
- Terry ignores parentheses:  
  \(8 - 2 + 5 = 6 + 5 = 11\).  

Difference \(H - T = 1 - 11 = -10\).  

ANSWER 10: A  

---

Problem 11:  
Compute:  
\(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

Notice these are all odd numbers from 81 to 99 stepping by 2.  
There are 10 terms.

Pair from ends:  
- 81 + 99 = 180  
- 83 + 97 = 180  
- 85 + 95 = 180  
- 87 + 93 = 180  
- 89 + 91 = 180  

There are 5 pairs, each sum 180:  
Sum = 5 × 180 = 900  

Now multiply by 2:  
2 × 900 = 1800  

ANSWER 11: E  

---

Problem 12:  
Total surveyed = 200  
Females = 96  
Females who listen = 58  
Males who don’t listen = 26  
Total listeners = 136  

We want: % of males who listen.

Step 1: Find males total.  
Males = 200 − 96 = 104

Step 2: Find males who don’t listen is given = 26.  
So males who listen = 104 − 26 = 78

Step 3: Convert to percent of males who listen:  
Percent = (78 / 104) × 100%  
Simplify 78/104 = 3/4 (divide by 26):  
So percent = (3/4) × 100% = 75%

ANSWER 12: E

Problem 1:
The original mixture is 30 liters. The amount of yellow tint is 30% of 30 liters: $0.30 \times 30 = 9$ liters.
Five liters of yellow tint are added, so the new amount of yellow tint is $9 + 5 = 14$ liters.
The new total volume of the mixture is $30 + 5 = 35$ liters.
The percentage of yellow tint is $\frac{14}{35} \times 100\%$.
Simplifying the fraction: $\frac{14}{35} = \frac{2}{5} = 0.4$.
$0.4 \times 100\% = 40\%$.
ANSWER 1: C

Problem 2:
Georgie has 6 choices for the window he enters. Since he must leave by a different window, he has $6 - 1 = 5$ choices for the window he leaves by.
The total number of ways is $6 \times 5 = 30$.
ANSWER 2: D

Problem 3:
Simplify each fraction:
$1/10 = 0.1$
$2/20 = 1/10 = 0.1$
$3/30 = 1/10 = 0.1$
Summing them: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides, which is $2 \times 12 = 24$.
Total spent: $12 + 24 = 36$.
Amount left: $50 - 36 = 14$.
ANSWER 4: B

Problem 5:
Two hundred thousand is $200,000$ (or $2 \times 10^5$).
$(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
$4 \times 10^{10} = 40,000,000,000$, which is forty billion.
ANSWER 5: E

Problem 6:
The total amount $1A2$ must be divisible by 11.
A number is divisible by 11 if the alternating sum of its digits is divisible by 11.
For $1A2$, the alternating sum is $1 - A + 2 = 3 - A$.
For this to be 0 (the only multiple of 11 possible for a single digit $A$), $3 - A = 0$, so $A = 3$.
Check: $132 / 11 = 12$. This works.
ANSWER 6: D

Problem 7:
Hexagon: Perimeter of a regular hexagon with side 5 is $6 \times 5 = 30$ cm. (Cannot make)
Square: Area is 36, so side is $\sqrt{36} = 6$. Perimeter is $4 \times 6 = 24$ cm. (Can make)
Triangle: Right triangle with legs 6 and 8. Hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. Perimeter is $6 + 8 + 10 = 24$ cm. (Can make)
ANSWER 7: D

Problem 8:
Reciprocals:
A: $1/(1/3) = 3$
B: $1/(2/5) = 2.5$
C: $1/1 = 1$
D: $1/5 = 0.2$
E: $1/1986 \approx 0.0005$
The largest is 3.
ANSWER 8: A

Problem 9:
Max heart rate = $220 - 26 = 194$.
Target heart rate = $80\%$ of $194 = 0.8 \times 194 = 155.2$.
To the nearest whole number, this is 155.
ANSWER 9: B

Problem 10:
$H = 8 - (2 + 5) = 8 - 7 = 1$.
$T = 8 - 2 + 5 = 6 + 5 = 11$.
$H - T = 1 - 11 = -10$.
ANSWER 10: A

Problem 11:
Sum inside parentheses: $81+83+85+87+89+91+93+95+97+99$.
This is an arithmetic progression with 10 terms.
Sum = $\frac{n}{2}(\text{first} + \text{last}) = \frac{10}{2}(81 + 99) = 5 \times 180 = 900$.
The expression is $2 \times 900 = 1800$.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96.
Males = $200 - 96 = 104$.
Males who don't listen = 26.
Males who listen = $104 - 26 = 78$.
Percentage of males who listen = $(78 / 104) \times 100\%$.
$\frac{78}{104} = \frac{39}{52} = \frac{3}{4} = 0.75 = 75\%$.
ANSWER 12: E

Problem 1: The original mixture has volume 30 L, of which 30% is yellow tint, so there are \(0.3 \times 30 = 9\) L of yellow tint. Adding 5 L of yellow tint produces a new total of 14 L of yellow tint and a new mixture volume of 35 L. The new percentage of yellow tint is therefore \((14/35) \times 100\% = 40\%\).

ANSWER 1: C

Problem 2: There are 6 choices for the entry window. For each of those choices there are then exactly 5 remaining choices for a different exit window, giving a total of \(6 \times 5 = 30\) ordered pairs of distinct windows.

ANSWER 2: D

Problem 3: Rewrite each term with a common denominator of 10:  
\(1/10 = 0.1\),  
\(2/20 = 1/10 = 0.1\),  
\(3/30 = 1/10 = 0.1\).  
Their sum is \(0.1 + 0.1 + 0.1 = 0.3\).

ANSWER 3: D

Problem 4: Twice the food cost is \(2 \times 12 = 24\) dollars spent on rides. The total spent is therefore \(12 + 24 = 36\) dollars, leaving \(50 - 36 = 14\) dollars.

ANSWER 4: B

Problem 5: Write the factors in scientific notation:  
\(200000 = 2 \times 10^5\), so  
\((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\),  
which is forty billion.

ANSWER 5: E

Problem 6: The three-digit total \(1A2\) must be divisible by 11. The alternating-sum divisibility rule requires that \(1 - A + 2 = 3 - A\) be a multiple of 11. The only admissible digit \(A\) (0-9) that satisfies the condition is \(A = 3\), giving the total 132, which is exactly \(11 \times 12\).

ANSWER 6: D

Problem 7: A regular hexagon with side 5 cm has perimeter 30 cm, which exceeds the 24 cm length of wire, so it is impossible. A square of area 36 cm² has side length 6 cm and perimeter 24 cm, which matches the wire length. A right triangle with legs 6 cm and 8 cm has hypotenuse 10 cm, so its perimeter is \(6 + 8 + 10 = 24\) cm, which also matches. Consequently only the square and the triangle are possible.

ANSWER 7: D

Problem 8: For positive numbers the reciprocal function is strictly decreasing, so the largest reciprocal belongs to the smallest number in the list, which is \(1/3\).

ANSWER 8: A

Problem 9: The theoretical maximum heart rate is \(220 - 26 = 194\) beats per minute. Eighty percent of that value is \(0.8 \times 194 = 155.2\), which rounds to the nearest whole number 155.

ANSWER 9: B

Problem 10: Harry evaluates the parentheses first and obtains \(8 - (2 + 5) = 1\). Terry omits the parentheses and obtains \(8 - 2 + 5 = 11\). Their difference is therefore \(1 - 11 = -10\).

ANSWER 10: A

Problem 11: The sum inside the parentheses consists of the ten consecutive odd integers from 81 to 99. Their average is \((81 + 99)/2 = 90\), so the sum equals \(10 \times 90 = 900\). Multiplying by the outer factor of 2 yields \(2 \times 900 = 1800\).

ANSWER 11: E

Problem 12: There are 200 people surveyed and 96 females, so there are \(200 - 96 = 104\) males. Of the 96 females, 58 listen, leaving \(96 - 58 = 38\) females who do not listen. Given that 26 males do not listen, the number of males who listen is \(104 - 26 = 78\). The required percentage is therefore \((78/104) \times 100\% = 75\%\).

ANSWER 12: E

Problem 1:
The problem asks us to find the percentage of yellow tint in a new mixture after adding 5 liters of yellow tint to the original 30 liters of paint mixture. The original mixture is 25% red tint, 30% yellow tint, and 45% water.

First, let's calculate the amount of yellow tint in the original mixture. 30% of 30 liters is $0.3 \times 30 = 9$ liters.

After adding 5 liters of yellow tint, the total amount of yellow tint becomes $9 + 5 = 14$ liters.

The total volume of the new mixture is $30 + 5 = 35$ liters.

To find the percentage of yellow tint in the new mixture, we divide the amount of yellow tint by the total volume and multiply by 100: $\frac{14}{35} \times 100 = 40\%$.

Let's examine the choices:
- A. 25: Too low because the percentage of yellow tint increased.
- B. 35: Not the correct calculation.
- C. 40: Matches our calculation.
- D. 45: Too high.
- E. 50: Too high.

The correct percentage is 40%.

ANSWER 1: C

Problem 2:
The problem asks us to determine the number of ways Georgie the Ghost can enter a haunted house through one of six windows and leave by a different window.

To enter, Georgie has 6 choices. After entering, there are 5 remaining windows to choose from for leaving.

The total number of ways to enter and leave is $6 \times 5 = 30$.

Let's examine the choices:
- A. 12: Too low.
- B. 15: Too low.
- C. 18: Too low.
- D. 30: Matches our calculation.
- E. 36: Too high because it would imply Georgie can leave by the same window.

The correct number of ways is 30.

ANSWER 2: D

Problem 3:
The problem requires us to simplify the expression $1⁄10 + 2⁄20 + 3⁄30$.

First, simplify each fraction:
- $1⁄10$ stays the same.
- $2⁄20 = 1⁄10$.
- $3⁄30 = 1⁄10$.

Adding them together: $1⁄10 + 1⁄10 + 1⁄10 = 3⁄10 = 0.3$.

Let's examine the choices:
- A. .1: Too low.
- B. .123: Not the correct sum.
- C. .2: Too low.
- D. .3: Matches our calculation.
- E. .6: Too high.

The correct sum is 0.3.

ANSWER 3: D

Problem 4:
The problem asks us to find out how many dollars Susan had left after spending money on food and rides at the carnival. She started with $50, spent $12 on food, and twice as much on rides.

First, calculate the amount spent on rides: $12 \times 2 = $24.

Total amount spent is $12 + $24 = $36.

The amount left is $50 - $36 = $14.

Let's examine the choices:
- A. 12: Too low.
- B. 14: Matches our calculation.
- C. 26: Too high.
- D. 38: Too high.
- E. 50: Incorrect because she spent money.

The correct amount left is $14.

ANSWER 4: B

Problem 5:
The problem asks us to calculate $200,000 \times 200,000$.

$200,000 = 2 \times 10^5$, so $200,000 \times 200,000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10} = 40,000,000,000$.

Let's examine the choices:
- A. four hundred thousand: Too low.
- B. four million: Too low.
- C. forty thousand: Too low.
- D. four hundred million: Too low.
- E. forty billion: Matches our calculation.

The correct result is forty billion.

ANSWER 5: E

Problem 6:
The problem requires us to find the missing digit A in the 3-digit number $1A2 that represents the total amount paid by 11 members of the Math Club.

The total amount paid is divisible by 11 since there are 11 members who paid the same amount.

To check divisibility by 11, we use the rule: the difference between the sum of digits at odd places and the sum of digits at even places should be divisible by 11.

For $1A2, the sum of digits at odd places is $1 + 2 = 3$, and the sum at even places is $A$. The difference is $3 - A$.

For $3 - A$ to be divisible by 11 (or 0, since $A$ is a single digit), $A = 3$ makes $3 - A = 0$, which is divisible by 11.

Let's examine the choices:
- A. 0: Not correct because $3 - 0 = 3$, not divisible by 11.
- B. 1: Not correct because $3 - 1 = 2$, not divisible by 11.
- C. 2: Not correct because $3 - 2 = 1$, not divisible by 11.
- D. 3: Correct because $3 - 3 = 0$, which is divisible by 11.
- E. 4: Not correct because $3 - 4 = -1$, not divisible by 11.

The missing digit A is 3, but we should verify if 132 is indeed divisible by 11. $132 / 11 = 12$, so it is.

ANSWER 6: D

Problem 7:
The problem asks us to determine which shapes Haruki can form using a 24 cm long wire.

1. A regular hexagon with side length 5 cm: The perimeter is $6 \times 5 = 30$ cm, which is more than 24 cm. So, Haruki cannot form this hexagon.

2. A square of area 36 cm$^2$: The side length is $\sqrt{36} = 6$ cm. The perimeter is $4 \times 6 = 24$ cm, which matches the wire length. So, Haruki can form this square.

3. A right triangle with legs 6 and 8 cm: Using the Pythagorean theorem, the hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm. The perimeter is $6 + 8 + 10 = 24$ cm, which matches the wire length. So, Haruki can form this triangle.

Let's examine the choices:
- A. Triangle only: Incorrect because Haruki can also form the square.
- B. Hexagon and square only: Incorrect because Haruki cannot form the hexagon.
- C. Hexagon and triangle only: Incorrect because Haruki cannot form the hexagon.
- D. Square and triangle only: Matches our findings.
- E. Hexagon, triangle, and square: Incorrect because Haruki cannot form the hexagon.

The correct shapes Haruki can form are the square and the triangle.

ANSWER 7: D

Problem 8:
The problem asks us to find which of the given numbers has the largest reciprocal.

The reciprocal of a number $x$ is $1/x$. The larger $x$ is, the smaller its reciprocal is, and vice versa.

Let's examine the choices:
- A. $1⁄3$: The reciprocal is 3.
- B. $2⁄5$: The reciprocal is $5/2 = 2.5$.
- C. 1: The reciprocal is 1.
- D. 5: The reciprocal is $1/5 = 0.2$.
- E. 1986: The reciprocal is $1/1986$, very small.

Comparing the reciprocals: $3 > 2.5 > 1 > 0.2 > 1/1986$. So, $1/3$ has the largest reciprocal.

Let's examine the choices again:
- A. $1⁄3$: Correct.
- B. $2⁄5$: Not correct.
- C. 1: Not correct.
- D. 5: Not correct.
- E. 1986: Not correct.

The number with the largest reciprocal is $1/3$.

ANSWER 8: A

Problem 9:
The proble

Problem 1:
Original mixture: 30 L. Yellow tint = 30% of 30 = 0.30 × 30 = 9 L.
Add 5 L yellow: new yellow = 9 + 5 = 14 L, new total volume = 30 + 5 = 35 L.
Percent yellow = (14/35) × 100% = 40%.
ANSWER 1: C

Problem 2:
6 choices for entry window, 5 remaining choices for a different exit window.
Total ways = 6 × 5 = 30.
ANSWER 2: D

Problem 3:
1/10 + 2/20 + 3/30 = 1/10 + 1/10 + 1/10 = 3/10 = 0.3.
ANSWER 3: D

Problem 4:
Spent on food: $12. Spent on rides: 2 × $12 = $24.
Total spent = $12 + $24 = $36. Left = $50 − $36 = $14.
ANSWER 4: B

Problem 5:
200,000 × 200,000 = (2×10⁵) × (2×10⁵) = 4×10¹⁰ = 40,000,000,000, which is forty billion.
ANSWER 5: E

Problem 6:
Total paid is $1A2, a multiple of 11. Divisibility by 11: (1 − A + 2) = 3 − A must be a multiple of 11.
With A a digit 0–9, 3 − A = 0 ⇒ A = 3. (Check: 132 ÷ 11 = 12.)
ANSWER 6: D

Problem 7:
Wire length = 24 cm.
- Regular hexagon, side 5 cm: perimeter = 6×5 = 30 cm > 24 cm → cannot make.
- Square, area 36 cm²: side = 6 cm, perimeter = 4×6 = 24 cm → can make.
- Right triangle, legs 6 and 8 cm: hypotenuse = 10 cm, perimeter = 6+8+10 = 24 cm → can make.
So square and triangle only.
ANSWER 7: D

Problem 8:
Reciprocals: 1/(1/3)=3, 1/(2/5)=2.5, 1/1=1, 1/5=0.2, 1/1986≈0.0005. Largest is 3, from 1/3.
ANSWER 8: A

Problem 9:
Maximum heart rate = 220 − 26 = 194 bpm.
Target = 80% of 194 = 0.8 × 194 = 155.2 → 155 to nearest whole number.
ANSWER 9: B

Problem 10:
Harry: H = 8 − (2+5) = 8 − 7 = 1.
Terry: T = 8 − 2 + 5 = 6 + 5 = 11.
H − T = 1 − 11 = −10.
ANSWER 10: A

Problem 11:
Sum inside parentheses: 10 terms, average = (81+99)/2 = 90, sum = 10×90 = 900.
2 × 900 = 1800.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96 ⇒ Males = 200 − 96 = 104.
Total listeners = 136, female listeners = 58 ⇒ male listeners = 136 − 58 = 78.
Percent of males who listen = (78/104) × 100% = 75%.
ANSWER 12: E

Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint.

**Original mixture (30 liters):**
- Red tint: 25% of 30 = 7.5 liters
- Yellow tint: 30% of 30 = 9 liters
- Water: 45% of 30 = 13.5 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total mixture: 30 + 5 = 35 liters

**New percentage of yellow tint:**
14 / 35 = 2/5 = 0.40 = 40%

Tempting wrong answers: Choice B (35) might come from just adding 5 to the original 30%, but we must recalculate based on the new total.

ANSWER 1: C

---

Problem 2:
Georgie enters through one window and leaves through a *different* window.

- Number of choices for the entry window: 6
- Number of choices for the exit window (must be different): 5
- Total ways: 6 × 5 = 30

Tempting wrong answers: Choice E (36) would be 6 × 6, which allows entering and leaving through the same window. Choice A (12) would be 6 + 6, which doesn't account for all combinations.

ANSWER 2: D

---

Problem 3:
Simplify each fraction first:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

Tempting wrong answers: Choice B (.123) might come from concatenating the digits 1, 2, 3 instead of actually computing. Choice C (.2) might come from miscounting the terms.

ANSWER 3: D

---

Problem 4:
Susan starts with $50.
- Spent on food: $12
- Spent on rides: twice the food amount = 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Amount left: $50 − $36 = $14

Tempting wrong answers: Choice C (26) might come from subtracting only the food ($50 − $24 = $26, forgetting food). Choice D (38) comes from subtracting only $12.

ANSWER 4: B

---

Problem 5:
200,000 × 200,000

Write in scientific notation:
(2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰

4 × 10¹⁰ = 40,000,000,000 = forty billion

Tempting wrong answers: Choice A (four hundred thousand) comes from adding instead of multiplying. Choice D (four hundred million) comes from miscounting zeros.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, so the total $1A2 must be divisible by 11.

**Divisibility rule for 11:** The alternating sum of digits must be a multiple of 11.
Alternating sum: 1 − A + 2 = 3 − A

For 3 − A to be divisible by 11 (and A to be a single digit 0–9):
3 − A = 0 → A = 3

**Check:** $132 ÷ 11 = $12 per member. This is a valid integer amount.

Tempting wrong answers: Other values of A don't make 3 − A a multiple of 11 within the digit range.

ANSWER 6: D

---

Problem 7:
Wire length = 24 cm. We check if each shape's perimeter ≤ 24 cm.

**Regular hexagon (side 5 cm):**
Perimeter = 6 × 5 = 30 cm. Since 30 > 24, he **cannot** make this.

**Square (area 36 cm²):**
Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Since 24 = 24, he **can** make this.

**Right triangle (legs 6 and 8 cm):**
Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Perimeter = 6 + 8 + 10 = 24 cm. Since 24 = 24, he **can** make this.

Haruki can make the square and triangle only.

Tempting wrong answers: Choice E includes the hexagon, but its perimeter exceeds the wire length.

ANSWER 7: D

---

Problem 8:
The reciprocal of a number x is 1/x. The *smaller* the positive number, the *larger* its reciprocal.

| Choice | Number | Reciprocal |
|--------|--------|------------|
| A | 1/3 | 3 |
| B | 2/5 | 5/2 = 2.5 |
| C | 1 | 1 |
| D | 5 | 1/5 = 0.2 |
| E | 1986 | 1/1986 ≈ 0.0005 |

The largest reciprocal is 3, which comes from 1/3.

Tempting wrong answers: Choice E (1986) is the largest number, so it has the *smallest* reciprocal.

ANSWER 8: A

---

Problem 9:
Athlete's age = 26 years.

**Step 1:** Maximum heart rate = 220 − 26 = 194 beats per minute.

**Step 2:** Target heart rate = 80% of 194 = 0.80 × 194 = 155.2

**Step 3:** Round to nearest whole number: 155

Tempting wrong answers: Choice C (176) might come from taking 80% of 220 without subtracting the age. Choice D (194) is the maximum heart rate itself.

ANSWER 9: B

---

Problem 10:
**Harry (correct calculation):**
H = 8 − (2 + 5) = 8 − 7 = 1

**Terry (ignores parentheses):**
T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:**
H − T = 1 − 11 = −10

Tempting wrong answers: Choice E (10) is |H − T|, but the question asks for H − T specifically, which is negative since Terry's answer is larger.

ANSWER 10: A

---

Problem 11:
We need: 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

The sum inside is an arithmetic sequence:
- First term: 81, Last term: 99
- Number of terms: 10
- Sum = (number of terms) × (first + last) / 2 = 10 × (81 + 99) / 2 = 10 × 180 / 2 = 10 × 90 = 900

Multiply by 2: 2 × 900 = 1800

Tempting wrong answers: Choice C (1700) might come from miscounting terms or miscalculating the average.

ANSWER 11: E

---

Problem 12:
Let me organize the data from the survey:

- Total surveyed: 200
- Females: 96 → Males: 200 − 96 = 104
- Total listeners: 136
- Females who listen: 58 → Males who listen: 136 − 58 = 78
- Males who don't listen: 26 (Check: 78 + 26 = 104 ✓)

**Percentage of males who listen:**
(78 / 104) × 100% = (3/4) × 100% = 75%

Tempting wrong answers: Choice C (52) might come from computing 104/200 (percentage of total who are male). Choice D (55) might come from 78/136 (percentage of listeners who are male).

ANSWER 12: E

 Problem 1:
We start with 30 liters of paint. The original amount of yellow tint is 30% of 30 liters:
\[
0.30 \times 30 = 9 \text{ liters.}
\]
Five liters of yellow tint are added, so the new amount of yellow tint is:
\[
9 + 5 = 14 \text{ liters.}
\]
The new total volume of the mixture is:
\[
30 + 5 = 35 \text{ liters.}
\]
The percent of yellow tint in the new mixture is:
\[
\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%.
\]
(Choice A is the original red percentage, D is the original water percentage, and E would require 17.5 liters of yellow, which we do not have.)
ANSWER 1: C

Problem 2:
Georgie chooses one of 6 windows to enter. Since he must leave through a different window, there are 5 remaining choices for his exit. By the multiplication principle:
\[
6 \times 5 = 30 \text{ ways.}
\]
(Choice E, 36, would count entering and leaving through the same window; the other choices are too small.)
ANSWER 2: D

Problem 3:
Simplify each fraction:
\[
\frac{2}{20} = \frac{1}{10}, \qquad \frac{3}{30} = \frac{1}{10}.
\]
So the sum is:
\[
\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3.
\]
(Choice A is just one term; C is 0.2; E is 0.6.)
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides:
\[
2 \times 12 = 24 \text{ dollars.}
\]
Total spent:
\[
12 + 24 = 36 \text{ dollars.}
\]
Amount left:
\[
50 - 36 = 14 \text{ dollars.}
\]
(Choice C is the total spent; D is the amount left if she had spent only $12.)
ANSWER 4: B

Problem 5:
\[
200{,}000 \times 200{,}000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}.
\]
\[
4 \times 10^{10} = 40{,}000{,}000{,}000,
\]
which is read as forty billion.
(Choice D is $4 \times 10^8$; B is $4 \times 10^6$.)
ANSWER 5: E

Problem 6:
The total amount paid is the three-digit number $1A2$, which equals:
\[
100 + 10A + 2 = 102 + 10A.
\]
Since 11 members each paid the same integer amount, $102 + 10A$ must be divisible by 11. Testing the choices:
- $A=0$: $102 \div 11 \approx 9.27$ (not an integer)
- $A=1$: $112 \div 11 \approx 10.18$ (not an integer)
- $A=2$: $122 \div 11 \approx 11.09$ (not an integer)
- $A=3$: $132 \div 11 = 12$ (integer!)
- $A=4$: $142 \div 11 \approx 12.9$ (not an integer)
Thus the missing digit is 3.
ANSWER 6: D

Problem 7:
The wire is 24 cm long, so a shape is possible only if its perimeter is at most 24 cm.
- **Regular hexagon, side 5 cm:** Perimeter $= 6 \times 5 = 30$ cm. $30 > 24$, so impossible.
- **Square of area 36 cm²:** Side $= \sqrt{36} = 6$ cm. Perimeter $= 4 \times 6 = 24$ cm. Exactly fits, so possible.
- **Right triangle with legs 6 and 8 cm:** Hypotenuse $= \sqrt{6^2 + 8^2} = \sqrt{100} = 10$ cm. Perimeter $= 6 + 8 + 10 = 24$ cm. Exactly fits, so possible.
Only the square and the triangle can be formed.
ANSWER 7: D

Problem 8:
For positive numbers, the reciprocal function $f(x) = \frac{1}{x}$ is decreasing, so the smallest number has the largest reciprocal. Compare the two smallest choices:
\[
\frac{1}{3} \approx 0.333\ldots, \qquad \frac{2}{5} = 0.4.
\]
Since $\frac{1}{3} < \frac{2}{5} < 1 < 5 < 1986$, the smallest number is $\frac{1}{3}$. Its reciprocal is $3$, which is larger than the reciprocals of all the other choices.
ANSWER 8: A

Problem 9:
First find the theoretical maximum heart rate:
\[
220 - 26 = 194 \text{ beats per minute.}
\]
The target rate is 80% of this:
\[
0.80 \times 194 = 155.2.
\]
Rounded to the nearest whole number, this is 155.
(Choice C is the maximum heart rate itself.)
ANSWER 9: B

Problem 10:
Harry computes correctly:
\[
H = 8 - (2 + 5) = 8 - 7 = 1.
\]
Terry ignores the parentheses:
\[
T = 8 - 2 + 5 = 6 + 5 = 11.
\]
The difference is:
\[
H - T = 1 - 11 = -10.
\]
ANSWER 10: A

Problem 11:
The ten numbers inside the parentheses form an arithmetic sequence: 81, 83, …, 99. The sum of an arithmetic sequence is:
\[
\frac{\text{number of terms}}{2} \times (\text{first} + \text{last}) = \frac{10}{2} \times (81 + 99) = 5 \times 180 = 900.
\]
Now multiply by the factor of 2 outside:
\[
2 \times 900 = 1800.
\]
(One can also pair them: $(81+99)=180$, $(83+97)=180$, etc., giving five pairs that each sum to 180.)
ANSWER 11: E

Problem 12:
Total surveyed = 200, and females surveyed = 96, so:
\[
\text{Males surveyed} = 200 - 96 = 104.
\]
Total listeners = 136, and females who listen = 58, so:
\[
\text{Males who listen} = 136 - 58 = 78.
\]
(We can check: total non-listeners = 64, males who don't listen = 26, so females who don't listen = 38; $58+38=96$ and $78+26=104$, which checks out.)

The percentage of males who listen is:
\[
\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%.
\]
ANSWER 12: E

Problem 1:
A mixture of 30 liters is 25% red, 30% yellow, and 45% water. Five liters of yellow tint are added. What is the percent of yellow tint in the new mixture?

**Setup:** Find the amount of yellow tint originally, add 5 liters, then find the new percentage.

**Original yellow tint:** 30% of 30 = 0.30 × 30 = 9 liters

**After adding 5 liters of yellow tint:**
- New yellow tint: 9 + 5 = 14 liters
- New total volume: 30 + 5 = 35 liters
- New percentage: 14/35 = 2/5 = 40%

**Checking wrong choices:** 25% would be if we just added the original percentages; 35% would be if we just added 5% to 30%; 45% and 50% are too high.

ANSWER 1: C

---

Problem 2:
Six windows. Georgie enters by one and leaves by a different one. How many ways?

**Setup:** Choose an entry window (6 choices), then choose a different exit window (5 choices).

**Calculation:** 6 × 5 = 30 ways

**Checking wrong choices:** 15 would be if we divided by 2 (treating order as irrelevant); 12 would be 6×2; 18 would be 6×3; 36 would be 6×6 (allowing same window).

ANSWER 2: D

---

Problem 3:
1/10 + 2/20 + 3/30 = ?

**Simplify each fraction:**
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

**Sum:** 1/10 + 1/10 + 1/10 = 3/10 = 0.3

**Checking wrong choices:** 0.1 would be just one term; 0.2 would be two terms; 0.6 would be adding numerators and denominators incorrectly.

ANSWER 3: D

---

Problem 4:
Susan had $50. She spent $12 on food and twice as much on rides. How much left?

**Setup:** Food = $12, Rides = 2 × $12 = $24

**Total spent:** $12 + $24 = $36

**Amount left:** $50 - $36 = $14

**Checking wrong choices:** 12 is just the food amount; 26 is 50-24 (forgetting food); 38 is 50-12 (forgetting rides).

ANSWER 4: B

---

Problem 5:
200,000 × 200,000 = ?

**Setup:** Express in scientific notation: 200,000 = 2 × 10⁵

**Calculation:** (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion

**Checking wrong choices:** Four hundred thousand is 4×10⁵; four million is 4×10⁶; forty thousand is 4×10⁴; four hundred million is 4×10⁸.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, totaling $1A2. Find the missing digit A.

**Setup:** The total $1A2 must be divisible by 11.

**Divisibility test for 11:** A number is divisible by 11 if the alternating sum of its digits is divisible by 11.

For 1A2: (1 + 2) - A = 3 - A must be divisible by 11.

Since A is a single digit (0-9), 3 - A can only equal 0 (the only multiple of 11 in range -6 to 3).

So A = 3, giving $132 = 11 × $12 ✓

**Checking:** With A=0: $102 ÷ 11 ≈ 9.27 (not integer); A=1: $112 ÷ 11 ≈ 10.18 (not integer); A=3: $132 ÷ 11 = 12 ✓

ANSWER 6: D

---

Problem 7:
Wire is 24 cm. Which shapes can be made?

**Regular hexagon with side 5 cm:** Perimeter = 6 × 5 = 30 cm. Need 30 cm, only have 24 cm. ❌

**Square with area 36 cm²:** Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Need 24 cm, have 24 cm. ✓

**Right triangle with legs 6 and 8 cm:** Hypotenuse = √(36 + 64) = √100 = 10 cm. Perimeter = 6 + 8 + 10 = 24 cm. Need 24 cm, have 24 cm. ✓

**Conclusion:** Can make the square and triangle, but not the hexagon.

ANSWER 7: D

---

Problem 8:
Which number has the largest reciprocal?

**Find each reciprocal:**
- A. 1/3 → reciprocal = 3
- B. 2/5 → reciprocal = 5/2 = 2.5
- C. 1 → reciprocal = 1
- D. 5 → reciprocal = 1/5 = 0.2
- E. 1986 → reciprocal ≈ 0.0005

**Key insight:** Smaller positive numbers have larger reciprocals. The smallest number listed is 1/3, so it has the largest reciprocal (3).

ANSWER 8: A

---

Problem 9:
Target heart rate = 80% of (220 - age). Find target for 26-year-old.

**Maximum heart rate:** 220 - 26 = 194 bpm

**Target heart rate:** 80% of 194 = 0.80 × 194 = 155.2 ≈ 155 bpm

**Checking wrong choices:** 134 would be about 69%; 176 would be about 91%; 194 is the max (100%); 243 is impossible.

ANSWER 9: B

---

Problem 10:
Harry calculates 8 − (2 + 5) correctly. Terry calculates 8 − 2 + 5. Find H − T.

**Harry's answer:** H = 8 − (2 + 5) = 8 − 7 = 1

**Terry's answer:** T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:** H − T = 1 − 11 = −10

**Checking wrong choices:** −6 would be if Terry got 7; 0 would be if they got the same answer; 6 or 10 would be T − H.

ANSWER 10: A

---

Problem 11:
2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99) = ?

**Setup:** This is 10 consecutive odd numbers. The average of the first and last is (81 + 99)/2 = 90.

**Sum of the 10 numbers:** 10 × 90 = 900

**Multiply by 2:** 2 × 900 = 1800

**Alternative verification:** Pair the numbers: (81+99) + (83+97) + (85+95) + (87+93) + (89+91) = 180 + 180 + 180 + 180 + 180 = 900. Then 2 × 900 = 1800.

ANSWER 11: E

---

Problem 12:
Find the percentage of males who listen to the station.

**Given information:**
- Total surveyed: 200
- Females: 96
- Females who listen: 58
- Males who don't listen: 26
- Total listeners: 136
- Total non-listeners: 64

**Find males:** Males = 200 - 96 = 104

**Find males who listen:** Males who listen = Total listeners - Females who listen = 136 - 58 = 78

**Verification:** Males who don't listen = 104 - 78 = 26 ✓ (matches given)

**Percentage of males who listen:** 78/104 × 100 = 75%

**Checking wrong choices:** 39% would be 78/200 (using total instead of males); 48% would be about 50/104; 52% would be about 54/104; 55% would be about 57/104.

ANSWER 12: E

**Problem 1**

The original mixture is 30 L with 30 % yellow tint → yellow = 0.30·30 = 9 L.  
Add 5 L of yellow → total yellow = 9 + 5 = 14 L.  
Total volume after addition = 30 + 5 = 35 L.  

Percent yellow = (14/35)·100 % = 40 %.

**Answer 1: C**

---

**Problem 2**

Choose an entry window (6 ways) and a different exit window (5 ways).  
Number of ways = 6 × 5 = 30.

**Answer 2: D**

---

**Problem 3**

\[
\frac{1}{10}=0.1,\qquad\frac{2}{20}=0.1,\qquad\frac{3}{30}=0.1
\]
Sum = 0.1 + 0.1 + 0.1 = 0.3.

**Answer 3: D**

---

**Problem 4**

Food cost = \$12.  
Rides cost = twice food = 2·12 = \$24.  
Total spent = 12 + 24 = \$36.  
Money left = 50 − 36 = \$14.

**Answer 4: B**

---

**Problem 5**

\(200{,}000 \times 200{,}000 = (2\times10^{5})^{2}=4\times10^{10}=40{,}000{,}000{,}000\).

That is 40 billion.

**Answer 5: E**

---

**Problem 6**

Total paid = \$1A2 = 100 + 10A + 2 = 102 + 10A.  
Since the 11 members each paid an integer amount, 102 + 10A must be divisible by 11.

\(102\equiv3\pmod{11}\).  
We need \(10A\equiv-3\equiv8\pmod{11}\).  
Because \(10\equiv-1\pmod{11}\), this is \(-A\equiv8\pmod{11}\) → \(A\equiv3\pmod{11}\).  
A digit, so \(A=3\).

**Answer 6: D**

---

**Problem 7**

- Hexagon: perimeter = 6·5 = 30 cm > 24 cm → impossible.  
- Square: side = √36 = 6 cm, perimeter = 4·6 = 24 cm → possible.  
- Right triangle: legs 6 cm, 8 cm, hypotenuse = √(6²+8²)=10 cm, perimeter = 6+8+10 = 24 cm → possible.

Only the square and triangle can be made.

**Answer 7: D**

---

**Problem 8**

Reciprocals:
- \(1/3 → 3\)  
- \(2/5 → 5/2 = 2.5\)  
- \(1 → 1\)  
- \(5 → 1/5 = 0.2\)  
- \(1986 → 1/1986 ≈ 0.0005\)

Largest reciprocal is 3, from \(1/3\).

**Answer 8: A**

---

**Problem 9**

Maximum heart rate = 220 − 26 = 194 bpm.  
Target = 80 % of 194 = 0.8·194 = 155.2 → rounded to 155 bpm.

**Answer 9: B**

---

**Problem 10**

Harry: \(8-(2+5)=8-7=1\).  
Terry (ignoring parentheses): \(8-2+5 = 6+5 = 11\).  
Difference \(H-T = 1-11 = -10\).

**Answer 10: A**

---

**Problem 11**

Sum of the ten terms: average = (81+99)/2 = 90, so sum = 90·10 = 900.  
Multiply by 2: \(2·900 = 1800\).

**Answer 11: E**

---

**Problem 12**

Total surveyed = 200.  
Females = 96 → Males = 200 − 96 = 104.  

Female listeners = 58 → Male listeners = 136 − 58 = 78.  

Percent of males who listen = (78/104)·100 % = (39/52)·100 % = 75 %.

**Answer 12: E**

---

**Final Answers**

ANSWER 1: C  
ANSWER 2: D  
ANSWER 3: D  
ANSWER 4: B  
ANSWER 5: E  
ANSWER 6: D  
ANSWER 7: D  
ANSWER 8: A  
ANSWER 9: B  
ANSWER 10: A  
ANSWER 11: E  
ANSWER 12: E

Problem 1:  
A 30-liter paint mixture is 25% red tint, 30% yellow tint, and 45% water. Five liters of yellow tint are added.  
Original yellow tint: $0.30 \times 30 = 9$ liters.  
After adding 5 liters: $9 + 5 = 14$ liters yellow.  
New total volume: $30 + 5 = 35$ liters.  
Percent yellow: $\frac{14}{35} = 0.40 = 40\%$.  
Tempting wrong choices: 35% (original yellow percent) or 45% (water percent).  
ANSWER 1: D  

Problem 2:  
Georgie enters through one of six windows and exits through a different one.  
Entry choices: 6. Exit choices: 5 (excluding entry window).  
Total ways: $6 \times 5 = 30$.  
Tempting wrong choices: 15 (if order didn’t matter) or 36 (if same window allowed).  
ANSWER 2: D  

Problem 3:  
$\frac{1}{10} + \frac{2}{20} + \frac{3}{30} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3$.  
Tempting wrong choices: 0.1 (one term) or 0.2 (two terms).  
ANSWER 3: D  

Problem 4:  
Susan starts with \$50.  
Food: \$12.  
Rides: $2 \times 12 = \$24$.  
Total spent: $12 + 24 = \$36$.  
Left: $50 - 36 = \$14$.  
Tempting wrong choices: \$26 (if spent \$24 total) or \$38 (if spent \$12 total).  
ANSWER 4: B  

Problem 5:  
$200,000 \times 200,000 = 40,000,000,000 = 40$ billion.  
Tempting wrong choices: 400,000 (added zeros) or 4 million (multiplied digits).  
ANSWER 5: E  

Problem 6:  
Total paid: \$1A2, an integer divisible by 11.  
$1A2 \mod 11 = 0$.  
$1 + 2 - A \equiv 0 \mod 11 \Rightarrow 3 - A \equiv 0 \mod 11 \Rightarrow A = 3$.  
Tempting wrong choices: 0, 1, 2, or 4 (incorrect divisibility).  
ANSWER 6: E (Wait, let's recheck: 1A2 must be divisible by 11. 132 ÷ 11 = 12, so A=3. But 132 is 11×12, so correct. However, the answer choices are A=0,B=1,C=2,D=3,E=4. So D is 3. So correct answer is D, not E. I made a mistake in the initial answer. Correcting:)  
**Correction**: $1A2 = 132$ when $A=3$, and $132 \div 11 = 12$, so $A=3$.  
ANSWER 6: D  

Problem 7:  
Wire length: 24 cm.  
- Regular hexagon with side 5 cm: Perimeter $6 \times 5 = 30$ cm (not 24).  
- Square with area 36 cm²: Side $6$ cm, perimeter $24$ cm.  
- Right triangle with legs 6 and 8 cm: Hypotenuse 10 cm, perimeter $6+8+10=24$ cm.  
Can make square and triangle.  
Tempting wrong choices: Hexagon (incorrect perimeter).  
ANSWER 7: D  

Problem 8:  
Largest reciprocal means smallest original number.  
$\frac{1}{3} \approx 0.333$, $\frac{2}{5} = 0.4$, $1$, $5$, $1986$.  
Smallest: $\frac{1}{3}$. Reciprocal: 3.  
Others: 2.5, 1, 0.2, 0.0005.  
Tempting wrong choices: $\frac{2}{5}$ (larger than $\frac{1}{3}$).  
ANSWER 8: A  

Problem 9:  
Maximum heart rate: $220 - 26 = 194$.  
Target: $0.80 \times 194 = 155.2 \approx 155$.  
Tempting wrong choices: 134 (70%), 176 (90%), or 243 (incorrect max).  
ANSWER 9: B  

Problem 10:  
Harry: $8 - (2 + 5) = 1$.  
Terry: $8 - 2 + 5 = 11$.  
Difference: $1 - 11 = -10$.  
Tempting wrong choices: 10 (absolute difference) or 0 (if parentheses ignored by both).  
ANSWER 10: A  

Problem 11:  
Sum: $81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99$.  
Average: 90, 10 terms: $90 \times 10 = 900$.  
$2 \times 900 = 1800$.  
Tempting wrong choices: 1600 (min+max=180, times 5=900, times 2=1800, but if miscount terms).  
ANSWER 11: E  

Problem 12:  
Total surveyed: 200.  
Females: 96, so males: $200 - 96 = 104$.  
Total listeners: 136.  
Female listeners: 58, so male listeners: $136 - 58 = 78$.  
Percent male listeners: $\frac{78}{104} = 0.75 = 75\%$.  
Tempting wrong choices: 39 (male non-listeners percent) or 52 (incorrect calculation).  
ANSWER 12: E

### Problem 1:
What is the percentage of yellow tint in the new 35-liter paint mixture, after adding 5 liters of yellow tint to an original 30-liter mixture that was 30% yellow?
First, calculate the original volume of yellow tint: 30% of 30 liters = \(0.3 \times 30 = 9\) liters. Add the 5 new liters of yellow tint to get \(9 + 5 = 14\) liters of yellow in the new mixture. The total new volume of the mixture is \(30 + 5 = 35\) liters. The percentage of yellow is \(\frac{14}{35} = 0.4 = 40\%\). Tempting wrong answers come from forgetting to update the total mixture volume when calculating the new percentage, or miscalculating the original yellow volume.
ANSWER 1: C

---

### Problem 2:
How many ways can Georgie enter through one window and leave through a different window, out of 6 total windows?
Georgie has 6 choices for an entry window. Since he must leave through a different window, he has 5 remaining choices for an exit window. The total number of ways is \(6 \times 5 = 30\). A common wrong answer is 15, which incorrectly treats entry/exit as unordered pairs, when order matters (entering window A and leaving B is different from entering B and leaving A).
ANSWER 2: D

---

### Problem 3:
Simplify \(\frac{1}{10} + \frac{2}{20} + \frac{3}{30}\):
Simplify each fraction first: \(\frac{2}{20} = \frac{1}{10}\) and \(\frac{3}{30} = \frac{1}{10}\). Add them together: \(\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3\). A common wrong answer is 0.1, which incorrectly adds numerators and denominators directly instead of simplifying first.
ANSWER 3: D

---

### Problem 4:
How much money does Susan have left after spending $12 on food and twice that amount on rides, starting with $50?
First calculate the amount spent on rides: twice $12 is \(2 \times 12 = 24\) dollars. Total spent is \(12 + 24 = 36\) dollars. The amount left is \(50 - 36 = 14\) dollars. A common wrong answer is 26, which incorrectly uses $12 instead of $24 for ride costs.
ANSWER 4: B

---

### Problem 5:
Calculate \(200,000 \times 200,000\):
Rewrite 200,000 as \(2 \times 10^5\). Multiplying two copies gives \((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\), which equals 40,000,000,000, or forty billion. Wrong answers come from miscounting the number of zeros in the product.
ANSWER 5: E

---

### Problem 6:
The 3-digit total cost \(1A2\) dollars is divisible by 11 (11 members paid the same integer amount). What is the missing digit A?
The divisibility rule for 11 states that alternating sum of digits must be divisible by 11. For 1A2, the alternating sum is \(1 - A + 2 = 3 - A\). The only value of A (0-9) that makes this sum divisible by 11 is A=3, since \(3-3=0\), which is divisible by 11. Verify: \(132 \div 11 = 12\), a valid integer.
ANSWER 6: D

---

### Problem 7:
Haruki has a 24cm wire. Which of the three shapes can he form? Calculate the perimeter of each shape:
1. Regular hexagon with side 5cm: perimeter = \(6 \times 5 = 30\)cm, which is longer than 24cm, so impossible.
2. Square with area 36cm²: side length = \(\sqrt{36}=6\)cm, perimeter = \(4 \times 6 = 24\)cm, which fits.
3. Right triangle with legs 6cm and 8cm: hypotenuse = 10cm (6-8-10 Pythagorean triple), perimeter = \(6+8+10=24\)cm, which fits.
Only the square and triangle can be formed.
ANSWER 7: D

---

### Problem 8:
Which number has the largest reciprocal? The reciprocal of \(x\) is \(\frac{1}{x}\), so smaller values of \(x\) produce larger reciprocals. List the values of all options:
A. \(\frac{1}{3} \approx 0.333\), B. \(\frac{2}{5}=0.4\), C.1, D.5, E.1986. The smallest value is \(\frac{1}{3}\), whose reciprocal is 3, the largest of all options.
ANSWER 8: A

---

### Problem 9:
Calculate the target heart rate for a 26-year-old athlete: target is 80% of max heart rate, where max = 220 - age.
First calculate max heart rate: \(220 - 26 = 194\) bpm. 80% of 194 is \(0.8 \times 194 = 155.2\), which rounds to 155 bpm. A common wrong answer is 176, which uses 220 instead of the correct max heart rate of 194.
ANSWER 9: B

---

### Problem 10:
Calculate \(H-T\), where H is Harry's correct answer to \(8-(2+5)\), and T is Terry's incorrect answer of \(8-2+5\).
Harry's answer: \(8 - 7 = 1 = H\). Terry's answer: \(6 + 5 = 11 = T\). The difference \(H-T = 1 - 11 = -10\). A common wrong answer is 10, which calculates \(T-H\) instead of \(H-T\).
ANSWER 10: A

---

### Problem 11:
Simplify \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\):
Pair the terms inside the parentheses to simplify addition: \((81+99)+(83+97)+(85+95)+(87+93)+(89+91) = 5 \times 180 = 900\). Multiply by 2: \(2 \times 900 = 1800\).
ANSWER 11: E

---

### Problem 12:
What percentage of males surveyed listen to the radio station? First calculate total number of males: total surveyed = 200, females = 96, so males = \(200 - 96 = 104\). Males who do not listen = 26, so males who listen = \(104 - 26 = 78\). The percentage of males who listen is \(\frac{78}{104} = 0.75 = 75\%\). A common wrong answer is 39%, which divides male listeners by the total surveyed population instead of the total male population.
ANSWER 12: E

### Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.
1.  Calculate original yellow tint volume: 25%? No wait, original is 30% yellow. So 30% of 30 L = $0.3 \times 30 = 9$ liters of yellow.
2.  Add 5 liters of yellow: total yellow = $9 + 5 = 14$ liters.
3.  Total new mixture volume: $30 + 5 = 35$ liters.
4.  Percent yellow: $\frac{14}{35} \times 100 = 40\%$.
ANSWER 1: C
---
### Problem 2:
We count the number of ways to choose an entry window and a different exit window from 6 windows.
1.  Entry window: 6 possible choices.
2.  Exit window: cannot be the entry window, so 5 remaining choices.
3.  Total ways: $6 \times 5 = 30$ (order matters because entry and exit are distinct actions).
ANSWER 2: D
---
### Problem 3:
Simplify and add the fractions:
1.  $\frac{1}{10} = 0.1$, $\frac{2}{20} = \frac{1}{10} = 0.1$, $\frac{3}{30} = \frac{1}{10} = 0.1$.
2.  Sum: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D
---
### Problem 4:
Calculate remaining money after expenses:
1.  Total initial money: $50.
2.  Money spent on rides: twice the food cost = $2 \times 12 = 24$ dollars.
3.  Total spent: $12 + 24 = 36$ dollars.
4.  Remaining: $50 - 36 = 14$ dollars.
ANSWER 4: B
---
### Problem 5:
Compute $200,000 \times 200,000$:
1.  Rewrite as $(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
2.  $10^{10} = 10,000,000,000$, so $4 \times 10^{10} = 40,000,000,000 = 40$ billion.
ANSWER 5: E
---
### Problem 6:
The total payment $1A2$ (a 3-digit number) must be divisible by 11 (since 11 people paid equal integer amounts).
1.  Use the divisibility rule for 11: the alternating sum of digits is divisible by 11. For $1A2$, this is $1 - A + 2 = 3 - A$.
2.  $A$ is a digit (0-9), so $3 - A$ must equal 0 (the only multiple of 11 in range: $3 - A = 11$ gives $A=-8$, $3-A=-11$ gives $A=14$, both invalid).
3.  $3 - A = 0 \implies A=3$. Verify: $132 \div 11 = 12$, which is an integer.
ANSWER 6: D
---
### Problem 7:
Check if the 24 cm wire can form each shape (perimeter of the shape must equal 24 cm):
1.  Regular hexagon (side 5 cm): Perimeter = $6 \times 5 = 30$ cm > 24 cm → cannot make.
2.  Square (area 36 cm²): Side length = $\sqrt{36} = 6$ cm. Perimeter = $4 \times 6 = 24$ cm → can make.
3.  Right triangle (legs 6, 8 cm): Hypotenuse = $\sqrt{6^2 + 8^2} = 10$ cm. Perimeter = $6 + 8 + 10 = 24$ cm → can make.
Only square and triangle can be made.
ANSWER 7: D
---
### Problem 8:
The reciprocal of a positive number $x$ is $\frac{1}{x}$; smaller $x$ gives larger reciprocal.
1.  List values: $\frac{1}{3} \approx 0.33$, $\frac{2}{5}=0.4$, $1$, $5$, $1986$.
2.  Smallest value is $\frac{1}{3}$, whose reciprocal is $3$ (largest among all reciprocals: $\frac{5}{2}=2.5$, $1$, $0.2$, $\frac{1}{1986} \approx 0.0005$).
ANSWER 8: A
---
### Problem 9:
Calculate target heart rate for a 26-year-old:
1.  Maximum heart rate: $220 - 26 = 194$ bpm.
2.  Target heart rate: 80% of 194 = $0.8 \times 194 = 155.2$, which rounds to 155.
ANSWER 9: B
---
### Problem 10:
Compute $H$ (Harry's correct answer) and $T$ (Terry's incorrect answer):
1.  Harry: $8 - (2 + 5) = 8 - 7 = 1$, so $H=1$.
2.  Terry (ignores parentheses, left-to-right): $8 - 2 + 5 = 6 + 5 = 11$, so $T=11$.
3.  Difference: $H - T = 1 - 11 = -10$.
ANSWER 10: A
---
### Problem 11:
Calculate $2 \times (81 + 83 + ... + 99)$:
1.  The sum inside is 10 consecutive odd numbers. Average of first and last term: $\frac{81 + 99}{2} = 90$.
2.  Sum of the 10 terms: $90 \times 10 = 900$.
3.  Multiply by 2: $2 \times 900 = 1800$.
ANSWER 11: E
---
### Problem 12:
Find the percentage of males who listen to the station:
1.  Total males surveyed: $200 - 96 = 104$.
2.  Males who do not listen: 26, so males who listen: $104 - 26 = 78$.
3.  Verify with total listeners: Total listeners = 136, female listeners = 58, so male listeners = $136 - 58 = 78$ (matches).
4.  Percentage: $\frac{78}{104} \times 100 = 75\%$.
ANSWER 12: E

Problem 1:
Original yellow = 30% of 30 = 9 L. Add 5 → 14 L yellow. New total = 35 L. Percent = 14/35 = 40%.

ANSWER 1: C

Problem 2:
Enter: 6 choices, leave: 5 different choices. 6×5 = 30.

ANSWER 2: D

Problem 3:
Each fraction = 0.1. Sum = 0.3.

ANSWER 3: D

Problem 4:
Food 12, rides twice = 24. Spent 36. Left = 50 − 36 = 14.

ANSWER 4: B

Problem 5:
200,000 × 200,000 = 40,000,000,000 = forty billion.

ANSWER 5: E

Problem 6:
1A2 divisible by 11. Alternating sum: 1 − A + 2 = 3 − A must be divisible by 11 → A = 3 (gives 0). Check 132/11 = 12. Yes.

ANSWER 6: D

Problem 7:
Hexagon: 6×5 = 30 > 24, no. Square area 36 → side 6, perimeter 24, yes. Triangle legs 6,8 → hypotenuse 10, perimeter 24, yes. Square and triangle only.

ANSWER 7: D

Problem 8:
Largest reciprocal = smallest number. Smallest is 1/3.

ANSWER 8: A

Problem 9:
Max = 220 − 26 = 194. Target = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

Problem 10:
H = 8 − 7 = 1. T = 8 − 2 + 5 = 11. H − T = 1 − 11 = −10.

ANSWER 10: A

Problem 11:
Sum of 81..99 odd numbers (10 terms): average (81+99)/2 = 90, ×10 = 900. ×2 = 1800.

ANSWER 11: E

Problem 12:
Males total = 200 − 96 = 104. Males who don't listen = 26 → males who listen = 104 − 26 = 78. Percent = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem systematically.

---

**Problem 1:**
Original mixture: 30 liters, 30% yellow = 9 liters yellow. Add 5 liters yellow → new yellow = 14 liters, new total = 35 liters.
Percent yellow = 14/35 = 40%.

ANSWER 1: C

---

**Problem 2:**
Georgie can enter through any of 6 windows, then leave through any of the remaining 5 windows.
6 × 5 = 30.

ANSWER 2: D

---

**Problem 3:**
1/10 = 0.1, 2/20 = 0.1, 3/30 = 0.1. Sum = 0.3.

ANSWER 3: D

---

**Problem 4:**
Food: $12. Rides: 2 × $12 = $24. Total spent: $12 + $24 = $36. Left: $50 − $36 = $14.

ANSWER 4: B

---

**Problem 5:**
200,000 × 200,000 = (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion.

ANSWER 5: E

---

**Problem 6:**
The number 1A2 must be divisible by 11. 1A2 ÷ 11 must be an integer.
11 × 10 = 110, 11 × 11 = 121 (no), 11 × 12 = 132 ✓ → A = 3.
Check: 132/11 = 12 ✓, each paid $12.

ANSWER 6: D

---

**Problem 7:**
Wire = 24 cm.
- Regular hexagon, side 5: perimeter = 6 × 5 = 30 cm > 24. ✗
- Square, area 36: side = 6, perimeter = 24 cm ✓
- Right triangle, legs 6 and 8: hypotenuse = 10, perimeter = 6+8+10 = 24 cm ✓

ANSWER 7: D

---

**Problem 8:**
The reciprocal of x is 1/x. The largest reciprocal comes from the smallest positive number.
Among 1/3, 2/5, 1, 5, 1986 — the smallest is 1/3, and its reciprocal is 3, which is the largest.

ANSWER 8: A

---

**Problem 9:**
Maximum heart rate = 220 − 26 = 194. Target = 80% × 194 = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

---

**Problem 10:**
Harry: 8 − (2 + 5) = 8 − 7 = 1, so H = 1.
Terry: 8 − 2 + 5 = 11, so T = 11.
H − T = 1 − 11 = −10.

ANSWER 10: A

---

**Problem 11:**
The ten odd numbers from 81 to 99: sum = (81+99)/2 × 10 = 90 × 10 = 900.
2 × 900 = 1800.

ANSWER 11: E

---

**Problem 12:**
Total surveyed = 200. Females = 96, so Males = 104.
Total non-listeners = 64. Males who don't listen = 26, so females who don't listen = 64 − 26 = 38.
Males who listen = 104 − 26 = 78.
Percentage = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem step-by-step.

## Problem 1:
**What is asked:** Find the percentage of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.

**Original mixture:**
- Yellow tint: 30% of 30 = 0.30 × 30 = 9 liters
- Total volume: 30 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total volume: 30 + 5 = 35 liters

**Percentage of yellow tint:**
$$\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%$$

ANSWER 1: C

## Problem 2:
**What is asked:** In how many ways can the ghost enter by one window and leave by a different window?

**Solution:**
- Choices for entering: 6 windows
- Choices for leaving: 5 windows (must be different from entry)
- Total ways: 6 × 5 = 30

ANSWER 2: D

## Problem 3:
**What is asked:** Calculate 1/10 + 2/20 + 3/30

**Solution:**
Simplify each fraction:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

ANSWER 3: D

## Problem 4:
**What is asked:** How much money does Susan have left?

**Solution:**
- Started with: $50
- Spent on food: $12
- Spent on rides: 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Left: $50 - $36 = $14

ANSWER 4: B

## Problem 5:
**What is asked:** Calculate 200,000 × 200,000

**Solution:**
$$200,000 × 200,000 = (2 × 10^5) × (2 × 10^5) = 4 × 10^{10} = 40,000,000,000 = 40 \text{ billion}$$

ANSWER 5: E

## Problem 6:
**What is asked:** Find the missing digit A in the 3-digit number 1A2, where 11 club members each paid an equal integer amount that totals 1A2.

**Solution:**
The number 1A2 must be divisible by 11.

Testing divisibility by 11 (alternating sum of digits):
- 1 - A + 2 = 0 (mod 11)
- 3 - A ≡ 0 (mod 11)
- A = 3

Check: 132 ÷ 11 = 12 ✓

ANSWER 6: D

## Problem 7:
**What is asked:** Which shapes can Haruki make with 24 cm of wire?

**Solution:**
Test each shape:

**Regular hexagon with side 5 cm:**
- Perimeter: 6 × 5 = 30 cm
- Need 30 cm, have 24 cm ✗

**Square with area 36 cm²:**
- Side length: √36 = 6 cm
- Perimeter: 4 × 6 = 24 cm ✓

**Right triangle with legs 6 and 8 cm:**
- Hypotenuse: √(6² + 8²) = √(36 + 64) = √100 = 10 cm
- Perimeter: 6 + 8 + 10 = 24 cm ✓

Only square and triangle can be made.

ANSWER 7: D

## Problem 8:
**What is asked:** Which number has the largest reciprocal?

**Solution:**
Find reciprocals:
- 1/(1/3) = 3
- 1/(2/5) = 5/2 = 2.5
- 1/1 = 1
- 1/5 = 0.2
- 1/1986 ≈ 0.0005

The largest reciprocal is 5 (from the number 1/5).

ANSWER 8: D

## Problem 9:
**What is asked:** Find the target heart rate for a 26-year-old athlete.

**Solution:**
- Maximum heart rate: 220 - 26 = 194 beats per minute
- Target heart rate: 80% of 194 = 0.80 × 194 = 155.2 ≈ 155

ANSWER 9: B

## Problem 10:
**What is asked:** Find H - T where H is Harry's answer and T is Terry's answer.

**Solution:**
**Harry's calculation (correct):**
$$H = 8 - (2 + 5) = 8 - 7 = 1$$

**Terry's calculation (ignoring parentheses):**
$$T = 8 - 2 + 5 = 6 + 5 = 11$$

**Difference:**
$$H - T = 1 - 11 = -10$$

ANSWER 10: A

## Problem 11:
**What is asked:** Calculate 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

**Solution:**
This is an arithmetic sequence with 10 terms. Sum = n(first + last)/2
$$81 + 83 + ... + 99 = \frac{10(81 + 99)}{2} = \frac{10 × 180}{2} = 900$$

$$2 × 900 = 1800$$

ANSWER 11: E

## Problem 12:
**What is asked:** What percentage of males surveyed listen to the station?

**Solution:**
- Total surveyed: 200
- Females: 96
- Males: 200 - 96 = 104

- Total non-listeners: 64
- Males who don't listen: 26
- Males who listen: 104 - 26 = 78

**Percentage of males who listen:**
$$\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%$$

ANSWER 12: E

Problem 1:

The original mixture is 30 liters total, with 30% yellow tint.

So the amount of yellow tint at first is:
- \(30\%\) of \(30 = 0.30 \times 30 = 9\) liters

Then 5 liters of yellow tint are added, so the new amount of yellow tint is:
- \(9 + 5 = 14\) liters

The new total mixture is:
- \(30 + 5 = 35\) liters

So the percent of yellow tint is:
- \(\frac{14}{35} = \frac{2}{5} = 0.4 = 40\%\)

Answer choice C matches 40%.

ANSWER 1: C

Problem 2:

Georgie can:
- choose 1 of the 6 windows to enter
- then choose a different window to leave

For the entry, there are 6 choices.

For the exit, there are 5 remaining choices, since it must be a different window.

So the total number of ways is:
- \(6 \times 5 = 30\)

ANSWER 2: D

Problem 3:

Compute each fraction:
- \(\frac{1}{10} = 0.1\)
- \(\frac{2}{20} = \frac{1}{10} = 0.1\)
- \(\frac{3}{30} = \frac{1}{10} = 0.1\)

Add them:
- \(0.1 + 0.1 + 0.1 = 0.3\)

So the value is \(0.3\).

ANSWER 3: D

Problem 4:

Susan had \$50.

She spent:
- \$12 on food
- twice as much on rides, so \(2 \times 12 = 24\)

Total spent:
- \(12 + 24 = 36\)

Money left:
- \(50 - 36 = 14\)

ANSWER 4: B

Problem 5:

Two hundred thousand is:
- \(200{,}000 = 2 \times 10^5\)

Multiply:
- \(200{,}000 \times 200{,}000 = (2 \times 10^5)(2 \times 10^5) = 4 \times 10^{10}\)

And \(4 \times 10^{10} = 40{,}000{,}000{,}000\), which is forty billion.

ANSWER 5: E

Problem 6:

The total is written as \(1A2\), and 11 members each paid the same integer amount.

So \(1A2\) must be divisible by 11.

Test the answer choices:
- 102 ÷ 11 is not an integer
- 112 ÷ 11 is not an integer
- 122 ÷ 11 = 11 remainder 1
- 132 ÷ 11 = 12 exactly
- 142 ÷ 11 is not an integer

So the number is 132, which means \(A = 3\).

ANSWER 6: D

Problem 7:

We check each shape.

### Hexagon
A regular hexagon has 6 equal sides.
If each side is 5 cm, perimeter is:
- \(6 \times 5 = 30\) cm

The wire is only 24 cm, so he cannot make the hexagon.

### Square
A square with area 36 has side length:
- \(\sqrt{36} = 6\) cm

Perimeter:
- \(4 \times 6 = 24\) cm

So he can make the square exactly.

### Right triangle
The legs are 6 and 8, so the hypotenuse is:
- \(\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)

Perimeter:
- \(6 + 8 + 10 = 24\) cm

So he can make the triangle exactly.

Therefore he can make the square and the triangle only.

ANSWER 7: D

Problem 8:

The reciprocal is largest when the original number is smallest, assuming positive numbers.

Compare:
- \(\frac13 \approx 0.333\)
- \(\frac25 = 0.4\)
- \(1\)
- \(5\)
- \(1986\)

The smallest number is \(\frac13\), so it has the largest reciprocal.

Its reciprocal is 3, which is larger than the reciprocals of all the others.

ANSWER 8: A

Problem 9:

Maximum heart rate:
- \(220 - 26 = 194\)

Target heart rate is 80% of that:
- \(0.8 \times 194 = 155.2\)

To the nearest whole number, that is 155.

ANSWER 9: B

Problem 10:

First, Harry uses the parentheses:
- \(8 - (2 + 5) = 8 - 7 = 1\)

So \(H = 1\).

Terry ignores parentheses:
- \(8 - 2 + 5 = 6 + 5 = 11\)

So \(T = 11\).

Difference:
- \(H - T = 1 - 11 = -10\)

ANSWER 10: A

Problem 11:

We need to evaluate:
- \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

These are 10 odd numbers from 81 to 99.

Instead of adding one by one, use the average:
- average of 81 and 99 is \( \frac{81+99}{2} = 90\)

Since there are 10 numbers, their sum is:
- \(10 \times 90 = 900\)

Now multiply by 2:
- \(2 \times 900 = 1800\)

ANSWER 11: E

Problem 12:

We want the percentage of males surveyed who listen.

From the table:
- Total surveyed = 200
- Females = 96
- So males = \(200 - 96 = 104\)

Males who don’t listen = 26, so males who do listen:
- \(104 - 26 = 78\)

Now find the percent of males who listen:
- \(\frac{78}{104} = \frac{3}{4} = 75\%\)

ANSWER 12: E

Problem 1:  
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?  

- Original yellow tint amount = 30 × 0.30 = 9 liters.  
- Add 5 liters of yellow tint → new yellow amount = 9 + 5 = 14 liters.  
- New total volume = 30 + 5 = 35 liters.  
- New percent yellow = (14 / 35) × 100% = (2/5) × 100% = 40%.  

So the correct choice is 40.  

ANSWER 1: C  

---

Problem 2:  
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?  

- Choose an entry window: 6 choices.  
- Choose a different exit window: 5 choices (cannot be the same as entry).  
- Total ways = 6 × 5 = 30.  

ANSWER 2: D  

---

Problem 3:  
Compute: \( \frac{1}{10} + \frac{2}{20} + \frac{3}{30} \).  

Convert each to a common denominator, 60:  
- \( \frac{1}{10} = \frac{6}{60} \)  
- \( \frac{2}{20} = \frac{2}{20} = \frac{6}{60} \) (since multiply top and bottom by 3)  
- \( \frac{3}{30} = \frac{3}{30} = \frac{6}{60} \) (since multiply top and bottom by 2)  

Add:  
\( \frac{6}{60} + \frac{6}{60} + \frac{6}{60} = \frac{18}{60} = \frac{3}{10} = 0.3 \)

So the value is .3.  

ANSWER 3: D  

---

Problem 4:  
Susan had $50 to spend. She spent $12 on food and twice as much on rides. How much left?  

- Food = 12  
- Rides = twice 12 = 24  
- Total spent = 12 + 24 = 36  
- Money left = 50 − 36 = 14  

ANSWER 4: B  

---

Problem 5:  
Two hundred thousand times two hundred thousand equals ?  

\(200{,}000 \times 200{,}000 = (2\times 10^5)(2\times 10^5) = 4 \times 10^{10}\).  
- \(10^{10} = 10{,}000{,}000{,}000\)  
- \(4 \times 10^{10} = 40{,}000{,}000{,}000 = 40\) billion  

ANSWER 5: E  

---

Problem 6:  
Eleven members each paid the same integer amount for a guest speaker. In all, they paid $1A2 (a 3-digit number). Find missing digit A.  

- Total = \(1A2\) means:  
  \(1A2 = 100 + 10A + 2 = 102 + 10A\).  
- This total is divided equally among 11 people:  
  So \(102 + 10A\) must be divisible by 11.  

Check \(102 + 10A \mod 11\):  
- \(102 \div 11 = 9\) remainder 3, so \(102 \equiv 3 \pmod{11}\).  
- \(10A \equiv -A \pmod{11}\) (since 10 ≡ −1 mod 11).  

So:  
\(102 + 10A \equiv 3 - A \equiv 0 \pmod{11}\)  
Thus \(A \equiv 3 \pmod{11}\).  
Since \(A\) is a digit 0–9, \(A=3\).  

ANSWER 6: D  

---

Problem 7:  
Wire length = 24 cm. Which shapes can be made (using all 24 cm of wire)?  

We need the perimeter (total boundary length) of each shape:

**(1) Regular hexagon with side length 5 cm**  
Perimeter = 6 × 5 = 30 cm → too long. Cannot.

**(2) Square with area 36 cm²**  
Side length = √36 = 6 cm  
Perimeter = 4 × 6 = 24 cm → fits. Can.

**(3) Right triangle with legs 6 and 8 cm**  
Right triangle sides are 6 and 8 legs, and hypotenuse:  
\( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)  
Perimeter = 6 + 8 + 10 = 24 cm → fits. Can.

So the shapes possible: **square and triangle only**.  

ANSWER 7: D  

---

Problem 8:  
Which number has the largest reciprocal? Recall: the reciprocal of \(x\) is \(1/x\).  
We compare \(1/3, 2/5, 1, 1/5, 1/1986\).  

Compute roughly:  
- \(1/3 \approx 0.333\)  
- \(2/5 = 0.4\)  
- \(1 = 1\)  
- \(1/5 = 0.2\)  
- \(1/1986\) is tiny  

Largest is 1 (reciprocal of 1).  

ANSWER 8: C  

---

Problem 9:  
Target heart rate = 80% of (maximum heart rate).  
Maximum heart rate = 220 − age.  
Age = 26.

- Maximum = 220 − 26 = 194  
- Target = 0.80 × 194 = 155.2  
Nearest whole number = 155  

ANSWER 9: B  

---

Problem 10:  
Expression: \(8 - (2 + 5)\)  
- Harry computes correctly:  
  \(2 + 5 = 7\), so \(8 - 7 = 1\).  
- Terry ignores parentheses:  
  \(8 - 2 + 5 = 6 + 5 = 11\).  

Difference \(H - T = 1 - 11 = -10\).  

ANSWER 10: A  

---

Problem 11:  
Compute:  
\(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

Notice these are all odd numbers from 81 to 99 stepping by 2.  
There are 10 terms.

Pair from ends:  
- 81 + 99 = 180  
- 83 + 97 = 180  
- 85 + 95 = 180  
- 87 + 93 = 180  
- 89 + 91 = 180  

There are 5 pairs, each sum 180:  
Sum = 5 × 180 = 900  

Now multiply by 2:  
2 × 900 = 1800  

ANSWER 11: E  

---

Problem 12:  
Total surveyed = 200  
Females = 96  
Females who listen = 58  
Males who don’t listen = 26  
Total listeners = 136  

We want: % of males who listen.

Step 1: Find males total.  
Males = 200 − 96 = 104

Step 2: Find males who don’t listen is given = 26.  
So males who listen = 104 − 26 = 78

Step 3: Convert to percent of males who listen:  
Percent = (78 / 104) × 100%  
Simplify 78/104 = 3/4 (divide by 26):  
So percent = (3/4) × 100% = 75%

ANSWER 12: E

Problem 1:
The original mixture is 30 liters. The amount of yellow tint is 30% of 30 liters: $0.30 \times 30 = 9$ liters.
Five liters of yellow tint are added, so the new amount of yellow tint is $9 + 5 = 14$ liters.
The new total volume of the mixture is $30 + 5 = 35$ liters.
The percentage of yellow tint is $\frac{14}{35} \times 100\%$.
Simplifying the fraction: $\frac{14}{35} = \frac{2}{5} = 0.4$.
$0.4 \times 100\% = 40\%$.
ANSWER 1: C

Problem 2:
Georgie has 6 choices for the window he enters. Since he must leave by a different window, he has $6 - 1 = 5$ choices for the window he leaves by.
The total number of ways is $6 \times 5 = 30$.
ANSWER 2: D

Problem 3:
Simplify each fraction:
$1/10 = 0.1$
$2/20 = 1/10 = 0.1$
$3/30 = 1/10 = 0.1$
Summing them: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides, which is $2 \times 12 = 24$.
Total spent: $12 + 24 = 36$.
Amount left: $50 - 36 = 14$.
ANSWER 4: B

Problem 5:
Two hundred thousand is $200,000$ (or $2 \times 10^5$).
$(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
$4 \times 10^{10} = 40,000,000,000$, which is forty billion.
ANSWER 5: E

Problem 6:
The total amount $1A2$ must be divisible by 11.
A number is divisible by 11 if the alternating sum of its digits is divisible by 11.
For $1A2$, the alternating sum is $1 - A + 2 = 3 - A$.
For this to be 0 (the only multiple of 11 possible for a single digit $A$), $3 - A = 0$, so $A = 3$.
Check: $132 / 11 = 12$. This works.
ANSWER 6: D

Problem 7:
Hexagon: Perimeter of a regular hexagon with side 5 is $6 \times 5 = 30$ cm. (Cannot make)
Square: Area is 36, so side is $\sqrt{36} = 6$. Perimeter is $4 \times 6 = 24$ cm. (Can make)
Triangle: Right triangle with legs 6 and 8. Hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. Perimeter is $6 + 8 + 10 = 24$ cm. (Can make)
ANSWER 7: D

Problem 8:
Reciprocals:
A: $1/(1/3) = 3$
B: $1/(2/5) = 2.5$
C: $1/1 = 1$
D: $1/5 = 0.2$
E: $1/1986 \approx 0.0005$
The largest is 3.
ANSWER 8: A

Problem 9:
Max heart rate = $220 - 26 = 194$.
Target heart rate = $80\%$ of $194 = 0.8 \times 194 = 155.2$.
To the nearest whole number, this is 155.
ANSWER 9: B

Problem 10:
$H = 8 - (2 + 5) = 8 - 7 = 1$.
$T = 8 - 2 + 5 = 6 + 5 = 11$.
$H - T = 1 - 11 = -10$.
ANSWER 10: A

Problem 11:
Sum inside parentheses: $81+83+85+87+89+91+93+95+97+99$.
This is an arithmetic progression with 10 terms.
Sum = $\frac{n}{2}(\text{first} + \text{last}) = \frac{10}{2}(81 + 99) = 5 \times 180 = 900$.
The expression is $2 \times 900 = 1800$.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96.
Males = $200 - 96 = 104$.
Males who don't listen = 26.
Males who listen = $104 - 26 = 78$.
Percentage of males who listen = $(78 / 104) \times 100\%$.
$\frac{78}{104} = \frac{39}{52} = \frac{3}{4} = 0.75 = 75\%$.
ANSWER 12: E

Problem 1: The original mixture has volume 30 L, of which 30% is yellow tint, so there are \(0.3 \times 30 = 9\) L of yellow tint. Adding 5 L of yellow tint produces a new total of 14 L of yellow tint and a new mixture volume of 35 L. The new percentage of yellow tint is therefore \((14/35) \times 100\% = 40\%\).

ANSWER 1: C

Problem 2: There are 6 choices for the entry window. For each of those choices there are then exactly 5 remaining choices for a different exit window, giving a total of \(6 \times 5 = 30\) ordered pairs of distinct windows.

ANSWER 2: D

Problem 3: Rewrite each term with a common denominator of 10:  
\(1/10 = 0.1\),  
\(2/20 = 1/10 = 0.1\),  
\(3/30 = 1/10 = 0.1\).  
Their sum is \(0.1 + 0.1 + 0.1 = 0.3\).

ANSWER 3: D

Problem 4: Twice the food cost is \(2 \times 12 = 24\) dollars spent on rides. The total spent is therefore \(12 + 24 = 36\) dollars, leaving \(50 - 36 = 14\) dollars.

ANSWER 4: B

Problem 5: Write the factors in scientific notation:  
\(200000 = 2 \times 10^5\), so  
\((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\),  
which is forty billion.

ANSWER 5: E

Problem 6: The three-digit total \(1A2\) must be divisible by 11. The alternating-sum divisibility rule requires that \(1 - A + 2 = 3 - A\) be a multiple of 11. The only admissible digit \(A\) (0-9) that satisfies the condition is \(A = 3\), giving the total 132, which is exactly \(11 \times 12\).

ANSWER 6: D

Problem 7: A regular hexagon with side 5 cm has perimeter 30 cm, which exceeds the 24 cm length of wire, so it is impossible. A square of area 36 cm² has side length 6 cm and perimeter 24 cm, which matches the wire length. A right triangle with legs 6 cm and 8 cm has hypotenuse 10 cm, so its perimeter is \(6 + 8 + 10 = 24\) cm, which also matches. Consequently only the square and the triangle are possible.

ANSWER 7: D

Problem 8: For positive numbers the reciprocal function is strictly decreasing, so the largest reciprocal belongs to the smallest number in the list, which is \(1/3\).

ANSWER 8: A

Problem 9: The theoretical maximum heart rate is \(220 - 26 = 194\) beats per minute. Eighty percent of that value is \(0.8 \times 194 = 155.2\), which rounds to the nearest whole number 155.

ANSWER 9: B

Problem 10: Harry evaluates the parentheses first and obtains \(8 - (2 + 5) = 1\). Terry omits the parentheses and obtains \(8 - 2 + 5 = 11\). Their difference is therefore \(1 - 11 = -10\).

ANSWER 10: A

Problem 11: The sum inside the parentheses consists of the ten consecutive odd integers from 81 to 99. Their average is \((81 + 99)/2 = 90\), so the sum equals \(10 \times 90 = 900\). Multiplying by the outer factor of 2 yields \(2 \times 900 = 1800\).

ANSWER 11: E

Problem 12: There are 200 people surveyed and 96 females, so there are \(200 - 96 = 104\) males. Of the 96 females, 58 listen, leaving \(96 - 58 = 38\) females who do not listen. Given that 26 males do not listen, the number of males who listen is \(104 - 26 = 78\). The required percentage is therefore \((78/104) \times 100\% = 75\%\).

ANSWER 12: E

Problem 1:
The problem asks us to find the percentage of yellow tint in a new mixture after adding 5 liters of yellow tint to the original 30 liters of paint mixture. The original mixture is 25% red tint, 30% yellow tint, and 45% water.

First, let's calculate the amount of yellow tint in the original mixture. 30% of 30 liters is $0.3 \times 30 = 9$ liters.

After adding 5 liters of yellow tint, the total amount of yellow tint becomes $9 + 5 = 14$ liters.

The total volume of the new mixture is $30 + 5 = 35$ liters.

To find the percentage of yellow tint in the new mixture, we divide the amount of yellow tint by the total volume and multiply by 100: $\frac{14}{35} \times 100 = 40\%$.

Let's examine the choices:
- A. 25: Too low because the percentage of yellow tint increased.
- B. 35: Not the correct calculation.
- C. 40: Matches our calculation.
- D. 45: Too high.
- E. 50: Too high.

The correct percentage is 40%.

ANSWER 1: C

Problem 2:
The problem asks us to determine the number of ways Georgie the Ghost can enter a haunted house through one of six windows and leave by a different window.

To enter, Georgie has 6 choices. After entering, there are 5 remaining windows to choose from for leaving.

The total number of ways to enter and leave is $6 \times 5 = 30$.

Let's examine the choices:
- A. 12: Too low.
- B. 15: Too low.
- C. 18: Too low.
- D. 30: Matches our calculation.
- E. 36: Too high because it would imply Georgie can leave by the same window.

The correct number of ways is 30.

ANSWER 2: D

Problem 3:
The problem requires us to simplify the expression $1⁄10 + 2⁄20 + 3⁄30$.

First, simplify each fraction:
- $1⁄10$ stays the same.
- $2⁄20 = 1⁄10$.
- $3⁄30 = 1⁄10$.

Adding them together: $1⁄10 + 1⁄10 + 1⁄10 = 3⁄10 = 0.3$.

Let's examine the choices:
- A. .1: Too low.
- B. .123: Not the correct sum.
- C. .2: Too low.
- D. .3: Matches our calculation.
- E. .6: Too high.

The correct sum is 0.3.

ANSWER 3: D

Problem 4:
The problem asks us to find out how many dollars Susan had left after spending money on food and rides at the carnival. She started with $50, spent $12 on food, and twice as much on rides.

First, calculate the amount spent on rides: $12 \times 2 = $24.

Total amount spent is $12 + $24 = $36.

The amount left is $50 - $36 = $14.

Let's examine the choices:
- A. 12: Too low.
- B. 14: Matches our calculation.
- C. 26: Too high.
- D. 38: Too high.
- E. 50: Incorrect because she spent money.

The correct amount left is $14.

ANSWER 4: B

Problem 5:
The problem asks us to calculate $200,000 \times 200,000$.

$200,000 = 2 \times 10^5$, so $200,000 \times 200,000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10} = 40,000,000,000$.

Let's examine the choices:
- A. four hundred thousand: Too low.
- B. four million: Too low.
- C. forty thousand: Too low.
- D. four hundred million: Too low.
- E. forty billion: Matches our calculation.

The correct result is forty billion.

ANSWER 5: E

Problem 6:
The problem requires us to find the missing digit A in the 3-digit number $1A2 that represents the total amount paid by 11 members of the Math Club.

The total amount paid is divisible by 11 since there are 11 members who paid the same amount.

To check divisibility by 11, we use the rule: the difference between the sum of digits at odd places and the sum of digits at even places should be divisible by 11.

For $1A2, the sum of digits at odd places is $1 + 2 = 3$, and the sum at even places is $A$. The difference is $3 - A$.

For $3 - A$ to be divisible by 11 (or 0, since $A$ is a single digit), $A = 3$ makes $3 - A = 0$, which is divisible by 11.

Let's examine the choices:
- A. 0: Not correct because $3 - 0 = 3$, not divisible by 11.
- B. 1: Not correct because $3 - 1 = 2$, not divisible by 11.
- C. 2: Not correct because $3 - 2 = 1$, not divisible by 11.
- D. 3: Correct because $3 - 3 = 0$, which is divisible by 11.
- E. 4: Not correct because $3 - 4 = -1$, not divisible by 11.

The missing digit A is 3, but we should verify if 132 is indeed divisible by 11. $132 / 11 = 12$, so it is.

ANSWER 6: D

Problem 7:
The problem asks us to determine which shapes Haruki can form using a 24 cm long wire.

1. A regular hexagon with side length 5 cm: The perimeter is $6 \times 5 = 30$ cm, which is more than 24 cm. So, Haruki cannot form this hexagon.

2. A square of area 36 cm$^2$: The side length is $\sqrt{36} = 6$ cm. The perimeter is $4 \times 6 = 24$ cm, which matches the wire length. So, Haruki can form this square.

3. A right triangle with legs 6 and 8 cm: Using the Pythagorean theorem, the hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm. The perimeter is $6 + 8 + 10 = 24$ cm, which matches the wire length. So, Haruki can form this triangle.

Let's examine the choices:
- A. Triangle only: Incorrect because Haruki can also form the square.
- B. Hexagon and square only: Incorrect because Haruki cannot form the hexagon.
- C. Hexagon and triangle only: Incorrect because Haruki cannot form the hexagon.
- D. Square and triangle only: Matches our findings.
- E. Hexagon, triangle, and square: Incorrect because Haruki cannot form the hexagon.

The correct shapes Haruki can form are the square and the triangle.

ANSWER 7: D

Problem 8:
The problem asks us to find which of the given numbers has the largest reciprocal.

The reciprocal of a number $x$ is $1/x$. The larger $x$ is, the smaller its reciprocal is, and vice versa.

Let's examine the choices:
- A. $1⁄3$: The reciprocal is 3.
- B. $2⁄5$: The reciprocal is $5/2 = 2.5$.
- C. 1: The reciprocal is 1.
- D. 5: The reciprocal is $1/5 = 0.2$.
- E. 1986: The reciprocal is $1/1986$, very small.

Comparing the reciprocals: $3 > 2.5 > 1 > 0.2 > 1/1986$. So, $1/3$ has the largest reciprocal.

Let's examine the choices again:
- A. $1⁄3$: Correct.
- B. $2⁄5$: Not correct.
- C. 1: Not correct.
- D. 5: Not correct.
- E. 1986: Not correct.

The number with the largest reciprocal is $1/3$.

ANSWER 8: A

Problem 9:
The proble

Problem 1:
Original mixture: 30 L. Yellow tint = 30% of 30 = 0.30 × 30 = 9 L.
Add 5 L yellow: new yellow = 9 + 5 = 14 L, new total volume = 30 + 5 = 35 L.
Percent yellow = (14/35) × 100% = 40%.
ANSWER 1: C

Problem 2:
6 choices for entry window, 5 remaining choices for a different exit window.
Total ways = 6 × 5 = 30.
ANSWER 2: D

Problem 3:
1/10 + 2/20 + 3/30 = 1/10 + 1/10 + 1/10 = 3/10 = 0.3.
ANSWER 3: D

Problem 4:
Spent on food: $12. Spent on rides: 2 × $12 = $24.
Total spent = $12 + $24 = $36. Left = $50 − $36 = $14.
ANSWER 4: B

Problem 5:
200,000 × 200,000 = (2×10⁵) × (2×10⁵) = 4×10¹⁰ = 40,000,000,000, which is forty billion.
ANSWER 5: E

Problem 6:
Total paid is $1A2, a multiple of 11. Divisibility by 11: (1 − A + 2) = 3 − A must be a multiple of 11.
With A a digit 0–9, 3 − A = 0 ⇒ A = 3. (Check: 132 ÷ 11 = 12.)
ANSWER 6: D

Problem 7:
Wire length = 24 cm.
- Regular hexagon, side 5 cm: perimeter = 6×5 = 30 cm > 24 cm → cannot make.
- Square, area 36 cm²: side = 6 cm, perimeter = 4×6 = 24 cm → can make.
- Right triangle, legs 6 and 8 cm: hypotenuse = 10 cm, perimeter = 6+8+10 = 24 cm → can make.
So square and triangle only.
ANSWER 7: D

Problem 8:
Reciprocals: 1/(1/3)=3, 1/(2/5)=2.5, 1/1=1, 1/5=0.2, 1/1986≈0.0005. Largest is 3, from 1/3.
ANSWER 8: A

Problem 9:
Maximum heart rate = 220 − 26 = 194 bpm.
Target = 80% of 194 = 0.8 × 194 = 155.2 → 155 to nearest whole number.
ANSWER 9: B

Problem 10:
Harry: H = 8 − (2+5) = 8 − 7 = 1.
Terry: T = 8 − 2 + 5 = 6 + 5 = 11.
H − T = 1 − 11 = −10.
ANSWER 10: A

Problem 11:
Sum inside parentheses: 10 terms, average = (81+99)/2 = 90, sum = 10×90 = 900.
2 × 900 = 1800.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96 ⇒ Males = 200 − 96 = 104.
Total listeners = 136, female listeners = 58 ⇒ male listeners = 136 − 58 = 78.
Percent of males who listen = (78/104) × 100% = 75%.
ANSWER 12: E

Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint.

**Original mixture (30 liters):**
- Red tint: 25% of 30 = 7.5 liters
- Yellow tint: 30% of 30 = 9 liters
- Water: 45% of 30 = 13.5 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total mixture: 30 + 5 = 35 liters

**New percentage of yellow tint:**
14 / 35 = 2/5 = 0.40 = 40%

Tempting wrong answers: Choice B (35) might come from just adding 5 to the original 30%, but we must recalculate based on the new total.

ANSWER 1: C

---

Problem 2:
Georgie enters through one window and leaves through a *different* window.

- Number of choices for the entry window: 6
- Number of choices for the exit window (must be different): 5
- Total ways: 6 × 5 = 30

Tempting wrong answers: Choice E (36) would be 6 × 6, which allows entering and leaving through the same window. Choice A (12) would be 6 + 6, which doesn't account for all combinations.

ANSWER 2: D

---

Problem 3:
Simplify each fraction first:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

Tempting wrong answers: Choice B (.123) might come from concatenating the digits 1, 2, 3 instead of actually computing. Choice C (.2) might come from miscounting the terms.

ANSWER 3: D

---

Problem 4:
Susan starts with $50.
- Spent on food: $12
- Spent on rides: twice the food amount = 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Amount left: $50 − $36 = $14

Tempting wrong answers: Choice C (26) might come from subtracting only the food ($50 − $24 = $26, forgetting food). Choice D (38) comes from subtracting only $12.

ANSWER 4: B

---

Problem 5:
200,000 × 200,000

Write in scientific notation:
(2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰

4 × 10¹⁰ = 40,000,000,000 = forty billion

Tempting wrong answers: Choice A (four hundred thousand) comes from adding instead of multiplying. Choice D (four hundred million) comes from miscounting zeros.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, so the total $1A2 must be divisible by 11.

**Divisibility rule for 11:** The alternating sum of digits must be a multiple of 11.
Alternating sum: 1 − A + 2 = 3 − A

For 3 − A to be divisible by 11 (and A to be a single digit 0–9):
3 − A = 0 → A = 3

**Check:** $132 ÷ 11 = $12 per member. This is a valid integer amount.

Tempting wrong answers: Other values of A don't make 3 − A a multiple of 11 within the digit range.

ANSWER 6: D

---

Problem 7:
Wire length = 24 cm. We check if each shape's perimeter ≤ 24 cm.

**Regular hexagon (side 5 cm):**
Perimeter = 6 × 5 = 30 cm. Since 30 > 24, he **cannot** make this.

**Square (area 36 cm²):**
Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Since 24 = 24, he **can** make this.

**Right triangle (legs 6 and 8 cm):**
Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Perimeter = 6 + 8 + 10 = 24 cm. Since 24 = 24, he **can** make this.

Haruki can make the square and triangle only.

Tempting wrong answers: Choice E includes the hexagon, but its perimeter exceeds the wire length.

ANSWER 7: D

---

Problem 8:
The reciprocal of a number x is 1/x. The *smaller* the positive number, the *larger* its reciprocal.

| Choice | Number | Reciprocal |
|--------|--------|------------|
| A | 1/3 | 3 |
| B | 2/5 | 5/2 = 2.5 |
| C | 1 | 1 |
| D | 5 | 1/5 = 0.2 |
| E | 1986 | 1/1986 ≈ 0.0005 |

The largest reciprocal is 3, which comes from 1/3.

Tempting wrong answers: Choice E (1986) is the largest number, so it has the *smallest* reciprocal.

ANSWER 8: A

---

Problem 9:
Athlete's age = 26 years.

**Step 1:** Maximum heart rate = 220 − 26 = 194 beats per minute.

**Step 2:** Target heart rate = 80% of 194 = 0.80 × 194 = 155.2

**Step 3:** Round to nearest whole number: 155

Tempting wrong answers: Choice C (176) might come from taking 80% of 220 without subtracting the age. Choice D (194) is the maximum heart rate itself.

ANSWER 9: B

---

Problem 10:
**Harry (correct calculation):**
H = 8 − (2 + 5) = 8 − 7 = 1

**Terry (ignores parentheses):**
T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:**
H − T = 1 − 11 = −10

Tempting wrong answers: Choice E (10) is |H − T|, but the question asks for H − T specifically, which is negative since Terry's answer is larger.

ANSWER 10: A

---

Problem 11:
We need: 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

The sum inside is an arithmetic sequence:
- First term: 81, Last term: 99
- Number of terms: 10
- Sum = (number of terms) × (first + last) / 2 = 10 × (81 + 99) / 2 = 10 × 180 / 2 = 10 × 90 = 900

Multiply by 2: 2 × 900 = 1800

Tempting wrong answers: Choice C (1700) might come from miscounting terms or miscalculating the average.

ANSWER 11: E

---

Problem 12:
Let me organize the data from the survey:

- Total surveyed: 200
- Females: 96 → Males: 200 − 96 = 104
- Total listeners: 136
- Females who listen: 58 → Males who listen: 136 − 58 = 78
- Males who don't listen: 26 (Check: 78 + 26 = 104 ✓)

**Percentage of males who listen:**
(78 / 104) × 100% = (3/4) × 100% = 75%

Tempting wrong answers: Choice C (52) might come from computing 104/200 (percentage of total who are male). Choice D (55) might come from 78/136 (percentage of listeners who are male).

ANSWER 12: E

 Problem 1:
We start with 30 liters of paint. The original amount of yellow tint is 30% of 30 liters:
\[
0.30 \times 30 = 9 \text{ liters.}
\]
Five liters of yellow tint are added, so the new amount of yellow tint is:
\[
9 + 5 = 14 \text{ liters.}
\]
The new total volume of the mixture is:
\[
30 + 5 = 35 \text{ liters.}
\]
The percent of yellow tint in the new mixture is:
\[
\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%.
\]
(Choice A is the original red percentage, D is the original water percentage, and E would require 17.5 liters of yellow, which we do not have.)
ANSWER 1: C

Problem 2:
Georgie chooses one of 6 windows to enter. Since he must leave through a different window, there are 5 remaining choices for his exit. By the multiplication principle:
\[
6 \times 5 = 30 \text{ ways.}
\]
(Choice E, 36, would count entering and leaving through the same window; the other choices are too small.)
ANSWER 2: D

Problem 3:
Simplify each fraction:
\[
\frac{2}{20} = \frac{1}{10}, \qquad \frac{3}{30} = \frac{1}{10}.
\]
So the sum is:
\[
\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3.
\]
(Choice A is just one term; C is 0.2; E is 0.6.)
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides:
\[
2 \times 12 = 24 \text{ dollars.}
\]
Total spent:
\[
12 + 24 = 36 \text{ dollars.}
\]
Amount left:
\[
50 - 36 = 14 \text{ dollars.}
\]
(Choice C is the total spent; D is the amount left if she had spent only $12.)
ANSWER 4: B

Problem 5:
\[
200{,}000 \times 200{,}000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}.
\]
\[
4 \times 10^{10} = 40{,}000{,}000{,}000,
\]
which is read as forty billion.
(Choice D is $4 \times 10^8$; B is $4 \times 10^6$.)
ANSWER 5: E

Problem 6:
The total amount paid is the three-digit number $1A2$, which equals:
\[
100 + 10A + 2 = 102 + 10A.
\]
Since 11 members each paid the same integer amount, $102 + 10A$ must be divisible by 11. Testing the choices:
- $A=0$: $102 \div 11 \approx 9.27$ (not an integer)
- $A=1$: $112 \div 11 \approx 10.18$ (not an integer)
- $A=2$: $122 \div 11 \approx 11.09$ (not an integer)
- $A=3$: $132 \div 11 = 12$ (integer!)
- $A=4$: $142 \div 11 \approx 12.9$ (not an integer)
Thus the missing digit is 3.
ANSWER 6: D

Problem 7:
The wire is 24 cm long, so a shape is possible only if its perimeter is at most 24 cm.
- **Regular hexagon, side 5 cm:** Perimeter $= 6 \times 5 = 30$ cm. $30 > 24$, so impossible.
- **Square of area 36 cm²:** Side $= \sqrt{36} = 6$ cm. Perimeter $= 4 \times 6 = 24$ cm. Exactly fits, so possible.
- **Right triangle with legs 6 and 8 cm:** Hypotenuse $= \sqrt{6^2 + 8^2} = \sqrt{100} = 10$ cm. Perimeter $= 6 + 8 + 10 = 24$ cm. Exactly fits, so possible.
Only the square and the triangle can be formed.
ANSWER 7: D

Problem 8:
For positive numbers, the reciprocal function $f(x) = \frac{1}{x}$ is decreasing, so the smallest number has the largest reciprocal. Compare the two smallest choices:
\[
\frac{1}{3} \approx 0.333\ldots, \qquad \frac{2}{5} = 0.4.
\]
Since $\frac{1}{3} < \frac{2}{5} < 1 < 5 < 1986$, the smallest number is $\frac{1}{3}$. Its reciprocal is $3$, which is larger than the reciprocals of all the other choices.
ANSWER 8: A

Problem 9:
First find the theoretical maximum heart rate:
\[
220 - 26 = 194 \text{ beats per minute.}
\]
The target rate is 80% of this:
\[
0.80 \times 194 = 155.2.
\]
Rounded to the nearest whole number, this is 155.
(Choice C is the maximum heart rate itself.)
ANSWER 9: B

Problem 10:
Harry computes correctly:
\[
H = 8 - (2 + 5) = 8 - 7 = 1.
\]
Terry ignores the parentheses:
\[
T = 8 - 2 + 5 = 6 + 5 = 11.
\]
The difference is:
\[
H - T = 1 - 11 = -10.
\]
ANSWER 10: A

Problem 11:
The ten numbers inside the parentheses form an arithmetic sequence: 81, 83, …, 99. The sum of an arithmetic sequence is:
\[
\frac{\text{number of terms}}{2} \times (\text{first} + \text{last}) = \frac{10}{2} \times (81 + 99) = 5 \times 180 = 900.
\]
Now multiply by the factor of 2 outside:
\[
2 \times 900 = 1800.
\]
(One can also pair them: $(81+99)=180$, $(83+97)=180$, etc., giving five pairs that each sum to 180.)
ANSWER 11: E

Problem 12:
Total surveyed = 200, and females surveyed = 96, so:
\[
\text{Males surveyed} = 200 - 96 = 104.
\]
Total listeners = 136, and females who listen = 58, so:
\[
\text{Males who listen} = 136 - 58 = 78.
\]
(We can check: total non-listeners = 64, males who don't listen = 26, so females who don't listen = 38; $58+38=96$ and $78+26=104$, which checks out.)

The percentage of males who listen is:
\[
\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%.
\]
ANSWER 12: E

Problem 1:
A mixture of 30 liters is 25% red, 30% yellow, and 45% water. Five liters of yellow tint are added. What is the percent of yellow tint in the new mixture?

**Setup:** Find the amount of yellow tint originally, add 5 liters, then find the new percentage.

**Original yellow tint:** 30% of 30 = 0.30 × 30 = 9 liters

**After adding 5 liters of yellow tint:**
- New yellow tint: 9 + 5 = 14 liters
- New total volume: 30 + 5 = 35 liters
- New percentage: 14/35 = 2/5 = 40%

**Checking wrong choices:** 25% would be if we just added the original percentages; 35% would be if we just added 5% to 30%; 45% and 50% are too high.

ANSWER 1: C

---

Problem 2:
Six windows. Georgie enters by one and leaves by a different one. How many ways?

**Setup:** Choose an entry window (6 choices), then choose a different exit window (5 choices).

**Calculation:** 6 × 5 = 30 ways

**Checking wrong choices:** 15 would be if we divided by 2 (treating order as irrelevant); 12 would be 6×2; 18 would be 6×3; 36 would be 6×6 (allowing same window).

ANSWER 2: D

---

Problem 3:
1/10 + 2/20 + 3/30 = ?

**Simplify each fraction:**
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

**Sum:** 1/10 + 1/10 + 1/10 = 3/10 = 0.3

**Checking wrong choices:** 0.1 would be just one term; 0.2 would be two terms; 0.6 would be adding numerators and denominators incorrectly.

ANSWER 3: D

---

Problem 4:
Susan had $50. She spent $12 on food and twice as much on rides. How much left?

**Setup:** Food = $12, Rides = 2 × $12 = $24

**Total spent:** $12 + $24 = $36

**Amount left:** $50 - $36 = $14

**Checking wrong choices:** 12 is just the food amount; 26 is 50-24 (forgetting food); 38 is 50-12 (forgetting rides).

ANSWER 4: B

---

Problem 5:
200,000 × 200,000 = ?

**Setup:** Express in scientific notation: 200,000 = 2 × 10⁵

**Calculation:** (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion

**Checking wrong choices:** Four hundred thousand is 4×10⁵; four million is 4×10⁶; forty thousand is 4×10⁴; four hundred million is 4×10⁸.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, totaling $1A2. Find the missing digit A.

**Setup:** The total $1A2 must be divisible by 11.

**Divisibility test for 11:** A number is divisible by 11 if the alternating sum of its digits is divisible by 11.

For 1A2: (1 + 2) - A = 3 - A must be divisible by 11.

Since A is a single digit (0-9), 3 - A can only equal 0 (the only multiple of 11 in range -6 to 3).

So A = 3, giving $132 = 11 × $12 ✓

**Checking:** With A=0: $102 ÷ 11 ≈ 9.27 (not integer); A=1: $112 ÷ 11 ≈ 10.18 (not integer); A=3: $132 ÷ 11 = 12 ✓

ANSWER 6: D

---

Problem 7:
Wire is 24 cm. Which shapes can be made?

**Regular hexagon with side 5 cm:** Perimeter = 6 × 5 = 30 cm. Need 30 cm, only have 24 cm. ❌

**Square with area 36 cm²:** Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Need 24 cm, have 24 cm. ✓

**Right triangle with legs 6 and 8 cm:** Hypotenuse = √(36 + 64) = √100 = 10 cm. Perimeter = 6 + 8 + 10 = 24 cm. Need 24 cm, have 24 cm. ✓

**Conclusion:** Can make the square and triangle, but not the hexagon.

ANSWER 7: D

---

Problem 8:
Which number has the largest reciprocal?

**Find each reciprocal:**
- A. 1/3 → reciprocal = 3
- B. 2/5 → reciprocal = 5/2 = 2.5
- C. 1 → reciprocal = 1
- D. 5 → reciprocal = 1/5 = 0.2
- E. 1986 → reciprocal ≈ 0.0005

**Key insight:** Smaller positive numbers have larger reciprocals. The smallest number listed is 1/3, so it has the largest reciprocal (3).

ANSWER 8: A

---

Problem 9:
Target heart rate = 80% of (220 - age). Find target for 26-year-old.

**Maximum heart rate:** 220 - 26 = 194 bpm

**Target heart rate:** 80% of 194 = 0.80 × 194 = 155.2 ≈ 155 bpm

**Checking wrong choices:** 134 would be about 69%; 176 would be about 91%; 194 is the max (100%); 243 is impossible.

ANSWER 9: B

---

Problem 10:
Harry calculates 8 − (2 + 5) correctly. Terry calculates 8 − 2 + 5. Find H − T.

**Harry's answer:** H = 8 − (2 + 5) = 8 − 7 = 1

**Terry's answer:** T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:** H − T = 1 − 11 = −10

**Checking wrong choices:** −6 would be if Terry got 7; 0 would be if they got the same answer; 6 or 10 would be T − H.

ANSWER 10: A

---

Problem 11:
2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99) = ?

**Setup:** This is 10 consecutive odd numbers. The average of the first and last is (81 + 99)/2 = 90.

**Sum of the 10 numbers:** 10 × 90 = 900

**Multiply by 2:** 2 × 900 = 1800

**Alternative verification:** Pair the numbers: (81+99) + (83+97) + (85+95) + (87+93) + (89+91) = 180 + 180 + 180 + 180 + 180 = 900. Then 2 × 900 = 1800.

ANSWER 11: E

---

Problem 12:
Find the percentage of males who listen to the station.

**Given information:**
- Total surveyed: 200
- Females: 96
- Females who listen: 58
- Males who don't listen: 26
- Total listeners: 136
- Total non-listeners: 64

**Find males:** Males = 200 - 96 = 104

**Find males who listen:** Males who listen = Total listeners - Females who listen = 136 - 58 = 78

**Verification:** Males who don't listen = 104 - 78 = 26 ✓ (matches given)

**Percentage of males who listen:** 78/104 × 100 = 75%

**Checking wrong choices:** 39% would be 78/200 (using total instead of males); 48% would be about 50/104; 52% would be about 54/104; 55% would be about 57/104.

ANSWER 12: E

**Problem 1**

The original mixture is 30 L with 30 % yellow tint → yellow = 0.30·30 = 9 L.  
Add 5 L of yellow → total yellow = 9 + 5 = 14 L.  
Total volume after addition = 30 + 5 = 35 L.  

Percent yellow = (14/35)·100 % = 40 %.

**Answer 1: C**

---

**Problem 2**

Choose an entry window (6 ways) and a different exit window (5 ways).  
Number of ways = 6 × 5 = 30.

**Answer 2: D**

---

**Problem 3**

\[
\frac{1}{10}=0.1,\qquad\frac{2}{20}=0.1,\qquad\frac{3}{30}=0.1
\]
Sum = 0.1 + 0.1 + 0.1 = 0.3.

**Answer 3: D**

---

**Problem 4**

Food cost = \$12.  
Rides cost = twice food = 2·12 = \$24.  
Total spent = 12 + 24 = \$36.  
Money left = 50 − 36 = \$14.

**Answer 4: B**

---

**Problem 5**

\(200{,}000 \times 200{,}000 = (2\times10^{5})^{2}=4\times10^{10}=40{,}000{,}000{,}000\).

That is 40 billion.

**Answer 5: E**

---

**Problem 6**

Total paid = \$1A2 = 100 + 10A + 2 = 102 + 10A.  
Since the 11 members each paid an integer amount, 102 + 10A must be divisible by 11.

\(102\equiv3\pmod{11}\).  
We need \(10A\equiv-3\equiv8\pmod{11}\).  
Because \(10\equiv-1\pmod{11}\), this is \(-A\equiv8\pmod{11}\) → \(A\equiv3\pmod{11}\).  
A digit, so \(A=3\).

**Answer 6: D**

---

**Problem 7**

- Hexagon: perimeter = 6·5 = 30 cm > 24 cm → impossible.  
- Square: side = √36 = 6 cm, perimeter = 4·6 = 24 cm → possible.  
- Right triangle: legs 6 cm, 8 cm, hypotenuse = √(6²+8²)=10 cm, perimeter = 6+8+10 = 24 cm → possible.

Only the square and triangle can be made.

**Answer 7: D**

---

**Problem 8**

Reciprocals:
- \(1/3 → 3\)  
- \(2/5 → 5/2 = 2.5\)  
- \(1 → 1\)  
- \(5 → 1/5 = 0.2\)  
- \(1986 → 1/1986 ≈ 0.0005\)

Largest reciprocal is 3, from \(1/3\).

**Answer 8: A**

---

**Problem 9**

Maximum heart rate = 220 − 26 = 194 bpm.  
Target = 80 % of 194 = 0.8·194 = 155.2 → rounded to 155 bpm.

**Answer 9: B**

---

**Problem 10**

Harry: \(8-(2+5)=8-7=1\).  
Terry (ignoring parentheses): \(8-2+5 = 6+5 = 11\).  
Difference \(H-T = 1-11 = -10\).

**Answer 10: A**

---

**Problem 11**

Sum of the ten terms: average = (81+99)/2 = 90, so sum = 90·10 = 900.  
Multiply by 2: \(2·900 = 1800\).

**Answer 11: E**

---

**Problem 12**

Total surveyed = 200.  
Females = 96 → Males = 200 − 96 = 104.  

Female listeners = 58 → Male listeners = 136 − 58 = 78.  

Percent of males who listen = (78/104)·100 % = (39/52)·100 % = 75 %.

**Answer 12: E**

---

**Final Answers**

ANSWER 1: C  
ANSWER 2: D  
ANSWER 3: D  
ANSWER 4: B  
ANSWER 5: E  
ANSWER 6: D  
ANSWER 7: D  
ANSWER 8: A  
ANSWER 9: B  
ANSWER 10: A  
ANSWER 11: E  
ANSWER 12: E

Problem 1:  
A 30-liter paint mixture is 25% red tint, 30% yellow tint, and 45% water. Five liters of yellow tint are added.  
Original yellow tint: $0.30 \times 30 = 9$ liters.  
After adding 5 liters: $9 + 5 = 14$ liters yellow.  
New total volume: $30 + 5 = 35$ liters.  
Percent yellow: $\frac{14}{35} = 0.40 = 40\%$.  
Tempting wrong choices: 35% (original yellow percent) or 45% (water percent).  
ANSWER 1: D  

Problem 2:  
Georgie enters through one of six windows and exits through a different one.  
Entry choices: 6. Exit choices: 5 (excluding entry window).  
Total ways: $6 \times 5 = 30$.  
Tempting wrong choices: 15 (if order didn’t matter) or 36 (if same window allowed).  
ANSWER 2: D  

Problem 3:  
$\frac{1}{10} + \frac{2}{20} + \frac{3}{30} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3$.  
Tempting wrong choices: 0.1 (one term) or 0.2 (two terms).  
ANSWER 3: D  

Problem 4:  
Susan starts with \$50.  
Food: \$12.  
Rides: $2 \times 12 = \$24$.  
Total spent: $12 + 24 = \$36$.  
Left: $50 - 36 = \$14$.  
Tempting wrong choices: \$26 (if spent \$24 total) or \$38 (if spent \$12 total).  
ANSWER 4: B  

Problem 5:  
$200,000 \times 200,000 = 40,000,000,000 = 40$ billion.  
Tempting wrong choices: 400,000 (added zeros) or 4 million (multiplied digits).  
ANSWER 5: E  

Problem 6:  
Total paid: \$1A2, an integer divisible by 11.  
$1A2 \mod 11 = 0$.  
$1 + 2 - A \equiv 0 \mod 11 \Rightarrow 3 - A \equiv 0 \mod 11 \Rightarrow A = 3$.  
Tempting wrong choices: 0, 1, 2, or 4 (incorrect divisibility).  
ANSWER 6: E (Wait, let's recheck: 1A2 must be divisible by 11. 132 ÷ 11 = 12, so A=3. But 132 is 11×12, so correct. However, the answer choices are A=0,B=1,C=2,D=3,E=4. So D is 3. So correct answer is D, not E. I made a mistake in the initial answer. Correcting:)  
**Correction**: $1A2 = 132$ when $A=3$, and $132 \div 11 = 12$, so $A=3$.  
ANSWER 6: D  

Problem 7:  
Wire length: 24 cm.  
- Regular hexagon with side 5 cm: Perimeter $6 \times 5 = 30$ cm (not 24).  
- Square with area 36 cm²: Side $6$ cm, perimeter $24$ cm.  
- Right triangle with legs 6 and 8 cm: Hypotenuse 10 cm, perimeter $6+8+10=24$ cm.  
Can make square and triangle.  
Tempting wrong choices: Hexagon (incorrect perimeter).  
ANSWER 7: D  

Problem 8:  
Largest reciprocal means smallest original number.  
$\frac{1}{3} \approx 0.333$, $\frac{2}{5} = 0.4$, $1$, $5$, $1986$.  
Smallest: $\frac{1}{3}$. Reciprocal: 3.  
Others: 2.5, 1, 0.2, 0.0005.  
Tempting wrong choices: $\frac{2}{5}$ (larger than $\frac{1}{3}$).  
ANSWER 8: A  

Problem 9:  
Maximum heart rate: $220 - 26 = 194$.  
Target: $0.80 \times 194 = 155.2 \approx 155$.  
Tempting wrong choices: 134 (70%), 176 (90%), or 243 (incorrect max).  
ANSWER 9: B  

Problem 10:  
Harry: $8 - (2 + 5) = 1$.  
Terry: $8 - 2 + 5 = 11$.  
Difference: $1 - 11 = -10$.  
Tempting wrong choices: 10 (absolute difference) or 0 (if parentheses ignored by both).  
ANSWER 10: A  

Problem 11:  
Sum: $81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99$.  
Average: 90, 10 terms: $90 \times 10 = 900$.  
$2 \times 900 = 1800$.  
Tempting wrong choices: 1600 (min+max=180, times 5=900, times 2=1800, but if miscount terms).  
ANSWER 11: E  

Problem 12:  
Total surveyed: 200.  
Females: 96, so males: $200 - 96 = 104$.  
Total listeners: 136.  
Female listeners: 58, so male listeners: $136 - 58 = 78$.  
Percent male listeners: $\frac{78}{104} = 0.75 = 75\%$.  
Tempting wrong choices: 39 (male non-listeners percent) or 52 (incorrect calculation).  
ANSWER 12: E

### Problem 1:
What is the percentage of yellow tint in the new 35-liter paint mixture, after adding 5 liters of yellow tint to an original 30-liter mixture that was 30% yellow?
First, calculate the original volume of yellow tint: 30% of 30 liters = \(0.3 \times 30 = 9\) liters. Add the 5 new liters of yellow tint to get \(9 + 5 = 14\) liters of yellow in the new mixture. The total new volume of the mixture is \(30 + 5 = 35\) liters. The percentage of yellow is \(\frac{14}{35} = 0.4 = 40\%\). Tempting wrong answers come from forgetting to update the total mixture volume when calculating the new percentage, or miscalculating the original yellow volume.
ANSWER 1: C

---

### Problem 2:
How many ways can Georgie enter through one window and leave through a different window, out of 6 total windows?
Georgie has 6 choices for an entry window. Since he must leave through a different window, he has 5 remaining choices for an exit window. The total number of ways is \(6 \times 5 = 30\). A common wrong answer is 15, which incorrectly treats entry/exit as unordered pairs, when order matters (entering window A and leaving B is different from entering B and leaving A).
ANSWER 2: D

---

### Problem 3:
Simplify \(\frac{1}{10} + \frac{2}{20} + \frac{3}{30}\):
Simplify each fraction first: \(\frac{2}{20} = \frac{1}{10}\) and \(\frac{3}{30} = \frac{1}{10}\). Add them together: \(\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3\). A common wrong answer is 0.1, which incorrectly adds numerators and denominators directly instead of simplifying first.
ANSWER 3: D

---

### Problem 4:
How much money does Susan have left after spending $12 on food and twice that amount on rides, starting with $50?
First calculate the amount spent on rides: twice $12 is \(2 \times 12 = 24\) dollars. Total spent is \(12 + 24 = 36\) dollars. The amount left is \(50 - 36 = 14\) dollars. A common wrong answer is 26, which incorrectly uses $12 instead of $24 for ride costs.
ANSWER 4: B

---

### Problem 5:
Calculate \(200,000 \times 200,000\):
Rewrite 200,000 as \(2 \times 10^5\). Multiplying two copies gives \((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\), which equals 40,000,000,000, or forty billion. Wrong answers come from miscounting the number of zeros in the product.
ANSWER 5: E

---

### Problem 6:
The 3-digit total cost \(1A2\) dollars is divisible by 11 (11 members paid the same integer amount). What is the missing digit A?
The divisibility rule for 11 states that alternating sum of digits must be divisible by 11. For 1A2, the alternating sum is \(1 - A + 2 = 3 - A\). The only value of A (0-9) that makes this sum divisible by 11 is A=3, since \(3-3=0\), which is divisible by 11. Verify: \(132 \div 11 = 12\), a valid integer.
ANSWER 6: D

---

### Problem 7:
Haruki has a 24cm wire. Which of the three shapes can he form? Calculate the perimeter of each shape:
1. Regular hexagon with side 5cm: perimeter = \(6 \times 5 = 30\)cm, which is longer than 24cm, so impossible.
2. Square with area 36cm²: side length = \(\sqrt{36}=6\)cm, perimeter = \(4 \times 6 = 24\)cm, which fits.
3. Right triangle with legs 6cm and 8cm: hypotenuse = 10cm (6-8-10 Pythagorean triple), perimeter = \(6+8+10=24\)cm, which fits.
Only the square and triangle can be formed.
ANSWER 7: D

---

### Problem 8:
Which number has the largest reciprocal? The reciprocal of \(x\) is \(\frac{1}{x}\), so smaller values of \(x\) produce larger reciprocals. List the values of all options:
A. \(\frac{1}{3} \approx 0.333\), B. \(\frac{2}{5}=0.4\), C.1, D.5, E.1986. The smallest value is \(\frac{1}{3}\), whose reciprocal is 3, the largest of all options.
ANSWER 8: A

---

### Problem 9:
Calculate the target heart rate for a 26-year-old athlete: target is 80% of max heart rate, where max = 220 - age.
First calculate max heart rate: \(220 - 26 = 194\) bpm. 80% of 194 is \(0.8 \times 194 = 155.2\), which rounds to 155 bpm. A common wrong answer is 176, which uses 220 instead of the correct max heart rate of 194.
ANSWER 9: B

---

### Problem 10:
Calculate \(H-T\), where H is Harry's correct answer to \(8-(2+5)\), and T is Terry's incorrect answer of \(8-2+5\).
Harry's answer: \(8 - 7 = 1 = H\). Terry's answer: \(6 + 5 = 11 = T\). The difference \(H-T = 1 - 11 = -10\). A common wrong answer is 10, which calculates \(T-H\) instead of \(H-T\).
ANSWER 10: A

---

### Problem 11:
Simplify \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\):
Pair the terms inside the parentheses to simplify addition: \((81+99)+(83+97)+(85+95)+(87+93)+(89+91) = 5 \times 180 = 900\). Multiply by 2: \(2 \times 900 = 1800\).
ANSWER 11: E

---

### Problem 12:
What percentage of males surveyed listen to the radio station? First calculate total number of males: total surveyed = 200, females = 96, so males = \(200 - 96 = 104\). Males who do not listen = 26, so males who listen = \(104 - 26 = 78\). The percentage of males who listen is \(\frac{78}{104} = 0.75 = 75\%\). A common wrong answer is 39%, which divides male listeners by the total surveyed population instead of the total male population.
ANSWER 12: E

### Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.
1.  Calculate original yellow tint volume: 25%? No wait, original is 30% yellow. So 30% of 30 L = $0.3 \times 30 = 9$ liters of yellow.
2.  Add 5 liters of yellow: total yellow = $9 + 5 = 14$ liters.
3.  Total new mixture volume: $30 + 5 = 35$ liters.
4.  Percent yellow: $\frac{14}{35} \times 100 = 40\%$.
ANSWER 1: C
---
### Problem 2:
We count the number of ways to choose an entry window and a different exit window from 6 windows.
1.  Entry window: 6 possible choices.
2.  Exit window: cannot be the entry window, so 5 remaining choices.
3.  Total ways: $6 \times 5 = 30$ (order matters because entry and exit are distinct actions).
ANSWER 2: D
---
### Problem 3:
Simplify and add the fractions:
1.  $\frac{1}{10} = 0.1$, $\frac{2}{20} = \frac{1}{10} = 0.1$, $\frac{3}{30} = \frac{1}{10} = 0.1$.
2.  Sum: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D
---
### Problem 4:
Calculate remaining money after expenses:
1.  Total initial money: $50.
2.  Money spent on rides: twice the food cost = $2 \times 12 = 24$ dollars.
3.  Total spent: $12 + 24 = 36$ dollars.
4.  Remaining: $50 - 36 = 14$ dollars.
ANSWER 4: B
---
### Problem 5:
Compute $200,000 \times 200,000$:
1.  Rewrite as $(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
2.  $10^{10} = 10,000,000,000$, so $4 \times 10^{10} = 40,000,000,000 = 40$ billion.
ANSWER 5: E
---
### Problem 6:
The total payment $1A2$ (a 3-digit number) must be divisible by 11 (since 11 people paid equal integer amounts).
1.  Use the divisibility rule for 11: the alternating sum of digits is divisible by 11. For $1A2$, this is $1 - A + 2 = 3 - A$.
2.  $A$ is a digit (0-9), so $3 - A$ must equal 0 (the only multiple of 11 in range: $3 - A = 11$ gives $A=-8$, $3-A=-11$ gives $A=14$, both invalid).
3.  $3 - A = 0 \implies A=3$. Verify: $132 \div 11 = 12$, which is an integer.
ANSWER 6: D
---
### Problem 7:
Check if the 24 cm wire can form each shape (perimeter of the shape must equal 24 cm):
1.  Regular hexagon (side 5 cm): Perimeter = $6 \times 5 = 30$ cm > 24 cm → cannot make.
2.  Square (area 36 cm²): Side length = $\sqrt{36} = 6$ cm. Perimeter = $4 \times 6 = 24$ cm → can make.
3.  Right triangle (legs 6, 8 cm): Hypotenuse = $\sqrt{6^2 + 8^2} = 10$ cm. Perimeter = $6 + 8 + 10 = 24$ cm → can make.
Only square and triangle can be made.
ANSWER 7: D
---
### Problem 8:
The reciprocal of a positive number $x$ is $\frac{1}{x}$; smaller $x$ gives larger reciprocal.
1.  List values: $\frac{1}{3} \approx 0.33$, $\frac{2}{5}=0.4$, $1$, $5$, $1986$.
2.  Smallest value is $\frac{1}{3}$, whose reciprocal is $3$ (largest among all reciprocals: $\frac{5}{2}=2.5$, $1$, $0.2$, $\frac{1}{1986} \approx 0.0005$).
ANSWER 8: A
---
### Problem 9:
Calculate target heart rate for a 26-year-old:
1.  Maximum heart rate: $220 - 26 = 194$ bpm.
2.  Target heart rate: 80% of 194 = $0.8 \times 194 = 155.2$, which rounds to 155.
ANSWER 9: B
---
### Problem 10:
Compute $H$ (Harry's correct answer) and $T$ (Terry's incorrect answer):
1.  Harry: $8 - (2 + 5) = 8 - 7 = 1$, so $H=1$.
2.  Terry (ignores parentheses, left-to-right): $8 - 2 + 5 = 6 + 5 = 11$, so $T=11$.
3.  Difference: $H - T = 1 - 11 = -10$.
ANSWER 10: A
---
### Problem 11:
Calculate $2 \times (81 + 83 + ... + 99)$:
1.  The sum inside is 10 consecutive odd numbers. Average of first and last term: $\frac{81 + 99}{2} = 90$.
2.  Sum of the 10 terms: $90 \times 10 = 900$.
3.  Multiply by 2: $2 \times 900 = 1800$.
ANSWER 11: E
---
### Problem 12:
Find the percentage of males who listen to the station:
1.  Total males surveyed: $200 - 96 = 104$.
2.  Males who do not listen: 26, so males who listen: $104 - 26 = 78$.
3.  Verify with total listeners: Total listeners = 136, female listeners = 58, so male listeners = $136 - 58 = 78$ (matches).
4.  Percentage: $\frac{78}{104} \times 100 = 75\%$.
ANSWER 12: E

Problem 1:
Original yellow = 30% of 30 = 9 L. Add 5 → 14 L yellow. New total = 35 L. Percent = 14/35 = 40%.

ANSWER 1: C

Problem 2:
Enter: 6 choices, leave: 5 different choices. 6×5 = 30.

ANSWER 2: D

Problem 3:
Each fraction = 0.1. Sum = 0.3.

ANSWER 3: D

Problem 4:
Food 12, rides twice = 24. Spent 36. Left = 50 − 36 = 14.

ANSWER 4: B

Problem 5:
200,000 × 200,000 = 40,000,000,000 = forty billion.

ANSWER 5: E

Problem 6:
1A2 divisible by 11. Alternating sum: 1 − A + 2 = 3 − A must be divisible by 11 → A = 3 (gives 0). Check 132/11 = 12. Yes.

ANSWER 6: D

Problem 7:
Hexagon: 6×5 = 30 > 24, no. Square area 36 → side 6, perimeter 24, yes. Triangle legs 6,8 → hypotenuse 10, perimeter 24, yes. Square and triangle only.

ANSWER 7: D

Problem 8:
Largest reciprocal = smallest number. Smallest is 1/3.

ANSWER 8: A

Problem 9:
Max = 220 − 26 = 194. Target = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

Problem 10:
H = 8 − 7 = 1. T = 8 − 2 + 5 = 11. H − T = 1 − 11 = −10.

ANSWER 10: A

Problem 11:
Sum of 81..99 odd numbers (10 terms): average (81+99)/2 = 90, ×10 = 900. ×2 = 1800.

ANSWER 11: E

Problem 12:
Males total = 200 − 96 = 104. Males who don't listen = 26 → males who listen = 104 − 26 = 78. Percent = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem systematically.

---

**Problem 1:**
Original mixture: 30 liters, 30% yellow = 9 liters yellow. Add 5 liters yellow → new yellow = 14 liters, new total = 35 liters.
Percent yellow = 14/35 = 40%.

ANSWER 1: C

---

**Problem 2:**
Georgie can enter through any of 6 windows, then leave through any of the remaining 5 windows.
6 × 5 = 30.

ANSWER 2: D

---

**Problem 3:**
1/10 = 0.1, 2/20 = 0.1, 3/30 = 0.1. Sum = 0.3.

ANSWER 3: D

---

**Problem 4:**
Food: $12. Rides: 2 × $12 = $24. Total spent: $12 + $24 = $36. Left: $50 − $36 = $14.

ANSWER 4: B

---

**Problem 5:**
200,000 × 200,000 = (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion.

ANSWER 5: E

---

**Problem 6:**
The number 1A2 must be divisible by 11. 1A2 ÷ 11 must be an integer.
11 × 10 = 110, 11 × 11 = 121 (no), 11 × 12 = 132 ✓ → A = 3.
Check: 132/11 = 12 ✓, each paid $12.

ANSWER 6: D

---

**Problem 7:**
Wire = 24 cm.
- Regular hexagon, side 5: perimeter = 6 × 5 = 30 cm > 24. ✗
- Square, area 36: side = 6, perimeter = 24 cm ✓
- Right triangle, legs 6 and 8: hypotenuse = 10, perimeter = 6+8+10 = 24 cm ✓

ANSWER 7: D

---

**Problem 8:**
The reciprocal of x is 1/x. The largest reciprocal comes from the smallest positive number.
Among 1/3, 2/5, 1, 5, 1986 — the smallest is 1/3, and its reciprocal is 3, which is the largest.

ANSWER 8: A

---

**Problem 9:**
Maximum heart rate = 220 − 26 = 194. Target = 80% × 194 = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

---

**Problem 10:**
Harry: 8 − (2 + 5) = 8 − 7 = 1, so H = 1.
Terry: 8 − 2 + 5 = 11, so T = 11.
H − T = 1 − 11 = −10.

ANSWER 10: A

---

**Problem 11:**
The ten odd numbers from 81 to 99: sum = (81+99)/2 × 10 = 90 × 10 = 900.
2 × 900 = 1800.

ANSWER 11: E

---

**Problem 12:**
Total surveyed = 200. Females = 96, so Males = 104.
Total non-listeners = 64. Males who don't listen = 26, so females who don't listen = 64 − 26 = 38.
Males who listen = 104 − 26 = 78.
Percentage = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem step-by-step.

## Problem 1:
**What is asked:** Find the percentage of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.

**Original mixture:**
- Yellow tint: 30% of 30 = 0.30 × 30 = 9 liters
- Total volume: 30 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total volume: 30 + 5 = 35 liters

**Percentage of yellow tint:**
$$\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%$$

ANSWER 1: C

## Problem 2:
**What is asked:** In how many ways can the ghost enter by one window and leave by a different window?

**Solution:**
- Choices for entering: 6 windows
- Choices for leaving: 5 windows (must be different from entry)
- Total ways: 6 × 5 = 30

ANSWER 2: D

## Problem 3:
**What is asked:** Calculate 1/10 + 2/20 + 3/30

**Solution:**
Simplify each fraction:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

ANSWER 3: D

## Problem 4:
**What is asked:** How much money does Susan have left?

**Solution:**
- Started with: $50
- Spent on food: $12
- Spent on rides: 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Left: $50 - $36 = $14

ANSWER 4: B

## Problem 5:
**What is asked:** Calculate 200,000 × 200,000

**Solution:**
$$200,000 × 200,000 = (2 × 10^5) × (2 × 10^5) = 4 × 10^{10} = 40,000,000,000 = 40 \text{ billion}$$

ANSWER 5: E

## Problem 6:
**What is asked:** Find the missing digit A in the 3-digit number 1A2, where 11 club members each paid an equal integer amount that totals 1A2.

**Solution:**
The number 1A2 must be divisible by 11.

Testing divisibility by 11 (alternating sum of digits):
- 1 - A + 2 = 0 (mod 11)
- 3 - A ≡ 0 (mod 11)
- A = 3

Check: 132 ÷ 11 = 12 ✓

ANSWER 6: D

## Problem 7:
**What is asked:** Which shapes can Haruki make with 24 cm of wire?

**Solution:**
Test each shape:

**Regular hexagon with side 5 cm:**
- Perimeter: 6 × 5 = 30 cm
- Need 30 cm, have 24 cm ✗

**Square with area 36 cm²:**
- Side length: √36 = 6 cm
- Perimeter: 4 × 6 = 24 cm ✓

**Right triangle with legs 6 and 8 cm:**
- Hypotenuse: √(6² + 8²) = √(36 + 64) = √100 = 10 cm
- Perimeter: 6 + 8 + 10 = 24 cm ✓

Only square and triangle can be made.

ANSWER 7: D

## Problem 8:
**What is asked:** Which number has the largest reciprocal?

**Solution:**
Find reciprocals:
- 1/(1/3) = 3
- 1/(2/5) = 5/2 = 2.5
- 1/1 = 1
- 1/5 = 0.2
- 1/1986 ≈ 0.0005

The largest reciprocal is 5 (from the number 1/5).

ANSWER 8: D

## Problem 9:
**What is asked:** Find the target heart rate for a 26-year-old athlete.

**Solution:**
- Maximum heart rate: 220 - 26 = 194 beats per minute
- Target heart rate: 80% of 194 = 0.80 × 194 = 155.2 ≈ 155

ANSWER 9: B

## Problem 10:
**What is asked:** Find H - T where H is Harry's answer and T is Terry's answer.

**Solution:**
**Harry's calculation (correct):**
$$H = 8 - (2 + 5) = 8 - 7 = 1$$

**Terry's calculation (ignoring parentheses):**
$$T = 8 - 2 + 5 = 6 + 5 = 11$$

**Difference:**
$$H - T = 1 - 11 = -10$$

ANSWER 10: A

## Problem 11:
**What is asked:** Calculate 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

**Solution:**
This is an arithmetic sequence with 10 terms. Sum = n(first + last)/2
$$81 + 83 + ... + 99 = \frac{10(81 + 99)}{2} = \frac{10 × 180}{2} = 900$$

$$2 × 900 = 1800$$

ANSWER 11: E

## Problem 12:
**What is asked:** What percentage of males surveyed listen to the station?

**Solution:**
- Total surveyed: 200
- Females: 96
- Males: 200 - 96 = 104

- Total non-listeners: 64
- Males who don't listen: 26
- Males who listen: 104 - 26 = 78

**Percentage of males who listen:**
$$\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%$$

ANSWER 12: E

Problem 1:

The original mixture is 30 liters total, with 30% yellow tint.

So the amount of yellow tint at first is:
- \(30\%\) of \(30 = 0.30 \times 30 = 9\) liters

Then 5 liters of yellow tint are added, so the new amount of yellow tint is:
- \(9 + 5 = 14\) liters

The new total mixture is:
- \(30 + 5 = 35\) liters

So the percent of yellow tint is:
- \(\frac{14}{35} = \frac{2}{5} = 0.4 = 40\%\)

Answer choice C matches 40%.

ANSWER 1: C

Problem 2:

Georgie can:
- choose 1 of the 6 windows to enter
- then choose a different window to leave

For the entry, there are 6 choices.

For the exit, there are 5 remaining choices, since it must be a different window.

So the total number of ways is:
- \(6 \times 5 = 30\)

ANSWER 2: D

Problem 3:

Compute each fraction:
- \(\frac{1}{10} = 0.1\)
- \(\frac{2}{20} = \frac{1}{10} = 0.1\)
- \(\frac{3}{30} = \frac{1}{10} = 0.1\)

Add them:
- \(0.1 + 0.1 + 0.1 = 0.3\)

So the value is \(0.3\).

ANSWER 3: D

Problem 4:

Susan had \$50.

She spent:
- \$12 on food
- twice as much on rides, so \(2 \times 12 = 24\)

Total spent:
- \(12 + 24 = 36\)

Money left:
- \(50 - 36 = 14\)

ANSWER 4: B

Problem 5:

Two hundred thousand is:
- \(200{,}000 = 2 \times 10^5\)

Multiply:
- \(200{,}000 \times 200{,}000 = (2 \times 10^5)(2 \times 10^5) = 4 \times 10^{10}\)

And \(4 \times 10^{10} = 40{,}000{,}000{,}000\), which is forty billion.

ANSWER 5: E

Problem 6:

The total is written as \(1A2\), and 11 members each paid the same integer amount.

So \(1A2\) must be divisible by 11.

Test the answer choices:
- 102 ÷ 11 is not an integer
- 112 ÷ 11 is not an integer
- 122 ÷ 11 = 11 remainder 1
- 132 ÷ 11 = 12 exactly
- 142 ÷ 11 is not an integer

So the number is 132, which means \(A = 3\).

ANSWER 6: D

Problem 7:

We check each shape.

### Hexagon
A regular hexagon has 6 equal sides.
If each side is 5 cm, perimeter is:
- \(6 \times 5 = 30\) cm

The wire is only 24 cm, so he cannot make the hexagon.

### Square
A square with area 36 has side length:
- \(\sqrt{36} = 6\) cm

Perimeter:
- \(4 \times 6 = 24\) cm

So he can make the square exactly.

### Right triangle
The legs are 6 and 8, so the hypotenuse is:
- \(\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)

Perimeter:
- \(6 + 8 + 10 = 24\) cm

So he can make the triangle exactly.

Therefore he can make the square and the triangle only.

ANSWER 7: D

Problem 8:

The reciprocal is largest when the original number is smallest, assuming positive numbers.

Compare:
- \(\frac13 \approx 0.333\)
- \(\frac25 = 0.4\)
- \(1\)
- \(5\)
- \(1986\)

The smallest number is \(\frac13\), so it has the largest reciprocal.

Its reciprocal is 3, which is larger than the reciprocals of all the others.

ANSWER 8: A

Problem 9:

Maximum heart rate:
- \(220 - 26 = 194\)

Target heart rate is 80% of that:
- \(0.8 \times 194 = 155.2\)

To the nearest whole number, that is 155.

ANSWER 9: B

Problem 10:

First, Harry uses the parentheses:
- \(8 - (2 + 5) = 8 - 7 = 1\)

So \(H = 1\).

Terry ignores parentheses:
- \(8 - 2 + 5 = 6 + 5 = 11\)

So \(T = 11\).

Difference:
- \(H - T = 1 - 11 = -10\)

ANSWER 10: A

Problem 11:

We need to evaluate:
- \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

These are 10 odd numbers from 81 to 99.

Instead of adding one by one, use the average:
- average of 81 and 99 is \( \frac{81+99}{2} = 90\)

Since there are 10 numbers, their sum is:
- \(10 \times 90 = 900\)

Now multiply by 2:
- \(2 \times 900 = 1800\)

ANSWER 11: E

Problem 12:

We want the percentage of males surveyed who listen.

From the table:
- Total surveyed = 200
- Females = 96
- So males = \(200 - 96 = 104\)

Males who don’t listen = 26, so males who do listen:
- \(104 - 26 = 78\)

Now find the percent of males who listen:
- \(\frac{78}{104} = \frac{3}{4} = 75\%\)

ANSWER 12: E

Problem 1:  
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?  

- Original yellow tint amount = 30 × 0.30 = 9 liters.  
- Add 5 liters of yellow tint → new yellow amount = 9 + 5 = 14 liters.  
- New total volume = 30 + 5 = 35 liters.  
- New percent yellow = (14 / 35) × 100% = (2/5) × 100% = 40%.  

So the correct choice is 40.  

ANSWER 1: C  

---

Problem 2:  
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?  

- Choose an entry window: 6 choices.  
- Choose a different exit window: 5 choices (cannot be the same as entry).  
- Total ways = 6 × 5 = 30.  

ANSWER 2: D  

---

Problem 3:  
Compute: \( \frac{1}{10} + \frac{2}{20} + \frac{3}{30} \).  

Convert each to a common denominator, 60:  
- \( \frac{1}{10} = \frac{6}{60} \)  
- \( \frac{2}{20} = \frac{2}{20} = \frac{6}{60} \) (since multiply top and bottom by 3)  
- \( \frac{3}{30} = \frac{3}{30} = \frac{6}{60} \) (since multiply top and bottom by 2)  

Add:  
\( \frac{6}{60} + \frac{6}{60} + \frac{6}{60} = \frac{18}{60} = \frac{3}{10} = 0.3 \)

So the value is .3.  

ANSWER 3: D  

---

Problem 4:  
Susan had $50 to spend. She spent $12 on food and twice as much on rides. How much left?  

- Food = 12  
- Rides = twice 12 = 24  
- Total spent = 12 + 24 = 36  
- Money left = 50 − 36 = 14  

ANSWER 4: B  

---

Problem 5:  
Two hundred thousand times two hundred thousand equals ?  

\(200{,}000 \times 200{,}000 = (2\times 10^5)(2\times 10^5) = 4 \times 10^{10}\).  
- \(10^{10} = 10{,}000{,}000{,}000\)  
- \(4 \times 10^{10} = 40{,}000{,}000{,}000 = 40\) billion  

ANSWER 5: E  

---

Problem 6:  
Eleven members each paid the same integer amount for a guest speaker. In all, they paid $1A2 (a 3-digit number). Find missing digit A.  

- Total = \(1A2\) means:  
  \(1A2 = 100 + 10A + 2 = 102 + 10A\).  
- This total is divided equally among 11 people:  
  So \(102 + 10A\) must be divisible by 11.  

Check \(102 + 10A \mod 11\):  
- \(102 \div 11 = 9\) remainder 3, so \(102 \equiv 3 \pmod{11}\).  
- \(10A \equiv -A \pmod{11}\) (since 10 ≡ −1 mod 11).  

So:  
\(102 + 10A \equiv 3 - A \equiv 0 \pmod{11}\)  
Thus \(A \equiv 3 \pmod{11}\).  
Since \(A\) is a digit 0–9, \(A=3\).  

ANSWER 6: D  

---

Problem 7:  
Wire length = 24 cm. Which shapes can be made (using all 24 cm of wire)?  

We need the perimeter (total boundary length) of each shape:

**(1) Regular hexagon with side length 5 cm**  
Perimeter = 6 × 5 = 30 cm → too long. Cannot.

**(2) Square with area 36 cm²**  
Side length = √36 = 6 cm  
Perimeter = 4 × 6 = 24 cm → fits. Can.

**(3) Right triangle with legs 6 and 8 cm**  
Right triangle sides are 6 and 8 legs, and hypotenuse:  
\( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)  
Perimeter = 6 + 8 + 10 = 24 cm → fits. Can.

So the shapes possible: **square and triangle only**.  

ANSWER 7: D  

---

Problem 8:  
Which number has the largest reciprocal? Recall: the reciprocal of \(x\) is \(1/x\).  
We compare \(1/3, 2/5, 1, 1/5, 1/1986\).  

Compute roughly:  
- \(1/3 \approx 0.333\)  
- \(2/5 = 0.4\)  
- \(1 = 1\)  
- \(1/5 = 0.2\)  
- \(1/1986\) is tiny  

Largest is 1 (reciprocal of 1).  

ANSWER 8: C  

---

Problem 9:  
Target heart rate = 80% of (maximum heart rate).  
Maximum heart rate = 220 − age.  
Age = 26.

- Maximum = 220 − 26 = 194  
- Target = 0.80 × 194 = 155.2  
Nearest whole number = 155  

ANSWER 9: B  

---

Problem 10:  
Expression: \(8 - (2 + 5)\)  
- Harry computes correctly:  
  \(2 + 5 = 7\), so \(8 - 7 = 1\).  
- Terry ignores parentheses:  
  \(8 - 2 + 5 = 6 + 5 = 11\).  

Difference \(H - T = 1 - 11 = -10\).  

ANSWER 10: A  

---

Problem 11:  
Compute:  
\(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

Notice these are all odd numbers from 81 to 99 stepping by 2.  
There are 10 terms.

Pair from ends:  
- 81 + 99 = 180  
- 83 + 97 = 180  
- 85 + 95 = 180  
- 87 + 93 = 180  
- 89 + 91 = 180  

There are 5 pairs, each sum 180:  
Sum = 5 × 180 = 900  

Now multiply by 2:  
2 × 900 = 1800  

ANSWER 11: E  

---

Problem 12:  
Total surveyed = 200  
Females = 96  
Females who listen = 58  
Males who don’t listen = 26  
Total listeners = 136  

We want: % of males who listen.

Step 1: Find males total.  
Males = 200 − 96 = 104

Step 2: Find males who don’t listen is given = 26.  
So males who listen = 104 − 26 = 78

Step 3: Convert to percent of males who listen:  
Percent = (78 / 104) × 100%  
Simplify 78/104 = 3/4 (divide by 26):  
So percent = (3/4) × 100% = 75%

ANSWER 12: E

Problem 1:
The original mixture is 30 liters. The amount of yellow tint is 30% of 30 liters: $0.30 \times 30 = 9$ liters.
Five liters of yellow tint are added, so the new amount of yellow tint is $9 + 5 = 14$ liters.
The new total volume of the mixture is $30 + 5 = 35$ liters.
The percentage of yellow tint is $\frac{14}{35} \times 100\%$.
Simplifying the fraction: $\frac{14}{35} = \frac{2}{5} = 0.4$.
$0.4 \times 100\% = 40\%$.
ANSWER 1: C

Problem 2:
Georgie has 6 choices for the window he enters. Since he must leave by a different window, he has $6 - 1 = 5$ choices for the window he leaves by.
The total number of ways is $6 \times 5 = 30$.
ANSWER 2: D

Problem 3:
Simplify each fraction:
$1/10 = 0.1$
$2/20 = 1/10 = 0.1$
$3/30 = 1/10 = 0.1$
Summing them: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides, which is $2 \times 12 = 24$.
Total spent: $12 + 24 = 36$.
Amount left: $50 - 36 = 14$.
ANSWER 4: B

Problem 5:
Two hundred thousand is $200,000$ (or $2 \times 10^5$).
$(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
$4 \times 10^{10} = 40,000,000,000$, which is forty billion.
ANSWER 5: E

Problem 6:
The total amount $1A2$ must be divisible by 11.
A number is divisible by 11 if the alternating sum of its digits is divisible by 11.
For $1A2$, the alternating sum is $1 - A + 2 = 3 - A$.
For this to be 0 (the only multiple of 11 possible for a single digit $A$), $3 - A = 0$, so $A = 3$.
Check: $132 / 11 = 12$. This works.
ANSWER 6: D

Problem 7:
Hexagon: Perimeter of a regular hexagon with side 5 is $6 \times 5 = 30$ cm. (Cannot make)
Square: Area is 36, so side is $\sqrt{36} = 6$. Perimeter is $4 \times 6 = 24$ cm. (Can make)
Triangle: Right triangle with legs 6 and 8. Hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. Perimeter is $6 + 8 + 10 = 24$ cm. (Can make)
ANSWER 7: D

Problem 8:
Reciprocals:
A: $1/(1/3) = 3$
B: $1/(2/5) = 2.5$
C: $1/1 = 1$
D: $1/5 = 0.2$
E: $1/1986 \approx 0.0005$
The largest is 3.
ANSWER 8: A

Problem 9:
Max heart rate = $220 - 26 = 194$.
Target heart rate = $80\%$ of $194 = 0.8 \times 194 = 155.2$.
To the nearest whole number, this is 155.
ANSWER 9: B

Problem 10:
$H = 8 - (2 + 5) = 8 - 7 = 1$.
$T = 8 - 2 + 5 = 6 + 5 = 11$.
$H - T = 1 - 11 = -10$.
ANSWER 10: A

Problem 11:
Sum inside parentheses: $81+83+85+87+89+91+93+95+97+99$.
This is an arithmetic progression with 10 terms.
Sum = $\frac{n}{2}(\text{first} + \text{last}) = \frac{10}{2}(81 + 99) = 5 \times 180 = 900$.
The expression is $2 \times 900 = 1800$.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96.
Males = $200 - 96 = 104$.
Males who don't listen = 26.
Males who listen = $104 - 26 = 78$.
Percentage of males who listen = $(78 / 104) \times 100\%$.
$\frac{78}{104} = \frac{39}{52} = \frac{3}{4} = 0.75 = 75\%$.
ANSWER 12: E

Problem 1: The original mixture has volume 30 L, of which 30% is yellow tint, so there are \(0.3 \times 30 = 9\) L of yellow tint. Adding 5 L of yellow tint produces a new total of 14 L of yellow tint and a new mixture volume of 35 L. The new percentage of yellow tint is therefore \((14/35) \times 100\% = 40\%\).

ANSWER 1: C

Problem 2: There are 6 choices for the entry window. For each of those choices there are then exactly 5 remaining choices for a different exit window, giving a total of \(6 \times 5 = 30\) ordered pairs of distinct windows.

ANSWER 2: D

Problem 3: Rewrite each term with a common denominator of 10:  
\(1/10 = 0.1\),  
\(2/20 = 1/10 = 0.1\),  
\(3/30 = 1/10 = 0.1\).  
Their sum is \(0.1 + 0.1 + 0.1 = 0.3\).

ANSWER 3: D

Problem 4: Twice the food cost is \(2 \times 12 = 24\) dollars spent on rides. The total spent is therefore \(12 + 24 = 36\) dollars, leaving \(50 - 36 = 14\) dollars.

ANSWER 4: B

Problem 5: Write the factors in scientific notation:  
\(200000 = 2 \times 10^5\), so  
\((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\),  
which is forty billion.

ANSWER 5: E

Problem 6: The three-digit total \(1A2\) must be divisible by 11. The alternating-sum divisibility rule requires that \(1 - A + 2 = 3 - A\) be a multiple of 11. The only admissible digit \(A\) (0-9) that satisfies the condition is \(A = 3\), giving the total 132, which is exactly \(11 \times 12\).

ANSWER 6: D

Problem 7: A regular hexagon with side 5 cm has perimeter 30 cm, which exceeds the 24 cm length of wire, so it is impossible. A square of area 36 cm² has side length 6 cm and perimeter 24 cm, which matches the wire length. A right triangle with legs 6 cm and 8 cm has hypotenuse 10 cm, so its perimeter is \(6 + 8 + 10 = 24\) cm, which also matches. Consequently only the square and the triangle are possible.

ANSWER 7: D

Problem 8: For positive numbers the reciprocal function is strictly decreasing, so the largest reciprocal belongs to the smallest number in the list, which is \(1/3\).

ANSWER 8: A

Problem 9: The theoretical maximum heart rate is \(220 - 26 = 194\) beats per minute. Eighty percent of that value is \(0.8 \times 194 = 155.2\), which rounds to the nearest whole number 155.

ANSWER 9: B

Problem 10: Harry evaluates the parentheses first and obtains \(8 - (2 + 5) = 1\). Terry omits the parentheses and obtains \(8 - 2 + 5 = 11\). Their difference is therefore \(1 - 11 = -10\).

ANSWER 10: A

Problem 11: The sum inside the parentheses consists of the ten consecutive odd integers from 81 to 99. Their average is \((81 + 99)/2 = 90\), so the sum equals \(10 \times 90 = 900\). Multiplying by the outer factor of 2 yields \(2 \times 900 = 1800\).

ANSWER 11: E

Problem 12: There are 200 people surveyed and 96 females, so there are \(200 - 96 = 104\) males. Of the 96 females, 58 listen, leaving \(96 - 58 = 38\) females who do not listen. Given that 26 males do not listen, the number of males who listen is \(104 - 26 = 78\). The required percentage is therefore \((78/104) \times 100\% = 75\%\).

ANSWER 12: E

Problem 1:
The problem asks us to find the percentage of yellow tint in a new mixture after adding 5 liters of yellow tint to the original 30 liters of paint mixture. The original mixture is 25% red tint, 30% yellow tint, and 45% water.

First, let's calculate the amount of yellow tint in the original mixture. 30% of 30 liters is $0.3 \times 30 = 9$ liters.

After adding 5 liters of yellow tint, the total amount of yellow tint becomes $9 + 5 = 14$ liters.

The total volume of the new mixture is $30 + 5 = 35$ liters.

To find the percentage of yellow tint in the new mixture, we divide the amount of yellow tint by the total volume and multiply by 100: $\frac{14}{35} \times 100 = 40\%$.

Let's examine the choices:
- A. 25: Too low because the percentage of yellow tint increased.
- B. 35: Not the correct calculation.
- C. 40: Matches our calculation.
- D. 45: Too high.
- E. 50: Too high.

The correct percentage is 40%.

ANSWER 1: C

Problem 2:
The problem asks us to determine the number of ways Georgie the Ghost can enter a haunted house through one of six windows and leave by a different window.

To enter, Georgie has 6 choices. After entering, there are 5 remaining windows to choose from for leaving.

The total number of ways to enter and leave is $6 \times 5 = 30$.

Let's examine the choices:
- A. 12: Too low.
- B. 15: Too low.
- C. 18: Too low.
- D. 30: Matches our calculation.
- E. 36: Too high because it would imply Georgie can leave by the same window.

The correct number of ways is 30.

ANSWER 2: D

Problem 3:
The problem requires us to simplify the expression $1⁄10 + 2⁄20 + 3⁄30$.

First, simplify each fraction:
- $1⁄10$ stays the same.
- $2⁄20 = 1⁄10$.
- $3⁄30 = 1⁄10$.

Adding them together: $1⁄10 + 1⁄10 + 1⁄10 = 3⁄10 = 0.3$.

Let's examine the choices:
- A. .1: Too low.
- B. .123: Not the correct sum.
- C. .2: Too low.
- D. .3: Matches our calculation.
- E. .6: Too high.

The correct sum is 0.3.

ANSWER 3: D

Problem 4:
The problem asks us to find out how many dollars Susan had left after spending money on food and rides at the carnival. She started with $50, spent $12 on food, and twice as much on rides.

First, calculate the amount spent on rides: $12 \times 2 = $24.

Total amount spent is $12 + $24 = $36.

The amount left is $50 - $36 = $14.

Let's examine the choices:
- A. 12: Too low.
- B. 14: Matches our calculation.
- C. 26: Too high.
- D. 38: Too high.
- E. 50: Incorrect because she spent money.

The correct amount left is $14.

ANSWER 4: B

Problem 5:
The problem asks us to calculate $200,000 \times 200,000$.

$200,000 = 2 \times 10^5$, so $200,000 \times 200,000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10} = 40,000,000,000$.

Let's examine the choices:
- A. four hundred thousand: Too low.
- B. four million: Too low.
- C. forty thousand: Too low.
- D. four hundred million: Too low.
- E. forty billion: Matches our calculation.

The correct result is forty billion.

ANSWER 5: E

Problem 6:
The problem requires us to find the missing digit A in the 3-digit number $1A2 that represents the total amount paid by 11 members of the Math Club.

The total amount paid is divisible by 11 since there are 11 members who paid the same amount.

To check divisibility by 11, we use the rule: the difference between the sum of digits at odd places and the sum of digits at even places should be divisible by 11.

For $1A2, the sum of digits at odd places is $1 + 2 = 3$, and the sum at even places is $A$. The difference is $3 - A$.

For $3 - A$ to be divisible by 11 (or 0, since $A$ is a single digit), $A = 3$ makes $3 - A = 0$, which is divisible by 11.

Let's examine the choices:
- A. 0: Not correct because $3 - 0 = 3$, not divisible by 11.
- B. 1: Not correct because $3 - 1 = 2$, not divisible by 11.
- C. 2: Not correct because $3 - 2 = 1$, not divisible by 11.
- D. 3: Correct because $3 - 3 = 0$, which is divisible by 11.
- E. 4: Not correct because $3 - 4 = -1$, not divisible by 11.

The missing digit A is 3, but we should verify if 132 is indeed divisible by 11. $132 / 11 = 12$, so it is.

ANSWER 6: D

Problem 7:
The problem asks us to determine which shapes Haruki can form using a 24 cm long wire.

1. A regular hexagon with side length 5 cm: The perimeter is $6 \times 5 = 30$ cm, which is more than 24 cm. So, Haruki cannot form this hexagon.

2. A square of area 36 cm$^2$: The side length is $\sqrt{36} = 6$ cm. The perimeter is $4 \times 6 = 24$ cm, which matches the wire length. So, Haruki can form this square.

3. A right triangle with legs 6 and 8 cm: Using the Pythagorean theorem, the hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm. The perimeter is $6 + 8 + 10 = 24$ cm, which matches the wire length. So, Haruki can form this triangle.

Let's examine the choices:
- A. Triangle only: Incorrect because Haruki can also form the square.
- B. Hexagon and square only: Incorrect because Haruki cannot form the hexagon.
- C. Hexagon and triangle only: Incorrect because Haruki cannot form the hexagon.
- D. Square and triangle only: Matches our findings.
- E. Hexagon, triangle, and square: Incorrect because Haruki cannot form the hexagon.

The correct shapes Haruki can form are the square and the triangle.

ANSWER 7: D

Problem 8:
The problem asks us to find which of the given numbers has the largest reciprocal.

The reciprocal of a number $x$ is $1/x$. The larger $x$ is, the smaller its reciprocal is, and vice versa.

Let's examine the choices:
- A. $1⁄3$: The reciprocal is 3.
- B. $2⁄5$: The reciprocal is $5/2 = 2.5$.
- C. 1: The reciprocal is 1.
- D. 5: The reciprocal is $1/5 = 0.2$.
- E. 1986: The reciprocal is $1/1986$, very small.

Comparing the reciprocals: $3 > 2.5 > 1 > 0.2 > 1/1986$. So, $1/3$ has the largest reciprocal.

Let's examine the choices again:
- A. $1⁄3$: Correct.
- B. $2⁄5$: Not correct.
- C. 1: Not correct.
- D. 5: Not correct.
- E. 1986: Not correct.

The number with the largest reciprocal is $1/3$.

ANSWER 8: A

Problem 9:
The proble

Problem 1:
Original mixture: 30 L. Yellow tint = 30% of 30 = 0.30 × 30 = 9 L.
Add 5 L yellow: new yellow = 9 + 5 = 14 L, new total volume = 30 + 5 = 35 L.
Percent yellow = (14/35) × 100% = 40%.
ANSWER 1: C

Problem 2:
6 choices for entry window, 5 remaining choices for a different exit window.
Total ways = 6 × 5 = 30.
ANSWER 2: D

Problem 3:
1/10 + 2/20 + 3/30 = 1/10 + 1/10 + 1/10 = 3/10 = 0.3.
ANSWER 3: D

Problem 4:
Spent on food: $12. Spent on rides: 2 × $12 = $24.
Total spent = $12 + $24 = $36. Left = $50 − $36 = $14.
ANSWER 4: B

Problem 5:
200,000 × 200,000 = (2×10⁵) × (2×10⁵) = 4×10¹⁰ = 40,000,000,000, which is forty billion.
ANSWER 5: E

Problem 6:
Total paid is $1A2, a multiple of 11. Divisibility by 11: (1 − A + 2) = 3 − A must be a multiple of 11.
With A a digit 0–9, 3 − A = 0 ⇒ A = 3. (Check: 132 ÷ 11 = 12.)
ANSWER 6: D

Problem 7:
Wire length = 24 cm.
- Regular hexagon, side 5 cm: perimeter = 6×5 = 30 cm > 24 cm → cannot make.
- Square, area 36 cm²: side = 6 cm, perimeter = 4×6 = 24 cm → can make.
- Right triangle, legs 6 and 8 cm: hypotenuse = 10 cm, perimeter = 6+8+10 = 24 cm → can make.
So square and triangle only.
ANSWER 7: D

Problem 8:
Reciprocals: 1/(1/3)=3, 1/(2/5)=2.5, 1/1=1, 1/5=0.2, 1/1986≈0.0005. Largest is 3, from 1/3.
ANSWER 8: A

Problem 9:
Maximum heart rate = 220 − 26 = 194 bpm.
Target = 80% of 194 = 0.8 × 194 = 155.2 → 155 to nearest whole number.
ANSWER 9: B

Problem 10:
Harry: H = 8 − (2+5) = 8 − 7 = 1.
Terry: T = 8 − 2 + 5 = 6 + 5 = 11.
H − T = 1 − 11 = −10.
ANSWER 10: A

Problem 11:
Sum inside parentheses: 10 terms, average = (81+99)/2 = 90, sum = 10×90 = 900.
2 × 900 = 1800.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96 ⇒ Males = 200 − 96 = 104.
Total listeners = 136, female listeners = 58 ⇒ male listeners = 136 − 58 = 78.
Percent of males who listen = (78/104) × 100% = 75%.
ANSWER 12: E

Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint.

**Original mixture (30 liters):**
- Red tint: 25% of 30 = 7.5 liters
- Yellow tint: 30% of 30 = 9 liters
- Water: 45% of 30 = 13.5 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total mixture: 30 + 5 = 35 liters

**New percentage of yellow tint:**
14 / 35 = 2/5 = 0.40 = 40%

Tempting wrong answers: Choice B (35) might come from just adding 5 to the original 30%, but we must recalculate based on the new total.

ANSWER 1: C

---

Problem 2:
Georgie enters through one window and leaves through a *different* window.

- Number of choices for the entry window: 6
- Number of choices for the exit window (must be different): 5
- Total ways: 6 × 5 = 30

Tempting wrong answers: Choice E (36) would be 6 × 6, which allows entering and leaving through the same window. Choice A (12) would be 6 + 6, which doesn't account for all combinations.

ANSWER 2: D

---

Problem 3:
Simplify each fraction first:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

Tempting wrong answers: Choice B (.123) might come from concatenating the digits 1, 2, 3 instead of actually computing. Choice C (.2) might come from miscounting the terms.

ANSWER 3: D

---

Problem 4:
Susan starts with $50.
- Spent on food: $12
- Spent on rides: twice the food amount = 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Amount left: $50 − $36 = $14

Tempting wrong answers: Choice C (26) might come from subtracting only the food ($50 − $24 = $26, forgetting food). Choice D (38) comes from subtracting only $12.

ANSWER 4: B

---

Problem 5:
200,000 × 200,000

Write in scientific notation:
(2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰

4 × 10¹⁰ = 40,000,000,000 = forty billion

Tempting wrong answers: Choice A (four hundred thousand) comes from adding instead of multiplying. Choice D (four hundred million) comes from miscounting zeros.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, so the total $1A2 must be divisible by 11.

**Divisibility rule for 11:** The alternating sum of digits must be a multiple of 11.
Alternating sum: 1 − A + 2 = 3 − A

For 3 − A to be divisible by 11 (and A to be a single digit 0–9):
3 − A = 0 → A = 3

**Check:** $132 ÷ 11 = $12 per member. This is a valid integer amount.

Tempting wrong answers: Other values of A don't make 3 − A a multiple of 11 within the digit range.

ANSWER 6: D

---

Problem 7:
Wire length = 24 cm. We check if each shape's perimeter ≤ 24 cm.

**Regular hexagon (side 5 cm):**
Perimeter = 6 × 5 = 30 cm. Since 30 > 24, he **cannot** make this.

**Square (area 36 cm²):**
Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Since 24 = 24, he **can** make this.

**Right triangle (legs 6 and 8 cm):**
Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Perimeter = 6 + 8 + 10 = 24 cm. Since 24 = 24, he **can** make this.

Haruki can make the square and triangle only.

Tempting wrong answers: Choice E includes the hexagon, but its perimeter exceeds the wire length.

ANSWER 7: D

---

Problem 8:
The reciprocal of a number x is 1/x. The *smaller* the positive number, the *larger* its reciprocal.

| Choice | Number | Reciprocal |
|--------|--------|------------|
| A | 1/3 | 3 |
| B | 2/5 | 5/2 = 2.5 |
| C | 1 | 1 |
| D | 5 | 1/5 = 0.2 |
| E | 1986 | 1/1986 ≈ 0.0005 |

The largest reciprocal is 3, which comes from 1/3.

Tempting wrong answers: Choice E (1986) is the largest number, so it has the *smallest* reciprocal.

ANSWER 8: A

---

Problem 9:
Athlete's age = 26 years.

**Step 1:** Maximum heart rate = 220 − 26 = 194 beats per minute.

**Step 2:** Target heart rate = 80% of 194 = 0.80 × 194 = 155.2

**Step 3:** Round to nearest whole number: 155

Tempting wrong answers: Choice C (176) might come from taking 80% of 220 without subtracting the age. Choice D (194) is the maximum heart rate itself.

ANSWER 9: B

---

Problem 10:
**Harry (correct calculation):**
H = 8 − (2 + 5) = 8 − 7 = 1

**Terry (ignores parentheses):**
T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:**
H − T = 1 − 11 = −10

Tempting wrong answers: Choice E (10) is |H − T|, but the question asks for H − T specifically, which is negative since Terry's answer is larger.

ANSWER 10: A

---

Problem 11:
We need: 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

The sum inside is an arithmetic sequence:
- First term: 81, Last term: 99
- Number of terms: 10
- Sum = (number of terms) × (first + last) / 2 = 10 × (81 + 99) / 2 = 10 × 180 / 2 = 10 × 90 = 900

Multiply by 2: 2 × 900 = 1800

Tempting wrong answers: Choice C (1700) might come from miscounting terms or miscalculating the average.

ANSWER 11: E

---

Problem 12:
Let me organize the data from the survey:

- Total surveyed: 200
- Females: 96 → Males: 200 − 96 = 104
- Total listeners: 136
- Females who listen: 58 → Males who listen: 136 − 58 = 78
- Males who don't listen: 26 (Check: 78 + 26 = 104 ✓)

**Percentage of males who listen:**
(78 / 104) × 100% = (3/4) × 100% = 75%

Tempting wrong answers: Choice C (52) might come from computing 104/200 (percentage of total who are male). Choice D (55) might come from 78/136 (percentage of listeners who are male).

ANSWER 12: E

 Problem 1:
We start with 30 liters of paint. The original amount of yellow tint is 30% of 30 liters:
\[
0.30 \times 30 = 9 \text{ liters.}
\]
Five liters of yellow tint are added, so the new amount of yellow tint is:
\[
9 + 5 = 14 \text{ liters.}
\]
The new total volume of the mixture is:
\[
30 + 5 = 35 \text{ liters.}
\]
The percent of yellow tint in the new mixture is:
\[
\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%.
\]
(Choice A is the original red percentage, D is the original water percentage, and E would require 17.5 liters of yellow, which we do not have.)
ANSWER 1: C

Problem 2:
Georgie chooses one of 6 windows to enter. Since he must leave through a different window, there are 5 remaining choices for his exit. By the multiplication principle:
\[
6 \times 5 = 30 \text{ ways.}
\]
(Choice E, 36, would count entering and leaving through the same window; the other choices are too small.)
ANSWER 2: D

Problem 3:
Simplify each fraction:
\[
\frac{2}{20} = \frac{1}{10}, \qquad \frac{3}{30} = \frac{1}{10}.
\]
So the sum is:
\[
\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3.
\]
(Choice A is just one term; C is 0.2; E is 0.6.)
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides:
\[
2 \times 12 = 24 \text{ dollars.}
\]
Total spent:
\[
12 + 24 = 36 \text{ dollars.}
\]
Amount left:
\[
50 - 36 = 14 \text{ dollars.}
\]
(Choice C is the total spent; D is the amount left if she had spent only $12.)
ANSWER 4: B

Problem 5:
\[
200{,}000 \times 200{,}000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}.
\]
\[
4 \times 10^{10} = 40{,}000{,}000{,}000,
\]
which is read as forty billion.
(Choice D is $4 \times 10^8$; B is $4 \times 10^6$.)
ANSWER 5: E

Problem 6:
The total amount paid is the three-digit number $1A2$, which equals:
\[
100 + 10A + 2 = 102 + 10A.
\]
Since 11 members each paid the same integer amount, $102 + 10A$ must be divisible by 11. Testing the choices:
- $A=0$: $102 \div 11 \approx 9.27$ (not an integer)
- $A=1$: $112 \div 11 \approx 10.18$ (not an integer)
- $A=2$: $122 \div 11 \approx 11.09$ (not an integer)
- $A=3$: $132 \div 11 = 12$ (integer!)
- $A=4$: $142 \div 11 \approx 12.9$ (not an integer)
Thus the missing digit is 3.
ANSWER 6: D

Problem 7:
The wire is 24 cm long, so a shape is possible only if its perimeter is at most 24 cm.
- **Regular hexagon, side 5 cm:** Perimeter $= 6 \times 5 = 30$ cm. $30 > 24$, so impossible.
- **Square of area 36 cm²:** Side $= \sqrt{36} = 6$ cm. Perimeter $= 4 \times 6 = 24$ cm. Exactly fits, so possible.
- **Right triangle with legs 6 and 8 cm:** Hypotenuse $= \sqrt{6^2 + 8^2} = \sqrt{100} = 10$ cm. Perimeter $= 6 + 8 + 10 = 24$ cm. Exactly fits, so possible.
Only the square and the triangle can be formed.
ANSWER 7: D

Problem 8:
For positive numbers, the reciprocal function $f(x) = \frac{1}{x}$ is decreasing, so the smallest number has the largest reciprocal. Compare the two smallest choices:
\[
\frac{1}{3} \approx 0.333\ldots, \qquad \frac{2}{5} = 0.4.
\]
Since $\frac{1}{3} < \frac{2}{5} < 1 < 5 < 1986$, the smallest number is $\frac{1}{3}$. Its reciprocal is $3$, which is larger than the reciprocals of all the other choices.
ANSWER 8: A

Problem 9:
First find the theoretical maximum heart rate:
\[
220 - 26 = 194 \text{ beats per minute.}
\]
The target rate is 80% of this:
\[
0.80 \times 194 = 155.2.
\]
Rounded to the nearest whole number, this is 155.
(Choice C is the maximum heart rate itself.)
ANSWER 9: B

Problem 10:
Harry computes correctly:
\[
H = 8 - (2 + 5) = 8 - 7 = 1.
\]
Terry ignores the parentheses:
\[
T = 8 - 2 + 5 = 6 + 5 = 11.
\]
The difference is:
\[
H - T = 1 - 11 = -10.
\]
ANSWER 10: A

Problem 11:
The ten numbers inside the parentheses form an arithmetic sequence: 81, 83, …, 99. The sum of an arithmetic sequence is:
\[
\frac{\text{number of terms}}{2} \times (\text{first} + \text{last}) = \frac{10}{2} \times (81 + 99) = 5 \times 180 = 900.
\]
Now multiply by the factor of 2 outside:
\[
2 \times 900 = 1800.
\]
(One can also pair them: $(81+99)=180$, $(83+97)=180$, etc., giving five pairs that each sum to 180.)
ANSWER 11: E

Problem 12:
Total surveyed = 200, and females surveyed = 96, so:
\[
\text{Males surveyed} = 200 - 96 = 104.
\]
Total listeners = 136, and females who listen = 58, so:
\[
\text{Males who listen} = 136 - 58 = 78.
\]
(We can check: total non-listeners = 64, males who don't listen = 26, so females who don't listen = 38; $58+38=96$ and $78+26=104$, which checks out.)

The percentage of males who listen is:
\[
\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%.
\]
ANSWER 12: E

Problem 1:
A mixture of 30 liters is 25% red, 30% yellow, and 45% water. Five liters of yellow tint are added. What is the percent of yellow tint in the new mixture?

**Setup:** Find the amount of yellow tint originally, add 5 liters, then find the new percentage.

**Original yellow tint:** 30% of 30 = 0.30 × 30 = 9 liters

**After adding 5 liters of yellow tint:**
- New yellow tint: 9 + 5 = 14 liters
- New total volume: 30 + 5 = 35 liters
- New percentage: 14/35 = 2/5 = 40%

**Checking wrong choices:** 25% would be if we just added the original percentages; 35% would be if we just added 5% to 30%; 45% and 50% are too high.

ANSWER 1: C

---

Problem 2:
Six windows. Georgie enters by one and leaves by a different one. How many ways?

**Setup:** Choose an entry window (6 choices), then choose a different exit window (5 choices).

**Calculation:** 6 × 5 = 30 ways

**Checking wrong choices:** 15 would be if we divided by 2 (treating order as irrelevant); 12 would be 6×2; 18 would be 6×3; 36 would be 6×6 (allowing same window).

ANSWER 2: D

---

Problem 3:
1/10 + 2/20 + 3/30 = ?

**Simplify each fraction:**
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

**Sum:** 1/10 + 1/10 + 1/10 = 3/10 = 0.3

**Checking wrong choices:** 0.1 would be just one term; 0.2 would be two terms; 0.6 would be adding numerators and denominators incorrectly.

ANSWER 3: D

---

Problem 4:
Susan had $50. She spent $12 on food and twice as much on rides. How much left?

**Setup:** Food = $12, Rides = 2 × $12 = $24

**Total spent:** $12 + $24 = $36

**Amount left:** $50 - $36 = $14

**Checking wrong choices:** 12 is just the food amount; 26 is 50-24 (forgetting food); 38 is 50-12 (forgetting rides).

ANSWER 4: B

---

Problem 5:
200,000 × 200,000 = ?

**Setup:** Express in scientific notation: 200,000 = 2 × 10⁵

**Calculation:** (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion

**Checking wrong choices:** Four hundred thousand is 4×10⁵; four million is 4×10⁶; forty thousand is 4×10⁴; four hundred million is 4×10⁸.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, totaling $1A2. Find the missing digit A.

**Setup:** The total $1A2 must be divisible by 11.

**Divisibility test for 11:** A number is divisible by 11 if the alternating sum of its digits is divisible by 11.

For 1A2: (1 + 2) - A = 3 - A must be divisible by 11.

Since A is a single digit (0-9), 3 - A can only equal 0 (the only multiple of 11 in range -6 to 3).

So A = 3, giving $132 = 11 × $12 ✓

**Checking:** With A=0: $102 ÷ 11 ≈ 9.27 (not integer); A=1: $112 ÷ 11 ≈ 10.18 (not integer); A=3: $132 ÷ 11 = 12 ✓

ANSWER 6: D

---

Problem 7:
Wire is 24 cm. Which shapes can be made?

**Regular hexagon with side 5 cm:** Perimeter = 6 × 5 = 30 cm. Need 30 cm, only have 24 cm. ❌

**Square with area 36 cm²:** Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Need 24 cm, have 24 cm. ✓

**Right triangle with legs 6 and 8 cm:** Hypotenuse = √(36 + 64) = √100 = 10 cm. Perimeter = 6 + 8 + 10 = 24 cm. Need 24 cm, have 24 cm. ✓

**Conclusion:** Can make the square and triangle, but not the hexagon.

ANSWER 7: D

---

Problem 8:
Which number has the largest reciprocal?

**Find each reciprocal:**
- A. 1/3 → reciprocal = 3
- B. 2/5 → reciprocal = 5/2 = 2.5
- C. 1 → reciprocal = 1
- D. 5 → reciprocal = 1/5 = 0.2
- E. 1986 → reciprocal ≈ 0.0005

**Key insight:** Smaller positive numbers have larger reciprocals. The smallest number listed is 1/3, so it has the largest reciprocal (3).

ANSWER 8: A

---

Problem 9:
Target heart rate = 80% of (220 - age). Find target for 26-year-old.

**Maximum heart rate:** 220 - 26 = 194 bpm

**Target heart rate:** 80% of 194 = 0.80 × 194 = 155.2 ≈ 155 bpm

**Checking wrong choices:** 134 would be about 69%; 176 would be about 91%; 194 is the max (100%); 243 is impossible.

ANSWER 9: B

---

Problem 10:
Harry calculates 8 − (2 + 5) correctly. Terry calculates 8 − 2 + 5. Find H − T.

**Harry's answer:** H = 8 − (2 + 5) = 8 − 7 = 1

**Terry's answer:** T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:** H − T = 1 − 11 = −10

**Checking wrong choices:** −6 would be if Terry got 7; 0 would be if they got the same answer; 6 or 10 would be T − H.

ANSWER 10: A

---

Problem 11:
2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99) = ?

**Setup:** This is 10 consecutive odd numbers. The average of the first and last is (81 + 99)/2 = 90.

**Sum of the 10 numbers:** 10 × 90 = 900

**Multiply by 2:** 2 × 900 = 1800

**Alternative verification:** Pair the numbers: (81+99) + (83+97) + (85+95) + (87+93) + (89+91) = 180 + 180 + 180 + 180 + 180 = 900. Then 2 × 900 = 1800.

ANSWER 11: E

---

Problem 12:
Find the percentage of males who listen to the station.

**Given information:**
- Total surveyed: 200
- Females: 96
- Females who listen: 58
- Males who don't listen: 26
- Total listeners: 136
- Total non-listeners: 64

**Find males:** Males = 200 - 96 = 104

**Find males who listen:** Males who listen = Total listeners - Females who listen = 136 - 58 = 78

**Verification:** Males who don't listen = 104 - 78 = 26 ✓ (matches given)

**Percentage of males who listen:** 78/104 × 100 = 75%

**Checking wrong choices:** 39% would be 78/200 (using total instead of males); 48% would be about 50/104; 52% would be about 54/104; 55% would be about 57/104.

ANSWER 12: E

**Problem 1**

The original mixture is 30 L with 30 % yellow tint → yellow = 0.30·30 = 9 L.  
Add 5 L of yellow → total yellow = 9 + 5 = 14 L.  
Total volume after addition = 30 + 5 = 35 L.  

Percent yellow = (14/35)·100 % = 40 %.

**Answer 1: C**

---

**Problem 2**

Choose an entry window (6 ways) and a different exit window (5 ways).  
Number of ways = 6 × 5 = 30.

**Answer 2: D**

---

**Problem 3**

\[
\frac{1}{10}=0.1,\qquad\frac{2}{20}=0.1,\qquad\frac{3}{30}=0.1
\]
Sum = 0.1 + 0.1 + 0.1 = 0.3.

**Answer 3: D**

---

**Problem 4**

Food cost = \$12.  
Rides cost = twice food = 2·12 = \$24.  
Total spent = 12 + 24 = \$36.  
Money left = 50 − 36 = \$14.

**Answer 4: B**

---

**Problem 5**

\(200{,}000 \times 200{,}000 = (2\times10^{5})^{2}=4\times10^{10}=40{,}000{,}000{,}000\).

That is 40 billion.

**Answer 5: E**

---

**Problem 6**

Total paid = \$1A2 = 100 + 10A + 2 = 102 + 10A.  
Since the 11 members each paid an integer amount, 102 + 10A must be divisible by 11.

\(102\equiv3\pmod{11}\).  
We need \(10A\equiv-3\equiv8\pmod{11}\).  
Because \(10\equiv-1\pmod{11}\), this is \(-A\equiv8\pmod{11}\) → \(A\equiv3\pmod{11}\).  
A digit, so \(A=3\).

**Answer 6: D**

---

**Problem 7**

- Hexagon: perimeter = 6·5 = 30 cm > 24 cm → impossible.  
- Square: side = √36 = 6 cm, perimeter = 4·6 = 24 cm → possible.  
- Right triangle: legs 6 cm, 8 cm, hypotenuse = √(6²+8²)=10 cm, perimeter = 6+8+10 = 24 cm → possible.

Only the square and triangle can be made.

**Answer 7: D**

---

**Problem 8**

Reciprocals:
- \(1/3 → 3\)  
- \(2/5 → 5/2 = 2.5\)  
- \(1 → 1\)  
- \(5 → 1/5 = 0.2\)  
- \(1986 → 1/1986 ≈ 0.0005\)

Largest reciprocal is 3, from \(1/3\).

**Answer 8: A**

---

**Problem 9**

Maximum heart rate = 220 − 26 = 194 bpm.  
Target = 80 % of 194 = 0.8·194 = 155.2 → rounded to 155 bpm.

**Answer 9: B**

---

**Problem 10**

Harry: \(8-(2+5)=8-7=1\).  
Terry (ignoring parentheses): \(8-2+5 = 6+5 = 11\).  
Difference \(H-T = 1-11 = -10\).

**Answer 10: A**

---

**Problem 11**

Sum of the ten terms: average = (81+99)/2 = 90, so sum = 90·10 = 900.  
Multiply by 2: \(2·900 = 1800\).

**Answer 11: E**

---

**Problem 12**

Total surveyed = 200.  
Females = 96 → Males = 200 − 96 = 104.  

Female listeners = 58 → Male listeners = 136 − 58 = 78.  

Percent of males who listen = (78/104)·100 % = (39/52)·100 % = 75 %.

**Answer 12: E**

---

**Final Answers**

ANSWER 1: C  
ANSWER 2: D  
ANSWER 3: D  
ANSWER 4: B  
ANSWER 5: E  
ANSWER 6: D  
ANSWER 7: D  
ANSWER 8: A  
ANSWER 9: B  
ANSWER 10: A  
ANSWER 11: E  
ANSWER 12: E

Problem 1:  
A 30-liter paint mixture is 25% red tint, 30% yellow tint, and 45% water. Five liters of yellow tint are added.  
Original yellow tint: $0.30 \times 30 = 9$ liters.  
After adding 5 liters: $9 + 5 = 14$ liters yellow.  
New total volume: $30 + 5 = 35$ liters.  
Percent yellow: $\frac{14}{35} = 0.40 = 40\%$.  
Tempting wrong choices: 35% (original yellow percent) or 45% (water percent).  
ANSWER 1: D  

Problem 2:  
Georgie enters through one of six windows and exits through a different one.  
Entry choices: 6. Exit choices: 5 (excluding entry window).  
Total ways: $6 \times 5 = 30$.  
Tempting wrong choices: 15 (if order didn’t matter) or 36 (if same window allowed).  
ANSWER 2: D  

Problem 3:  
$\frac{1}{10} + \frac{2}{20} + \frac{3}{30} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3$.  
Tempting wrong choices: 0.1 (one term) or 0.2 (two terms).  
ANSWER 3: D  

Problem 4:  
Susan starts with \$50.  
Food: \$12.  
Rides: $2 \times 12 = \$24$.  
Total spent: $12 + 24 = \$36$.  
Left: $50 - 36 = \$14$.  
Tempting wrong choices: \$26 (if spent \$24 total) or \$38 (if spent \$12 total).  
ANSWER 4: B  

Problem 5:  
$200,000 \times 200,000 = 40,000,000,000 = 40$ billion.  
Tempting wrong choices: 400,000 (added zeros) or 4 million (multiplied digits).  
ANSWER 5: E  

Problem 6:  
Total paid: \$1A2, an integer divisible by 11.  
$1A2 \mod 11 = 0$.  
$1 + 2 - A \equiv 0 \mod 11 \Rightarrow 3 - A \equiv 0 \mod 11 \Rightarrow A = 3$.  
Tempting wrong choices: 0, 1, 2, or 4 (incorrect divisibility).  
ANSWER 6: E (Wait, let's recheck: 1A2 must be divisible by 11. 132 ÷ 11 = 12, so A=3. But 132 is 11×12, so correct. However, the answer choices are A=0,B=1,C=2,D=3,E=4. So D is 3. So correct answer is D, not E. I made a mistake in the initial answer. Correcting:)  
**Correction**: $1A2 = 132$ when $A=3$, and $132 \div 11 = 12$, so $A=3$.  
ANSWER 6: D  

Problem 7:  
Wire length: 24 cm.  
- Regular hexagon with side 5 cm: Perimeter $6 \times 5 = 30$ cm (not 24).  
- Square with area 36 cm²: Side $6$ cm, perimeter $24$ cm.  
- Right triangle with legs 6 and 8 cm: Hypotenuse 10 cm, perimeter $6+8+10=24$ cm.  
Can make square and triangle.  
Tempting wrong choices: Hexagon (incorrect perimeter).  
ANSWER 7: D  

Problem 8:  
Largest reciprocal means smallest original number.  
$\frac{1}{3} \approx 0.333$, $\frac{2}{5} = 0.4$, $1$, $5$, $1986$.  
Smallest: $\frac{1}{3}$. Reciprocal: 3.  
Others: 2.5, 1, 0.2, 0.0005.  
Tempting wrong choices: $\frac{2}{5}$ (larger than $\frac{1}{3}$).  
ANSWER 8: A  

Problem 9:  
Maximum heart rate: $220 - 26 = 194$.  
Target: $0.80 \times 194 = 155.2 \approx 155$.  
Tempting wrong choices: 134 (70%), 176 (90%), or 243 (incorrect max).  
ANSWER 9: B  

Problem 10:  
Harry: $8 - (2 + 5) = 1$.  
Terry: $8 - 2 + 5 = 11$.  
Difference: $1 - 11 = -10$.  
Tempting wrong choices: 10 (absolute difference) or 0 (if parentheses ignored by both).  
ANSWER 10: A  

Problem 11:  
Sum: $81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99$.  
Average: 90, 10 terms: $90 \times 10 = 900$.  
$2 \times 900 = 1800$.  
Tempting wrong choices: 1600 (min+max=180, times 5=900, times 2=1800, but if miscount terms).  
ANSWER 11: E  

Problem 12:  
Total surveyed: 200.  
Females: 96, so males: $200 - 96 = 104$.  
Total listeners: 136.  
Female listeners: 58, so male listeners: $136 - 58 = 78$.  
Percent male listeners: $\frac{78}{104} = 0.75 = 75\%$.  
Tempting wrong choices: 39 (male non-listeners percent) or 52 (incorrect calculation).  
ANSWER 12: E

### Problem 1:
What is the percentage of yellow tint in the new 35-liter paint mixture, after adding 5 liters of yellow tint to an original 30-liter mixture that was 30% yellow?
First, calculate the original volume of yellow tint: 30% of 30 liters = \(0.3 \times 30 = 9\) liters. Add the 5 new liters of yellow tint to get \(9 + 5 = 14\) liters of yellow in the new mixture. The total new volume of the mixture is \(30 + 5 = 35\) liters. The percentage of yellow is \(\frac{14}{35} = 0.4 = 40\%\). Tempting wrong answers come from forgetting to update the total mixture volume when calculating the new percentage, or miscalculating the original yellow volume.
ANSWER 1: C

---

### Problem 2:
How many ways can Georgie enter through one window and leave through a different window, out of 6 total windows?
Georgie has 6 choices for an entry window. Since he must leave through a different window, he has 5 remaining choices for an exit window. The total number of ways is \(6 \times 5 = 30\). A common wrong answer is 15, which incorrectly treats entry/exit as unordered pairs, when order matters (entering window A and leaving B is different from entering B and leaving A).
ANSWER 2: D

---

### Problem 3:
Simplify \(\frac{1}{10} + \frac{2}{20} + \frac{3}{30}\):
Simplify each fraction first: \(\frac{2}{20} = \frac{1}{10}\) and \(\frac{3}{30} = \frac{1}{10}\). Add them together: \(\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3\). A common wrong answer is 0.1, which incorrectly adds numerators and denominators directly instead of simplifying first.
ANSWER 3: D

---

### Problem 4:
How much money does Susan have left after spending $12 on food and twice that amount on rides, starting with $50?
First calculate the amount spent on rides: twice $12 is \(2 \times 12 = 24\) dollars. Total spent is \(12 + 24 = 36\) dollars. The amount left is \(50 - 36 = 14\) dollars. A common wrong answer is 26, which incorrectly uses $12 instead of $24 for ride costs.
ANSWER 4: B

---

### Problem 5:
Calculate \(200,000 \times 200,000\):
Rewrite 200,000 as \(2 \times 10^5\). Multiplying two copies gives \((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\), which equals 40,000,000,000, or forty billion. Wrong answers come from miscounting the number of zeros in the product.
ANSWER 5: E

---

### Problem 6:
The 3-digit total cost \(1A2\) dollars is divisible by 11 (11 members paid the same integer amount). What is the missing digit A?
The divisibility rule for 11 states that alternating sum of digits must be divisible by 11. For 1A2, the alternating sum is \(1 - A + 2 = 3 - A\). The only value of A (0-9) that makes this sum divisible by 11 is A=3, since \(3-3=0\), which is divisible by 11. Verify: \(132 \div 11 = 12\), a valid integer.
ANSWER 6: D

---

### Problem 7:
Haruki has a 24cm wire. Which of the three shapes can he form? Calculate the perimeter of each shape:
1. Regular hexagon with side 5cm: perimeter = \(6 \times 5 = 30\)cm, which is longer than 24cm, so impossible.
2. Square with area 36cm²: side length = \(\sqrt{36}=6\)cm, perimeter = \(4 \times 6 = 24\)cm, which fits.
3. Right triangle with legs 6cm and 8cm: hypotenuse = 10cm (6-8-10 Pythagorean triple), perimeter = \(6+8+10=24\)cm, which fits.
Only the square and triangle can be formed.
ANSWER 7: D

---

### Problem 8:
Which number has the largest reciprocal? The reciprocal of \(x\) is \(\frac{1}{x}\), so smaller values of \(x\) produce larger reciprocals. List the values of all options:
A. \(\frac{1}{3} \approx 0.333\), B. \(\frac{2}{5}=0.4\), C.1, D.5, E.1986. The smallest value is \(\frac{1}{3}\), whose reciprocal is 3, the largest of all options.
ANSWER 8: A

---

### Problem 9:
Calculate the target heart rate for a 26-year-old athlete: target is 80% of max heart rate, where max = 220 - age.
First calculate max heart rate: \(220 - 26 = 194\) bpm. 80% of 194 is \(0.8 \times 194 = 155.2\), which rounds to 155 bpm. A common wrong answer is 176, which uses 220 instead of the correct max heart rate of 194.
ANSWER 9: B

---

### Problem 10:
Calculate \(H-T\), where H is Harry's correct answer to \(8-(2+5)\), and T is Terry's incorrect answer of \(8-2+5\).
Harry's answer: \(8 - 7 = 1 = H\). Terry's answer: \(6 + 5 = 11 = T\). The difference \(H-T = 1 - 11 = -10\). A common wrong answer is 10, which calculates \(T-H\) instead of \(H-T\).
ANSWER 10: A

---

### Problem 11:
Simplify \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\):
Pair the terms inside the parentheses to simplify addition: \((81+99)+(83+97)+(85+95)+(87+93)+(89+91) = 5 \times 180 = 900\). Multiply by 2: \(2 \times 900 = 1800\).
ANSWER 11: E

---

### Problem 12:
What percentage of males surveyed listen to the radio station? First calculate total number of males: total surveyed = 200, females = 96, so males = \(200 - 96 = 104\). Males who do not listen = 26, so males who listen = \(104 - 26 = 78\). The percentage of males who listen is \(\frac{78}{104} = 0.75 = 75\%\). A common wrong answer is 39%, which divides male listeners by the total surveyed population instead of the total male population.
ANSWER 12: E

### Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.
1.  Calculate original yellow tint volume: 25%? No wait, original is 30% yellow. So 30% of 30 L = $0.3 \times 30 = 9$ liters of yellow.
2.  Add 5 liters of yellow: total yellow = $9 + 5 = 14$ liters.
3.  Total new mixture volume: $30 + 5 = 35$ liters.
4.  Percent yellow: $\frac{14}{35} \times 100 = 40\%$.
ANSWER 1: C
---
### Problem 2:
We count the number of ways to choose an entry window and a different exit window from 6 windows.
1.  Entry window: 6 possible choices.
2.  Exit window: cannot be the entry window, so 5 remaining choices.
3.  Total ways: $6 \times 5 = 30$ (order matters because entry and exit are distinct actions).
ANSWER 2: D
---
### Problem 3:
Simplify and add the fractions:
1.  $\frac{1}{10} = 0.1$, $\frac{2}{20} = \frac{1}{10} = 0.1$, $\frac{3}{30} = \frac{1}{10} = 0.1$.
2.  Sum: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D
---
### Problem 4:
Calculate remaining money after expenses:
1.  Total initial money: $50.
2.  Money spent on rides: twice the food cost = $2 \times 12 = 24$ dollars.
3.  Total spent: $12 + 24 = 36$ dollars.
4.  Remaining: $50 - 36 = 14$ dollars.
ANSWER 4: B
---
### Problem 5:
Compute $200,000 \times 200,000$:
1.  Rewrite as $(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
2.  $10^{10} = 10,000,000,000$, so $4 \times 10^{10} = 40,000,000,000 = 40$ billion.
ANSWER 5: E
---
### Problem 6:
The total payment $1A2$ (a 3-digit number) must be divisible by 11 (since 11 people paid equal integer amounts).
1.  Use the divisibility rule for 11: the alternating sum of digits is divisible by 11. For $1A2$, this is $1 - A + 2 = 3 - A$.
2.  $A$ is a digit (0-9), so $3 - A$ must equal 0 (the only multiple of 11 in range: $3 - A = 11$ gives $A=-8$, $3-A=-11$ gives $A=14$, both invalid).
3.  $3 - A = 0 \implies A=3$. Verify: $132 \div 11 = 12$, which is an integer.
ANSWER 6: D
---
### Problem 7:
Check if the 24 cm wire can form each shape (perimeter of the shape must equal 24 cm):
1.  Regular hexagon (side 5 cm): Perimeter = $6 \times 5 = 30$ cm > 24 cm → cannot make.
2.  Square (area 36 cm²): Side length = $\sqrt{36} = 6$ cm. Perimeter = $4 \times 6 = 24$ cm → can make.
3.  Right triangle (legs 6, 8 cm): Hypotenuse = $\sqrt{6^2 + 8^2} = 10$ cm. Perimeter = $6 + 8 + 10 = 24$ cm → can make.
Only square and triangle can be made.
ANSWER 7: D
---
### Problem 8:
The reciprocal of a positive number $x$ is $\frac{1}{x}$; smaller $x$ gives larger reciprocal.
1.  List values: $\frac{1}{3} \approx 0.33$, $\frac{2}{5}=0.4$, $1$, $5$, $1986$.
2.  Smallest value is $\frac{1}{3}$, whose reciprocal is $3$ (largest among all reciprocals: $\frac{5}{2}=2.5$, $1$, $0.2$, $\frac{1}{1986} \approx 0.0005$).
ANSWER 8: A
---
### Problem 9:
Calculate target heart rate for a 26-year-old:
1.  Maximum heart rate: $220 - 26 = 194$ bpm.
2.  Target heart rate: 80% of 194 = $0.8 \times 194 = 155.2$, which rounds to 155.
ANSWER 9: B
---
### Problem 10:
Compute $H$ (Harry's correct answer) and $T$ (Terry's incorrect answer):
1.  Harry: $8 - (2 + 5) = 8 - 7 = 1$, so $H=1$.
2.  Terry (ignores parentheses, left-to-right): $8 - 2 + 5 = 6 + 5 = 11$, so $T=11$.
3.  Difference: $H - T = 1 - 11 = -10$.
ANSWER 10: A
---
### Problem 11:
Calculate $2 \times (81 + 83 + ... + 99)$:
1.  The sum inside is 10 consecutive odd numbers. Average of first and last term: $\frac{81 + 99}{2} = 90$.
2.  Sum of the 10 terms: $90 \times 10 = 900$.
3.  Multiply by 2: $2 \times 900 = 1800$.
ANSWER 11: E
---
### Problem 12:
Find the percentage of males who listen to the station:
1.  Total males surveyed: $200 - 96 = 104$.
2.  Males who do not listen: 26, so males who listen: $104 - 26 = 78$.
3.  Verify with total listeners: Total listeners = 136, female listeners = 58, so male listeners = $136 - 58 = 78$ (matches).
4.  Percentage: $\frac{78}{104} \times 100 = 75\%$.
ANSWER 12: E

Problem 1:
Original yellow = 30% of 30 = 9 L. Add 5 → 14 L yellow. New total = 35 L. Percent = 14/35 = 40%.

ANSWER 1: C

Problem 2:
Enter: 6 choices, leave: 5 different choices. 6×5 = 30.

ANSWER 2: D

Problem 3:
Each fraction = 0.1. Sum = 0.3.

ANSWER 3: D

Problem 4:
Food 12, rides twice = 24. Spent 36. Left = 50 − 36 = 14.

ANSWER 4: B

Problem 5:
200,000 × 200,000 = 40,000,000,000 = forty billion.

ANSWER 5: E

Problem 6:
1A2 divisible by 11. Alternating sum: 1 − A + 2 = 3 − A must be divisible by 11 → A = 3 (gives 0). Check 132/11 = 12. Yes.

ANSWER 6: D

Problem 7:
Hexagon: 6×5 = 30 > 24, no. Square area 36 → side 6, perimeter 24, yes. Triangle legs 6,8 → hypotenuse 10, perimeter 24, yes. Square and triangle only.

ANSWER 7: D

Problem 8:
Largest reciprocal = smallest number. Smallest is 1/3.

ANSWER 8: A

Problem 9:
Max = 220 − 26 = 194. Target = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

Problem 10:
H = 8 − 7 = 1. T = 8 − 2 + 5 = 11. H − T = 1 − 11 = −10.

ANSWER 10: A

Problem 11:
Sum of 81..99 odd numbers (10 terms): average (81+99)/2 = 90, ×10 = 900. ×2 = 1800.

ANSWER 11: E

Problem 12:
Males total = 200 − 96 = 104. Males who don't listen = 26 → males who listen = 104 − 26 = 78. Percent = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem systematically.

---

**Problem 1:**
Original mixture: 30 liters, 30% yellow = 9 liters yellow. Add 5 liters yellow → new yellow = 14 liters, new total = 35 liters.
Percent yellow = 14/35 = 40%.

ANSWER 1: C

---

**Problem 2:**
Georgie can enter through any of 6 windows, then leave through any of the remaining 5 windows.
6 × 5 = 30.

ANSWER 2: D

---

**Problem 3:**
1/10 = 0.1, 2/20 = 0.1, 3/30 = 0.1. Sum = 0.3.

ANSWER 3: D

---

**Problem 4:**
Food: $12. Rides: 2 × $12 = $24. Total spent: $12 + $24 = $36. Left: $50 − $36 = $14.

ANSWER 4: B

---

**Problem 5:**
200,000 × 200,000 = (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion.

ANSWER 5: E

---

**Problem 6:**
The number 1A2 must be divisible by 11. 1A2 ÷ 11 must be an integer.
11 × 10 = 110, 11 × 11 = 121 (no), 11 × 12 = 132 ✓ → A = 3.
Check: 132/11 = 12 ✓, each paid $12.

ANSWER 6: D

---

**Problem 7:**
Wire = 24 cm.
- Regular hexagon, side 5: perimeter = 6 × 5 = 30 cm > 24. ✗
- Square, area 36: side = 6, perimeter = 24 cm ✓
- Right triangle, legs 6 and 8: hypotenuse = 10, perimeter = 6+8+10 = 24 cm ✓

ANSWER 7: D

---

**Problem 8:**
The reciprocal of x is 1/x. The largest reciprocal comes from the smallest positive number.
Among 1/3, 2/5, 1, 5, 1986 — the smallest is 1/3, and its reciprocal is 3, which is the largest.

ANSWER 8: A

---

**Problem 9:**
Maximum heart rate = 220 − 26 = 194. Target = 80% × 194 = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

---

**Problem 10:**
Harry: 8 − (2 + 5) = 8 − 7 = 1, so H = 1.
Terry: 8 − 2 + 5 = 11, so T = 11.
H − T = 1 − 11 = −10.

ANSWER 10: A

---

**Problem 11:**
The ten odd numbers from 81 to 99: sum = (81+99)/2 × 10 = 90 × 10 = 900.
2 × 900 = 1800.

ANSWER 11: E

---

**Problem 12:**
Total surveyed = 200. Females = 96, so Males = 104.
Total non-listeners = 64. Males who don't listen = 26, so females who don't listen = 64 − 26 = 38.
Males who listen = 104 − 26 = 78.
Percentage = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem step-by-step.

## Problem 1:
**What is asked:** Find the percentage of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.

**Original mixture:**
- Yellow tint: 30% of 30 = 0.30 × 30 = 9 liters
- Total volume: 30 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total volume: 30 + 5 = 35 liters

**Percentage of yellow tint:**
$$\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%$$

ANSWER 1: C

## Problem 2:
**What is asked:** In how many ways can the ghost enter by one window and leave by a different window?

**Solution:**
- Choices for entering: 6 windows
- Choices for leaving: 5 windows (must be different from entry)
- Total ways: 6 × 5 = 30

ANSWER 2: D

## Problem 3:
**What is asked:** Calculate 1/10 + 2/20 + 3/30

**Solution:**
Simplify each fraction:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

ANSWER 3: D

## Problem 4:
**What is asked:** How much money does Susan have left?

**Solution:**
- Started with: $50
- Spent on food: $12
- Spent on rides: 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Left: $50 - $36 = $14

ANSWER 4: B

## Problem 5:
**What is asked:** Calculate 200,000 × 200,000

**Solution:**
$$200,000 × 200,000 = (2 × 10^5) × (2 × 10^5) = 4 × 10^{10} = 40,000,000,000 = 40 \text{ billion}$$

ANSWER 5: E

## Problem 6:
**What is asked:** Find the missing digit A in the 3-digit number 1A2, where 11 club members each paid an equal integer amount that totals 1A2.

**Solution:**
The number 1A2 must be divisible by 11.

Testing divisibility by 11 (alternating sum of digits):
- 1 - A + 2 = 0 (mod 11)
- 3 - A ≡ 0 (mod 11)
- A = 3

Check: 132 ÷ 11 = 12 ✓

ANSWER 6: D

## Problem 7:
**What is asked:** Which shapes can Haruki make with 24 cm of wire?

**Solution:**
Test each shape:

**Regular hexagon with side 5 cm:**
- Perimeter: 6 × 5 = 30 cm
- Need 30 cm, have 24 cm ✗

**Square with area 36 cm²:**
- Side length: √36 = 6 cm
- Perimeter: 4 × 6 = 24 cm ✓

**Right triangle with legs 6 and 8 cm:**
- Hypotenuse: √(6² + 8²) = √(36 + 64) = √100 = 10 cm
- Perimeter: 6 + 8 + 10 = 24 cm ✓

Only square and triangle can be made.

ANSWER 7: D

## Problem 8:
**What is asked:** Which number has the largest reciprocal?

**Solution:**
Find reciprocals:
- 1/(1/3) = 3
- 1/(2/5) = 5/2 = 2.5
- 1/1 = 1
- 1/5 = 0.2
- 1/1986 ≈ 0.0005

The largest reciprocal is 5 (from the number 1/5).

ANSWER 8: D

## Problem 9:
**What is asked:** Find the target heart rate for a 26-year-old athlete.

**Solution:**
- Maximum heart rate: 220 - 26 = 194 beats per minute
- Target heart rate: 80% of 194 = 0.80 × 194 = 155.2 ≈ 155

ANSWER 9: B

## Problem 10:
**What is asked:** Find H - T where H is Harry's answer and T is Terry's answer.

**Solution:**
**Harry's calculation (correct):**
$$H = 8 - (2 + 5) = 8 - 7 = 1$$

**Terry's calculation (ignoring parentheses):**
$$T = 8 - 2 + 5 = 6 + 5 = 11$$

**Difference:**
$$H - T = 1 - 11 = -10$$

ANSWER 10: A

## Problem 11:
**What is asked:** Calculate 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

**Solution:**
This is an arithmetic sequence with 10 terms. Sum = n(first + last)/2
$$81 + 83 + ... + 99 = \frac{10(81 + 99)}{2} = \frac{10 × 180}{2} = 900$$

$$2 × 900 = 1800$$

ANSWER 11: E

## Problem 12:
**What is asked:** What percentage of males surveyed listen to the station?

**Solution:**
- Total surveyed: 200
- Females: 96
- Males: 200 - 96 = 104

- Total non-listeners: 64
- Males who don't listen: 26
- Males who listen: 104 - 26 = 78

**Percentage of males who listen:**
$$\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%$$

ANSWER 12: E

Problem 1:

The original mixture is 30 liters total, with 30% yellow tint.

So the amount of yellow tint at first is:
- \(30\%\) of \(30 = 0.30 \times 30 = 9\) liters

Then 5 liters of yellow tint are added, so the new amount of yellow tint is:
- \(9 + 5 = 14\) liters

The new total mixture is:
- \(30 + 5 = 35\) liters

So the percent of yellow tint is:
- \(\frac{14}{35} = \frac{2}{5} = 0.4 = 40\%\)

Answer choice C matches 40%.

ANSWER 1: C

Problem 2:

Georgie can:
- choose 1 of the 6 windows to enter
- then choose a different window to leave

For the entry, there are 6 choices.

For the exit, there are 5 remaining choices, since it must be a different window.

So the total number of ways is:
- \(6 \times 5 = 30\)

ANSWER 2: D

Problem 3:

Compute each fraction:
- \(\frac{1}{10} = 0.1\)
- \(\frac{2}{20} = \frac{1}{10} = 0.1\)
- \(\frac{3}{30} = \frac{1}{10} = 0.1\)

Add them:
- \(0.1 + 0.1 + 0.1 = 0.3\)

So the value is \(0.3\).

ANSWER 3: D

Problem 4:

Susan had \$50.

She spent:
- \$12 on food
- twice as much on rides, so \(2 \times 12 = 24\)

Total spent:
- \(12 + 24 = 36\)

Money left:
- \(50 - 36 = 14\)

ANSWER 4: B

Problem 5:

Two hundred thousand is:
- \(200{,}000 = 2 \times 10^5\)

Multiply:
- \(200{,}000 \times 200{,}000 = (2 \times 10^5)(2 \times 10^5) = 4 \times 10^{10}\)

And \(4 \times 10^{10} = 40{,}000{,}000{,}000\), which is forty billion.

ANSWER 5: E

Problem 6:

The total is written as \(1A2\), and 11 members each paid the same integer amount.

So \(1A2\) must be divisible by 11.

Test the answer choices:
- 102 ÷ 11 is not an integer
- 112 ÷ 11 is not an integer
- 122 ÷ 11 = 11 remainder 1
- 132 ÷ 11 = 12 exactly
- 142 ÷ 11 is not an integer

So the number is 132, which means \(A = 3\).

ANSWER 6: D

Problem 7:

We check each shape.

### Hexagon
A regular hexagon has 6 equal sides.
If each side is 5 cm, perimeter is:
- \(6 \times 5 = 30\) cm

The wire is only 24 cm, so he cannot make the hexagon.

### Square
A square with area 36 has side length:
- \(\sqrt{36} = 6\) cm

Perimeter:
- \(4 \times 6 = 24\) cm

So he can make the square exactly.

### Right triangle
The legs are 6 and 8, so the hypotenuse is:
- \(\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)

Perimeter:
- \(6 + 8 + 10 = 24\) cm

So he can make the triangle exactly.

Therefore he can make the square and the triangle only.

ANSWER 7: D

Problem 8:

The reciprocal is largest when the original number is smallest, assuming positive numbers.

Compare:
- \(\frac13 \approx 0.333\)
- \(\frac25 = 0.4\)
- \(1\)
- \(5\)
- \(1986\)

The smallest number is \(\frac13\), so it has the largest reciprocal.

Its reciprocal is 3, which is larger than the reciprocals of all the others.

ANSWER 8: A

Problem 9:

Maximum heart rate:
- \(220 - 26 = 194\)

Target heart rate is 80% of that:
- \(0.8 \times 194 = 155.2\)

To the nearest whole number, that is 155.

ANSWER 9: B

Problem 10:

First, Harry uses the parentheses:
- \(8 - (2 + 5) = 8 - 7 = 1\)

So \(H = 1\).

Terry ignores parentheses:
- \(8 - 2 + 5 = 6 + 5 = 11\)

So \(T = 11\).

Difference:
- \(H - T = 1 - 11 = -10\)

ANSWER 10: A

Problem 11:

We need to evaluate:
- \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

These are 10 odd numbers from 81 to 99.

Instead of adding one by one, use the average:
- average of 81 and 99 is \( \frac{81+99}{2} = 90\)

Since there are 10 numbers, their sum is:
- \(10 \times 90 = 900\)

Now multiply by 2:
- \(2 \times 900 = 1800\)

ANSWER 11: E

Problem 12:

We want the percentage of males surveyed who listen.

From the table:
- Total surveyed = 200
- Females = 96
- So males = \(200 - 96 = 104\)

Males who don’t listen = 26, so males who do listen:
- \(104 - 26 = 78\)

Now find the percent of males who listen:
- \(\frac{78}{104} = \frac{3}{4} = 75\%\)

ANSWER 12: E

Problem 1:  
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?  

- Original yellow tint amount = 30 × 0.30 = 9 liters.  
- Add 5 liters of yellow tint → new yellow amount = 9 + 5 = 14 liters.  
- New total volume = 30 + 5 = 35 liters.  
- New percent yellow = (14 / 35) × 100% = (2/5) × 100% = 40%.  

So the correct choice is 40.  

ANSWER 1: C  

---

Problem 2:  
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?  

- Choose an entry window: 6 choices.  
- Choose a different exit window: 5 choices (cannot be the same as entry).  
- Total ways = 6 × 5 = 30.  

ANSWER 2: D  

---

Problem 3:  
Compute: \( \frac{1}{10} + \frac{2}{20} + \frac{3}{30} \).  

Convert each to a common denominator, 60:  
- \( \frac{1}{10} = \frac{6}{60} \)  
- \( \frac{2}{20} = \frac{2}{20} = \frac{6}{60} \) (since multiply top and bottom by 3)  
- \( \frac{3}{30} = \frac{3}{30} = \frac{6}{60} \) (since multiply top and bottom by 2)  

Add:  
\( \frac{6}{60} + \frac{6}{60} + \frac{6}{60} = \frac{18}{60} = \frac{3}{10} = 0.3 \)

So the value is .3.  

ANSWER 3: D  

---

Problem 4:  
Susan had $50 to spend. She spent $12 on food and twice as much on rides. How much left?  

- Food = 12  
- Rides = twice 12 = 24  
- Total spent = 12 + 24 = 36  
- Money left = 50 − 36 = 14  

ANSWER 4: B  

---

Problem 5:  
Two hundred thousand times two hundred thousand equals ?  

\(200{,}000 \times 200{,}000 = (2\times 10^5)(2\times 10^5) = 4 \times 10^{10}\).  
- \(10^{10} = 10{,}000{,}000{,}000\)  
- \(4 \times 10^{10} = 40{,}000{,}000{,}000 = 40\) billion  

ANSWER 5: E  

---

Problem 6:  
Eleven members each paid the same integer amount for a guest speaker. In all, they paid $1A2 (a 3-digit number). Find missing digit A.  

- Total = \(1A2\) means:  
  \(1A2 = 100 + 10A + 2 = 102 + 10A\).  
- This total is divided equally among 11 people:  
  So \(102 + 10A\) must be divisible by 11.  

Check \(102 + 10A \mod 11\):  
- \(102 \div 11 = 9\) remainder 3, so \(102 \equiv 3 \pmod{11}\).  
- \(10A \equiv -A \pmod{11}\) (since 10 ≡ −1 mod 11).  

So:  
\(102 + 10A \equiv 3 - A \equiv 0 \pmod{11}\)  
Thus \(A \equiv 3 \pmod{11}\).  
Since \(A\) is a digit 0–9, \(A=3\).  

ANSWER 6: D  

---

Problem 7:  
Wire length = 24 cm. Which shapes can be made (using all 24 cm of wire)?  

We need the perimeter (total boundary length) of each shape:

**(1) Regular hexagon with side length 5 cm**  
Perimeter = 6 × 5 = 30 cm → too long. Cannot.

**(2) Square with area 36 cm²**  
Side length = √36 = 6 cm  
Perimeter = 4 × 6 = 24 cm → fits. Can.

**(3) Right triangle with legs 6 and 8 cm**  
Right triangle sides are 6 and 8 legs, and hypotenuse:  
\( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)  
Perimeter = 6 + 8 + 10 = 24 cm → fits. Can.

So the shapes possible: **square and triangle only**.  

ANSWER 7: D  

---

Problem 8:  
Which number has the largest reciprocal? Recall: the reciprocal of \(x\) is \(1/x\).  
We compare \(1/3, 2/5, 1, 1/5, 1/1986\).  

Compute roughly:  
- \(1/3 \approx 0.333\)  
- \(2/5 = 0.4\)  
- \(1 = 1\)  
- \(1/5 = 0.2\)  
- \(1/1986\) is tiny  

Largest is 1 (reciprocal of 1).  

ANSWER 8: C  

---

Problem 9:  
Target heart rate = 80% of (maximum heart rate).  
Maximum heart rate = 220 − age.  
Age = 26.

- Maximum = 220 − 26 = 194  
- Target = 0.80 × 194 = 155.2  
Nearest whole number = 155  

ANSWER 9: B  

---

Problem 10:  
Expression: \(8 - (2 + 5)\)  
- Harry computes correctly:  
  \(2 + 5 = 7\), so \(8 - 7 = 1\).  
- Terry ignores parentheses:  
  \(8 - 2 + 5 = 6 + 5 = 11\).  

Difference \(H - T = 1 - 11 = -10\).  

ANSWER 10: A  

---

Problem 11:  
Compute:  
\(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

Notice these are all odd numbers from 81 to 99 stepping by 2.  
There are 10 terms.

Pair from ends:  
- 81 + 99 = 180  
- 83 + 97 = 180  
- 85 + 95 = 180  
- 87 + 93 = 180  
- 89 + 91 = 180  

There are 5 pairs, each sum 180:  
Sum = 5 × 180 = 900  

Now multiply by 2:  
2 × 900 = 1800  

ANSWER 11: E  

---

Problem 12:  
Total surveyed = 200  
Females = 96  
Females who listen = 58  
Males who don’t listen = 26  
Total listeners = 136  

We want: % of males who listen.

Step 1: Find males total.  
Males = 200 − 96 = 104

Step 2: Find males who don’t listen is given = 26.  
So males who listen = 104 − 26 = 78

Step 3: Convert to percent of males who listen:  
Percent = (78 / 104) × 100%  
Simplify 78/104 = 3/4 (divide by 26):  
So percent = (3/4) × 100% = 75%

ANSWER 12: E

Problem 1:
The original mixture is 30 liters. The amount of yellow tint is 30% of 30 liters: $0.30 \times 30 = 9$ liters.
Five liters of yellow tint are added, so the new amount of yellow tint is $9 + 5 = 14$ liters.
The new total volume of the mixture is $30 + 5 = 35$ liters.
The percentage of yellow tint is $\frac{14}{35} \times 100\%$.
Simplifying the fraction: $\frac{14}{35} = \frac{2}{5} = 0.4$.
$0.4 \times 100\% = 40\%$.
ANSWER 1: C

Problem 2:
Georgie has 6 choices for the window he enters. Since he must leave by a different window, he has $6 - 1 = 5$ choices for the window he leaves by.
The total number of ways is $6 \times 5 = 30$.
ANSWER 2: D

Problem 3:
Simplify each fraction:
$1/10 = 0.1$
$2/20 = 1/10 = 0.1$
$3/30 = 1/10 = 0.1$
Summing them: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides, which is $2 \times 12 = 24$.
Total spent: $12 + 24 = 36$.
Amount left: $50 - 36 = 14$.
ANSWER 4: B

Problem 5:
Two hundred thousand is $200,000$ (or $2 \times 10^5$).
$(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
$4 \times 10^{10} = 40,000,000,000$, which is forty billion.
ANSWER 5: E

Problem 6:
The total amount $1A2$ must be divisible by 11.
A number is divisible by 11 if the alternating sum of its digits is divisible by 11.
For $1A2$, the alternating sum is $1 - A + 2 = 3 - A$.
For this to be 0 (the only multiple of 11 possible for a single digit $A$), $3 - A = 0$, so $A = 3$.
Check: $132 / 11 = 12$. This works.
ANSWER 6: D

Problem 7:
Hexagon: Perimeter of a regular hexagon with side 5 is $6 \times 5 = 30$ cm. (Cannot make)
Square: Area is 36, so side is $\sqrt{36} = 6$. Perimeter is $4 \times 6 = 24$ cm. (Can make)
Triangle: Right triangle with legs 6 and 8. Hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. Perimeter is $6 + 8 + 10 = 24$ cm. (Can make)
ANSWER 7: D

Problem 8:
Reciprocals:
A: $1/(1/3) = 3$
B: $1/(2/5) = 2.5$
C: $1/1 = 1$
D: $1/5 = 0.2$
E: $1/1986 \approx 0.0005$
The largest is 3.
ANSWER 8: A

Problem 9:
Max heart rate = $220 - 26 = 194$.
Target heart rate = $80\%$ of $194 = 0.8 \times 194 = 155.2$.
To the nearest whole number, this is 155.
ANSWER 9: B

Problem 10:
$H = 8 - (2 + 5) = 8 - 7 = 1$.
$T = 8 - 2 + 5 = 6 + 5 = 11$.
$H - T = 1 - 11 = -10$.
ANSWER 10: A

Problem 11:
Sum inside parentheses: $81+83+85+87+89+91+93+95+97+99$.
This is an arithmetic progression with 10 terms.
Sum = $\frac{n}{2}(\text{first} + \text{last}) = \frac{10}{2}(81 + 99) = 5 \times 180 = 900$.
The expression is $2 \times 900 = 1800$.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96.
Males = $200 - 96 = 104$.
Males who don't listen = 26.
Males who listen = $104 - 26 = 78$.
Percentage of males who listen = $(78 / 104) \times 100\%$.
$\frac{78}{104} = \frac{39}{52} = \frac{3}{4} = 0.75 = 75\%$.
ANSWER 12: E

Problem 1: The original mixture has volume 30 L, of which 30% is yellow tint, so there are \(0.3 \times 30 = 9\) L of yellow tint. Adding 5 L of yellow tint produces a new total of 14 L of yellow tint and a new mixture volume of 35 L. The new percentage of yellow tint is therefore \((14/35) \times 100\% = 40\%\).

ANSWER 1: C

Problem 2: There are 6 choices for the entry window. For each of those choices there are then exactly 5 remaining choices for a different exit window, giving a total of \(6 \times 5 = 30\) ordered pairs of distinct windows.

ANSWER 2: D

Problem 3: Rewrite each term with a common denominator of 10:  
\(1/10 = 0.1\),  
\(2/20 = 1/10 = 0.1\),  
\(3/30 = 1/10 = 0.1\).  
Their sum is \(0.1 + 0.1 + 0.1 = 0.3\).

ANSWER 3: D

Problem 4: Twice the food cost is \(2 \times 12 = 24\) dollars spent on rides. The total spent is therefore \(12 + 24 = 36\) dollars, leaving \(50 - 36 = 14\) dollars.

ANSWER 4: B

Problem 5: Write the factors in scientific notation:  
\(200000 = 2 \times 10^5\), so  
\((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\),  
which is forty billion.

ANSWER 5: E

Problem 6: The three-digit total \(1A2\) must be divisible by 11. The alternating-sum divisibility rule requires that \(1 - A + 2 = 3 - A\) be a multiple of 11. The only admissible digit \(A\) (0-9) that satisfies the condition is \(A = 3\), giving the total 132, which is exactly \(11 \times 12\).

ANSWER 6: D

Problem 7: A regular hexagon with side 5 cm has perimeter 30 cm, which exceeds the 24 cm length of wire, so it is impossible. A square of area 36 cm² has side length 6 cm and perimeter 24 cm, which matches the wire length. A right triangle with legs 6 cm and 8 cm has hypotenuse 10 cm, so its perimeter is \(6 + 8 + 10 = 24\) cm, which also matches. Consequently only the square and the triangle are possible.

ANSWER 7: D

Problem 8: For positive numbers the reciprocal function is strictly decreasing, so the largest reciprocal belongs to the smallest number in the list, which is \(1/3\).

ANSWER 8: A

Problem 9: The theoretical maximum heart rate is \(220 - 26 = 194\) beats per minute. Eighty percent of that value is \(0.8 \times 194 = 155.2\), which rounds to the nearest whole number 155.

ANSWER 9: B

Problem 10: Harry evaluates the parentheses first and obtains \(8 - (2 + 5) = 1\). Terry omits the parentheses and obtains \(8 - 2 + 5 = 11\). Their difference is therefore \(1 - 11 = -10\).

ANSWER 10: A

Problem 11: The sum inside the parentheses consists of the ten consecutive odd integers from 81 to 99. Their average is \((81 + 99)/2 = 90\), so the sum equals \(10 \times 90 = 900\). Multiplying by the outer factor of 2 yields \(2 \times 900 = 1800\).

ANSWER 11: E

Problem 12: There are 200 people surveyed and 96 females, so there are \(200 - 96 = 104\) males. Of the 96 females, 58 listen, leaving \(96 - 58 = 38\) females who do not listen. Given that 26 males do not listen, the number of males who listen is \(104 - 26 = 78\). The required percentage is therefore \((78/104) \times 100\% = 75\%\).

ANSWER 12: E

Problem 1:
The problem asks us to find the percentage of yellow tint in a new mixture after adding 5 liters of yellow tint to the original 30 liters of paint mixture. The original mixture is 25% red tint, 30% yellow tint, and 45% water.

First, let's calculate the amount of yellow tint in the original mixture. 30% of 30 liters is $0.3 \times 30 = 9$ liters.

After adding 5 liters of yellow tint, the total amount of yellow tint becomes $9 + 5 = 14$ liters.

The total volume of the new mixture is $30 + 5 = 35$ liters.

To find the percentage of yellow tint in the new mixture, we divide the amount of yellow tint by the total volume and multiply by 100: $\frac{14}{35} \times 100 = 40\%$.

Let's examine the choices:
- A. 25: Too low because the percentage of yellow tint increased.
- B. 35: Not the correct calculation.
- C. 40: Matches our calculation.
- D. 45: Too high.
- E. 50: Too high.

The correct percentage is 40%.

ANSWER 1: C

Problem 2:
The problem asks us to determine the number of ways Georgie the Ghost can enter a haunted house through one of six windows and leave by a different window.

To enter, Georgie has 6 choices. After entering, there are 5 remaining windows to choose from for leaving.

The total number of ways to enter and leave is $6 \times 5 = 30$.

Let's examine the choices:
- A. 12: Too low.
- B. 15: Too low.
- C. 18: Too low.
- D. 30: Matches our calculation.
- E. 36: Too high because it would imply Georgie can leave by the same window.

The correct number of ways is 30.

ANSWER 2: D

Problem 3:
The problem requires us to simplify the expression $1⁄10 + 2⁄20 + 3⁄30$.

First, simplify each fraction:
- $1⁄10$ stays the same.
- $2⁄20 = 1⁄10$.
- $3⁄30 = 1⁄10$.

Adding them together: $1⁄10 + 1⁄10 + 1⁄10 = 3⁄10 = 0.3$.

Let's examine the choices:
- A. .1: Too low.
- B. .123: Not the correct sum.
- C. .2: Too low.
- D. .3: Matches our calculation.
- E. .6: Too high.

The correct sum is 0.3.

ANSWER 3: D

Problem 4:
The problem asks us to find out how many dollars Susan had left after spending money on food and rides at the carnival. She started with $50, spent $12 on food, and twice as much on rides.

First, calculate the amount spent on rides: $12 \times 2 = $24.

Total amount spent is $12 + $24 = $36.

The amount left is $50 - $36 = $14.

Let's examine the choices:
- A. 12: Too low.
- B. 14: Matches our calculation.
- C. 26: Too high.
- D. 38: Too high.
- E. 50: Incorrect because she spent money.

The correct amount left is $14.

ANSWER 4: B

Problem 5:
The problem asks us to calculate $200,000 \times 200,000$.

$200,000 = 2 \times 10^5$, so $200,000 \times 200,000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10} = 40,000,000,000$.

Let's examine the choices:
- A. four hundred thousand: Too low.
- B. four million: Too low.
- C. forty thousand: Too low.
- D. four hundred million: Too low.
- E. forty billion: Matches our calculation.

The correct result is forty billion.

ANSWER 5: E

Problem 6:
The problem requires us to find the missing digit A in the 3-digit number $1A2 that represents the total amount paid by 11 members of the Math Club.

The total amount paid is divisible by 11 since there are 11 members who paid the same amount.

To check divisibility by 11, we use the rule: the difference between the sum of digits at odd places and the sum of digits at even places should be divisible by 11.

For $1A2, the sum of digits at odd places is $1 + 2 = 3$, and the sum at even places is $A$. The difference is $3 - A$.

For $3 - A$ to be divisible by 11 (or 0, since $A$ is a single digit), $A = 3$ makes $3 - A = 0$, which is divisible by 11.

Let's examine the choices:
- A. 0: Not correct because $3 - 0 = 3$, not divisible by 11.
- B. 1: Not correct because $3 - 1 = 2$, not divisible by 11.
- C. 2: Not correct because $3 - 2 = 1$, not divisible by 11.
- D. 3: Correct because $3 - 3 = 0$, which is divisible by 11.
- E. 4: Not correct because $3 - 4 = -1$, not divisible by 11.

The missing digit A is 3, but we should verify if 132 is indeed divisible by 11. $132 / 11 = 12$, so it is.

ANSWER 6: D

Problem 7:
The problem asks us to determine which shapes Haruki can form using a 24 cm long wire.

1. A regular hexagon with side length 5 cm: The perimeter is $6 \times 5 = 30$ cm, which is more than 24 cm. So, Haruki cannot form this hexagon.

2. A square of area 36 cm$^2$: The side length is $\sqrt{36} = 6$ cm. The perimeter is $4 \times 6 = 24$ cm, which matches the wire length. So, Haruki can form this square.

3. A right triangle with legs 6 and 8 cm: Using the Pythagorean theorem, the hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm. The perimeter is $6 + 8 + 10 = 24$ cm, which matches the wire length. So, Haruki can form this triangle.

Let's examine the choices:
- A. Triangle only: Incorrect because Haruki can also form the square.
- B. Hexagon and square only: Incorrect because Haruki cannot form the hexagon.
- C. Hexagon and triangle only: Incorrect because Haruki cannot form the hexagon.
- D. Square and triangle only: Matches our findings.
- E. Hexagon, triangle, and square: Incorrect because Haruki cannot form the hexagon.

The correct shapes Haruki can form are the square and the triangle.

ANSWER 7: D

Problem 8:
The problem asks us to find which of the given numbers has the largest reciprocal.

The reciprocal of a number $x$ is $1/x$. The larger $x$ is, the smaller its reciprocal is, and vice versa.

Let's examine the choices:
- A. $1⁄3$: The reciprocal is 3.
- B. $2⁄5$: The reciprocal is $5/2 = 2.5$.
- C. 1: The reciprocal is 1.
- D. 5: The reciprocal is $1/5 = 0.2$.
- E. 1986: The reciprocal is $1/1986$, very small.

Comparing the reciprocals: $3 > 2.5 > 1 > 0.2 > 1/1986$. So, $1/3$ has the largest reciprocal.

Let's examine the choices again:
- A. $1⁄3$: Correct.
- B. $2⁄5$: Not correct.
- C. 1: Not correct.
- D. 5: Not correct.
- E. 1986: Not correct.

The number with the largest reciprocal is $1/3$.

ANSWER 8: A

Problem 9:
The proble

Problem 1:
Original mixture: 30 L. Yellow tint = 30% of 30 = 0.30 × 30 = 9 L.
Add 5 L yellow: new yellow = 9 + 5 = 14 L, new total volume = 30 + 5 = 35 L.
Percent yellow = (14/35) × 100% = 40%.
ANSWER 1: C

Problem 2:
6 choices for entry window, 5 remaining choices for a different exit window.
Total ways = 6 × 5 = 30.
ANSWER 2: D

Problem 3:
1/10 + 2/20 + 3/30 = 1/10 + 1/10 + 1/10 = 3/10 = 0.3.
ANSWER 3: D

Problem 4:
Spent on food: $12. Spent on rides: 2 × $12 = $24.
Total spent = $12 + $24 = $36. Left = $50 − $36 = $14.
ANSWER 4: B

Problem 5:
200,000 × 200,000 = (2×10⁵) × (2×10⁵) = 4×10¹⁰ = 40,000,000,000, which is forty billion.
ANSWER 5: E

Problem 6:
Total paid is $1A2, a multiple of 11. Divisibility by 11: (1 − A + 2) = 3 − A must be a multiple of 11.
With A a digit 0–9, 3 − A = 0 ⇒ A = 3. (Check: 132 ÷ 11 = 12.)
ANSWER 6: D

Problem 7:
Wire length = 24 cm.
- Regular hexagon, side 5 cm: perimeter = 6×5 = 30 cm > 24 cm → cannot make.
- Square, area 36 cm²: side = 6 cm, perimeter = 4×6 = 24 cm → can make.
- Right triangle, legs 6 and 8 cm: hypotenuse = 10 cm, perimeter = 6+8+10 = 24 cm → can make.
So square and triangle only.
ANSWER 7: D

Problem 8:
Reciprocals: 1/(1/3)=3, 1/(2/5)=2.5, 1/1=1, 1/5=0.2, 1/1986≈0.0005. Largest is 3, from 1/3.
ANSWER 8: A

Problem 9:
Maximum heart rate = 220 − 26 = 194 bpm.
Target = 80% of 194 = 0.8 × 194 = 155.2 → 155 to nearest whole number.
ANSWER 9: B

Problem 10:
Harry: H = 8 − (2+5) = 8 − 7 = 1.
Terry: T = 8 − 2 + 5 = 6 + 5 = 11.
H − T = 1 − 11 = −10.
ANSWER 10: A

Problem 11:
Sum inside parentheses: 10 terms, average = (81+99)/2 = 90, sum = 10×90 = 900.
2 × 900 = 1800.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96 ⇒ Males = 200 − 96 = 104.
Total listeners = 136, female listeners = 58 ⇒ male listeners = 136 − 58 = 78.
Percent of males who listen = (78/104) × 100% = 75%.
ANSWER 12: E

Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint.

**Original mixture (30 liters):**
- Red tint: 25% of 30 = 7.5 liters
- Yellow tint: 30% of 30 = 9 liters
- Water: 45% of 30 = 13.5 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total mixture: 30 + 5 = 35 liters

**New percentage of yellow tint:**
14 / 35 = 2/5 = 0.40 = 40%

Tempting wrong answers: Choice B (35) might come from just adding 5 to the original 30%, but we must recalculate based on the new total.

ANSWER 1: C

---

Problem 2:
Georgie enters through one window and leaves through a *different* window.

- Number of choices for the entry window: 6
- Number of choices for the exit window (must be different): 5
- Total ways: 6 × 5 = 30

Tempting wrong answers: Choice E (36) would be 6 × 6, which allows entering and leaving through the same window. Choice A (12) would be 6 + 6, which doesn't account for all combinations.

ANSWER 2: D

---

Problem 3:
Simplify each fraction first:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

Tempting wrong answers: Choice B (.123) might come from concatenating the digits 1, 2, 3 instead of actually computing. Choice C (.2) might come from miscounting the terms.

ANSWER 3: D

---

Problem 4:
Susan starts with $50.
- Spent on food: $12
- Spent on rides: twice the food amount = 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Amount left: $50 − $36 = $14

Tempting wrong answers: Choice C (26) might come from subtracting only the food ($50 − $24 = $26, forgetting food). Choice D (38) comes from subtracting only $12.

ANSWER 4: B

---

Problem 5:
200,000 × 200,000

Write in scientific notation:
(2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰

4 × 10¹⁰ = 40,000,000,000 = forty billion

Tempting wrong answers: Choice A (four hundred thousand) comes from adding instead of multiplying. Choice D (four hundred million) comes from miscounting zeros.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, so the total $1A2 must be divisible by 11.

**Divisibility rule for 11:** The alternating sum of digits must be a multiple of 11.
Alternating sum: 1 − A + 2 = 3 − A

For 3 − A to be divisible by 11 (and A to be a single digit 0–9):
3 − A = 0 → A = 3

**Check:** $132 ÷ 11 = $12 per member. This is a valid integer amount.

Tempting wrong answers: Other values of A don't make 3 − A a multiple of 11 within the digit range.

ANSWER 6: D

---

Problem 7:
Wire length = 24 cm. We check if each shape's perimeter ≤ 24 cm.

**Regular hexagon (side 5 cm):**
Perimeter = 6 × 5 = 30 cm. Since 30 > 24, he **cannot** make this.

**Square (area 36 cm²):**
Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Since 24 = 24, he **can** make this.

**Right triangle (legs 6 and 8 cm):**
Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Perimeter = 6 + 8 + 10 = 24 cm. Since 24 = 24, he **can** make this.

Haruki can make the square and triangle only.

Tempting wrong answers: Choice E includes the hexagon, but its perimeter exceeds the wire length.

ANSWER 7: D

---

Problem 8:
The reciprocal of a number x is 1/x. The *smaller* the positive number, the *larger* its reciprocal.

| Choice | Number | Reciprocal |
|--------|--------|------------|
| A | 1/3 | 3 |
| B | 2/5 | 5/2 = 2.5 |
| C | 1 | 1 |
| D | 5 | 1/5 = 0.2 |
| E | 1986 | 1/1986 ≈ 0.0005 |

The largest reciprocal is 3, which comes from 1/3.

Tempting wrong answers: Choice E (1986) is the largest number, so it has the *smallest* reciprocal.

ANSWER 8: A

---

Problem 9:
Athlete's age = 26 years.

**Step 1:** Maximum heart rate = 220 − 26 = 194 beats per minute.

**Step 2:** Target heart rate = 80% of 194 = 0.80 × 194 = 155.2

**Step 3:** Round to nearest whole number: 155

Tempting wrong answers: Choice C (176) might come from taking 80% of 220 without subtracting the age. Choice D (194) is the maximum heart rate itself.

ANSWER 9: B

---

Problem 10:
**Harry (correct calculation):**
H = 8 − (2 + 5) = 8 − 7 = 1

**Terry (ignores parentheses):**
T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:**
H − T = 1 − 11 = −10

Tempting wrong answers: Choice E (10) is |H − T|, but the question asks for H − T specifically, which is negative since Terry's answer is larger.

ANSWER 10: A

---

Problem 11:
We need: 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

The sum inside is an arithmetic sequence:
- First term: 81, Last term: 99
- Number of terms: 10
- Sum = (number of terms) × (first + last) / 2 = 10 × (81 + 99) / 2 = 10 × 180 / 2 = 10 × 90 = 900

Multiply by 2: 2 × 900 = 1800

Tempting wrong answers: Choice C (1700) might come from miscounting terms or miscalculating the average.

ANSWER 11: E

---

Problem 12:
Let me organize the data from the survey:

- Total surveyed: 200
- Females: 96 → Males: 200 − 96 = 104
- Total listeners: 136
- Females who listen: 58 → Males who listen: 136 − 58 = 78
- Males who don't listen: 26 (Check: 78 + 26 = 104 ✓)

**Percentage of males who listen:**
(78 / 104) × 100% = (3/4) × 100% = 75%

Tempting wrong answers: Choice C (52) might come from computing 104/200 (percentage of total who are male). Choice D (55) might come from 78/136 (percentage of listeners who are male).

ANSWER 12: E

 Problem 1:
We start with 30 liters of paint. The original amount of yellow tint is 30% of 30 liters:
\[
0.30 \times 30 = 9 \text{ liters.}
\]
Five liters of yellow tint are added, so the new amount of yellow tint is:
\[
9 + 5 = 14 \text{ liters.}
\]
The new total volume of the mixture is:
\[
30 + 5 = 35 \text{ liters.}
\]
The percent of yellow tint in the new mixture is:
\[
\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%.
\]
(Choice A is the original red percentage, D is the original water percentage, and E would require 17.5 liters of yellow, which we do not have.)
ANSWER 1: C

Problem 2:
Georgie chooses one of 6 windows to enter. Since he must leave through a different window, there are 5 remaining choices for his exit. By the multiplication principle:
\[
6 \times 5 = 30 \text{ ways.}
\]
(Choice E, 36, would count entering and leaving through the same window; the other choices are too small.)
ANSWER 2: D

Problem 3:
Simplify each fraction:
\[
\frac{2}{20} = \frac{1}{10}, \qquad \frac{3}{30} = \frac{1}{10}.
\]
So the sum is:
\[
\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3.
\]
(Choice A is just one term; C is 0.2; E is 0.6.)
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides:
\[
2 \times 12 = 24 \text{ dollars.}
\]
Total spent:
\[
12 + 24 = 36 \text{ dollars.}
\]
Amount left:
\[
50 - 36 = 14 \text{ dollars.}
\]
(Choice C is the total spent; D is the amount left if she had spent only $12.)
ANSWER 4: B

Problem 5:
\[
200{,}000 \times 200{,}000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}.
\]
\[
4 \times 10^{10} = 40{,}000{,}000{,}000,
\]
which is read as forty billion.
(Choice D is $4 \times 10^8$; B is $4 \times 10^6$.)
ANSWER 5: E

Problem 6:
The total amount paid is the three-digit number $1A2$, which equals:
\[
100 + 10A + 2 = 102 + 10A.
\]
Since 11 members each paid the same integer amount, $102 + 10A$ must be divisible by 11. Testing the choices:
- $A=0$: $102 \div 11 \approx 9.27$ (not an integer)
- $A=1$: $112 \div 11 \approx 10.18$ (not an integer)
- $A=2$: $122 \div 11 \approx 11.09$ (not an integer)
- $A=3$: $132 \div 11 = 12$ (integer!)
- $A=4$: $142 \div 11 \approx 12.9$ (not an integer)
Thus the missing digit is 3.
ANSWER 6: D

Problem 7:
The wire is 24 cm long, so a shape is possible only if its perimeter is at most 24 cm.
- **Regular hexagon, side 5 cm:** Perimeter $= 6 \times 5 = 30$ cm. $30 > 24$, so impossible.
- **Square of area 36 cm²:** Side $= \sqrt{36} = 6$ cm. Perimeter $= 4 \times 6 = 24$ cm. Exactly fits, so possible.
- **Right triangle with legs 6 and 8 cm:** Hypotenuse $= \sqrt{6^2 + 8^2} = \sqrt{100} = 10$ cm. Perimeter $= 6 + 8 + 10 = 24$ cm. Exactly fits, so possible.
Only the square and the triangle can be formed.
ANSWER 7: D

Problem 8:
For positive numbers, the reciprocal function $f(x) = \frac{1}{x}$ is decreasing, so the smallest number has the largest reciprocal. Compare the two smallest choices:
\[
\frac{1}{3} \approx 0.333\ldots, \qquad \frac{2}{5} = 0.4.
\]
Since $\frac{1}{3} < \frac{2}{5} < 1 < 5 < 1986$, the smallest number is $\frac{1}{3}$. Its reciprocal is $3$, which is larger than the reciprocals of all the other choices.
ANSWER 8: A

Problem 9:
First find the theoretical maximum heart rate:
\[
220 - 26 = 194 \text{ beats per minute.}
\]
The target rate is 80% of this:
\[
0.80 \times 194 = 155.2.
\]
Rounded to the nearest whole number, this is 155.
(Choice C is the maximum heart rate itself.)
ANSWER 9: B

Problem 10:
Harry computes correctly:
\[
H = 8 - (2 + 5) = 8 - 7 = 1.
\]
Terry ignores the parentheses:
\[
T = 8 - 2 + 5 = 6 + 5 = 11.
\]
The difference is:
\[
H - T = 1 - 11 = -10.
\]
ANSWER 10: A

Problem 11:
The ten numbers inside the parentheses form an arithmetic sequence: 81, 83, …, 99. The sum of an arithmetic sequence is:
\[
\frac{\text{number of terms}}{2} \times (\text{first} + \text{last}) = \frac{10}{2} \times (81 + 99) = 5 \times 180 = 900.
\]
Now multiply by the factor of 2 outside:
\[
2 \times 900 = 1800.
\]
(One can also pair them: $(81+99)=180$, $(83+97)=180$, etc., giving five pairs that each sum to 180.)
ANSWER 11: E

Problem 12:
Total surveyed = 200, and females surveyed = 96, so:
\[
\text{Males surveyed} = 200 - 96 = 104.
\]
Total listeners = 136, and females who listen = 58, so:
\[
\text{Males who listen} = 136 - 58 = 78.
\]
(We can check: total non-listeners = 64, males who don't listen = 26, so females who don't listen = 38; $58+38=96$ and $78+26=104$, which checks out.)

The percentage of males who listen is:
\[
\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%.
\]
ANSWER 12: E

Problem 1:
A mixture of 30 liters is 25% red, 30% yellow, and 45% water. Five liters of yellow tint are added. What is the percent of yellow tint in the new mixture?

**Setup:** Find the amount of yellow tint originally, add 5 liters, then find the new percentage.

**Original yellow tint:** 30% of 30 = 0.30 × 30 = 9 liters

**After adding 5 liters of yellow tint:**
- New yellow tint: 9 + 5 = 14 liters
- New total volume: 30 + 5 = 35 liters
- New percentage: 14/35 = 2/5 = 40%

**Checking wrong choices:** 25% would be if we just added the original percentages; 35% would be if we just added 5% to 30%; 45% and 50% are too high.

ANSWER 1: C

---

Problem 2:
Six windows. Georgie enters by one and leaves by a different one. How many ways?

**Setup:** Choose an entry window (6 choices), then choose a different exit window (5 choices).

**Calculation:** 6 × 5 = 30 ways

**Checking wrong choices:** 15 would be if we divided by 2 (treating order as irrelevant); 12 would be 6×2; 18 would be 6×3; 36 would be 6×6 (allowing same window).

ANSWER 2: D

---

Problem 3:
1/10 + 2/20 + 3/30 = ?

**Simplify each fraction:**
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

**Sum:** 1/10 + 1/10 + 1/10 = 3/10 = 0.3

**Checking wrong choices:** 0.1 would be just one term; 0.2 would be two terms; 0.6 would be adding numerators and denominators incorrectly.

ANSWER 3: D

---

Problem 4:
Susan had $50. She spent $12 on food and twice as much on rides. How much left?

**Setup:** Food = $12, Rides = 2 × $12 = $24

**Total spent:** $12 + $24 = $36

**Amount left:** $50 - $36 = $14

**Checking wrong choices:** 12 is just the food amount; 26 is 50-24 (forgetting food); 38 is 50-12 (forgetting rides).

ANSWER 4: B

---

Problem 5:
200,000 × 200,000 = ?

**Setup:** Express in scientific notation: 200,000 = 2 × 10⁵

**Calculation:** (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion

**Checking wrong choices:** Four hundred thousand is 4×10⁵; four million is 4×10⁶; forty thousand is 4×10⁴; four hundred million is 4×10⁸.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, totaling $1A2. Find the missing digit A.

**Setup:** The total $1A2 must be divisible by 11.

**Divisibility test for 11:** A number is divisible by 11 if the alternating sum of its digits is divisible by 11.

For 1A2: (1 + 2) - A = 3 - A must be divisible by 11.

Since A is a single digit (0-9), 3 - A can only equal 0 (the only multiple of 11 in range -6 to 3).

So A = 3, giving $132 = 11 × $12 ✓

**Checking:** With A=0: $102 ÷ 11 ≈ 9.27 (not integer); A=1: $112 ÷ 11 ≈ 10.18 (not integer); A=3: $132 ÷ 11 = 12 ✓

ANSWER 6: D

---

Problem 7:
Wire is 24 cm. Which shapes can be made?

**Regular hexagon with side 5 cm:** Perimeter = 6 × 5 = 30 cm. Need 30 cm, only have 24 cm. ❌

**Square with area 36 cm²:** Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Need 24 cm, have 24 cm. ✓

**Right triangle with legs 6 and 8 cm:** Hypotenuse = √(36 + 64) = √100 = 10 cm. Perimeter = 6 + 8 + 10 = 24 cm. Need 24 cm, have 24 cm. ✓

**Conclusion:** Can make the square and triangle, but not the hexagon.

ANSWER 7: D

---

Problem 8:
Which number has the largest reciprocal?

**Find each reciprocal:**
- A. 1/3 → reciprocal = 3
- B. 2/5 → reciprocal = 5/2 = 2.5
- C. 1 → reciprocal = 1
- D. 5 → reciprocal = 1/5 = 0.2
- E. 1986 → reciprocal ≈ 0.0005

**Key insight:** Smaller positive numbers have larger reciprocals. The smallest number listed is 1/3, so it has the largest reciprocal (3).

ANSWER 8: A

---

Problem 9:
Target heart rate = 80% of (220 - age). Find target for 26-year-old.

**Maximum heart rate:** 220 - 26 = 194 bpm

**Target heart rate:** 80% of 194 = 0.80 × 194 = 155.2 ≈ 155 bpm

**Checking wrong choices:** 134 would be about 69%; 176 would be about 91%; 194 is the max (100%); 243 is impossible.

ANSWER 9: B

---

Problem 10:
Harry calculates 8 − (2 + 5) correctly. Terry calculates 8 − 2 + 5. Find H − T.

**Harry's answer:** H = 8 − (2 + 5) = 8 − 7 = 1

**Terry's answer:** T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:** H − T = 1 − 11 = −10

**Checking wrong choices:** −6 would be if Terry got 7; 0 would be if they got the same answer; 6 or 10 would be T − H.

ANSWER 10: A

---

Problem 11:
2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99) = ?

**Setup:** This is 10 consecutive odd numbers. The average of the first and last is (81 + 99)/2 = 90.

**Sum of the 10 numbers:** 10 × 90 = 900

**Multiply by 2:** 2 × 900 = 1800

**Alternative verification:** Pair the numbers: (81+99) + (83+97) + (85+95) + (87+93) + (89+91) = 180 + 180 + 180 + 180 + 180 = 900. Then 2 × 900 = 1800.

ANSWER 11: E

---

Problem 12:
Find the percentage of males who listen to the station.

**Given information:**
- Total surveyed: 200
- Females: 96
- Females who listen: 58
- Males who don't listen: 26
- Total listeners: 136
- Total non-listeners: 64

**Find males:** Males = 200 - 96 = 104

**Find males who listen:** Males who listen = Total listeners - Females who listen = 136 - 58 = 78

**Verification:** Males who don't listen = 104 - 78 = 26 ✓ (matches given)

**Percentage of males who listen:** 78/104 × 100 = 75%

**Checking wrong choices:** 39% would be 78/200 (using total instead of males); 48% would be about 50/104; 52% would be about 54/104; 55% would be about 57/104.

ANSWER 12: E

**Problem 1**

The original mixture is 30 L with 30 % yellow tint → yellow = 0.30·30 = 9 L.  
Add 5 L of yellow → total yellow = 9 + 5 = 14 L.  
Total volume after addition = 30 + 5 = 35 L.  

Percent yellow = (14/35)·100 % = 40 %.

**Answer 1: C**

---

**Problem 2**

Choose an entry window (6 ways) and a different exit window (5 ways).  
Number of ways = 6 × 5 = 30.

**Answer 2: D**

---

**Problem 3**

\[
\frac{1}{10}=0.1,\qquad\frac{2}{20}=0.1,\qquad\frac{3}{30}=0.1
\]
Sum = 0.1 + 0.1 + 0.1 = 0.3.

**Answer 3: D**

---

**Problem 4**

Food cost = \$12.  
Rides cost = twice food = 2·12 = \$24.  
Total spent = 12 + 24 = \$36.  
Money left = 50 − 36 = \$14.

**Answer 4: B**

---

**Problem 5**

\(200{,}000 \times 200{,}000 = (2\times10^{5})^{2}=4\times10^{10}=40{,}000{,}000{,}000\).

That is 40 billion.

**Answer 5: E**

---

**Problem 6**

Total paid = \$1A2 = 100 + 10A + 2 = 102 + 10A.  
Since the 11 members each paid an integer amount, 102 + 10A must be divisible by 11.

\(102\equiv3\pmod{11}\).  
We need \(10A\equiv-3\equiv8\pmod{11}\).  
Because \(10\equiv-1\pmod{11}\), this is \(-A\equiv8\pmod{11}\) → \(A\equiv3\pmod{11}\).  
A digit, so \(A=3\).

**Answer 6: D**

---

**Problem 7**

- Hexagon: perimeter = 6·5 = 30 cm > 24 cm → impossible.  
- Square: side = √36 = 6 cm, perimeter = 4·6 = 24 cm → possible.  
- Right triangle: legs 6 cm, 8 cm, hypotenuse = √(6²+8²)=10 cm, perimeter = 6+8+10 = 24 cm → possible.

Only the square and triangle can be made.

**Answer 7: D**

---

**Problem 8**

Reciprocals:
- \(1/3 → 3\)  
- \(2/5 → 5/2 = 2.5\)  
- \(1 → 1\)  
- \(5 → 1/5 = 0.2\)  
- \(1986 → 1/1986 ≈ 0.0005\)

Largest reciprocal is 3, from \(1/3\).

**Answer 8: A**

---

**Problem 9**

Maximum heart rate = 220 − 26 = 194 bpm.  
Target = 80 % of 194 = 0.8·194 = 155.2 → rounded to 155 bpm.

**Answer 9: B**

---

**Problem 10**

Harry: \(8-(2+5)=8-7=1\).  
Terry (ignoring parentheses): \(8-2+5 = 6+5 = 11\).  
Difference \(H-T = 1-11 = -10\).

**Answer 10: A**

---

**Problem 11**

Sum of the ten terms: average = (81+99)/2 = 90, so sum = 90·10 = 900.  
Multiply by 2: \(2·900 = 1800\).

**Answer 11: E**

---

**Problem 12**

Total surveyed = 200.  
Females = 96 → Males = 200 − 96 = 104.  

Female listeners = 58 → Male listeners = 136 − 58 = 78.  

Percent of males who listen = (78/104)·100 % = (39/52)·100 % = 75 %.

**Answer 12: E**

---

**Final Answers**

ANSWER 1: C  
ANSWER 2: D  
ANSWER 3: D  
ANSWER 4: B  
ANSWER 5: E  
ANSWER 6: D  
ANSWER 7: D  
ANSWER 8: A  
ANSWER 9: B  
ANSWER 10: A  
ANSWER 11: E  
ANSWER 12: E

Problem 1:  
A 30-liter paint mixture is 25% red tint, 30% yellow tint, and 45% water. Five liters of yellow tint are added.  
Original yellow tint: $0.30 \times 30 = 9$ liters.  
After adding 5 liters: $9 + 5 = 14$ liters yellow.  
New total volume: $30 + 5 = 35$ liters.  
Percent yellow: $\frac{14}{35} = 0.40 = 40\%$.  
Tempting wrong choices: 35% (original yellow percent) or 45% (water percent).  
ANSWER 1: D  

Problem 2:  
Georgie enters through one of six windows and exits through a different one.  
Entry choices: 6. Exit choices: 5 (excluding entry window).  
Total ways: $6 \times 5 = 30$.  
Tempting wrong choices: 15 (if order didn’t matter) or 36 (if same window allowed).  
ANSWER 2: D  

Problem 3:  
$\frac{1}{10} + \frac{2}{20} + \frac{3}{30} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3$.  
Tempting wrong choices: 0.1 (one term) or 0.2 (two terms).  
ANSWER 3: D  

Problem 4:  
Susan starts with \$50.  
Food: \$12.  
Rides: $2 \times 12 = \$24$.  
Total spent: $12 + 24 = \$36$.  
Left: $50 - 36 = \$14$.  
Tempting wrong choices: \$26 (if spent \$24 total) or \$38 (if spent \$12 total).  
ANSWER 4: B  

Problem 5:  
$200,000 \times 200,000 = 40,000,000,000 = 40$ billion.  
Tempting wrong choices: 400,000 (added zeros) or 4 million (multiplied digits).  
ANSWER 5: E  

Problem 6:  
Total paid: \$1A2, an integer divisible by 11.  
$1A2 \mod 11 = 0$.  
$1 + 2 - A \equiv 0 \mod 11 \Rightarrow 3 - A \equiv 0 \mod 11 \Rightarrow A = 3$.  
Tempting wrong choices: 0, 1, 2, or 4 (incorrect divisibility).  
ANSWER 6: E (Wait, let's recheck: 1A2 must be divisible by 11. 132 ÷ 11 = 12, so A=3. But 132 is 11×12, so correct. However, the answer choices are A=0,B=1,C=2,D=3,E=4. So D is 3. So correct answer is D, not E. I made a mistake in the initial answer. Correcting:)  
**Correction**: $1A2 = 132$ when $A=3$, and $132 \div 11 = 12$, so $A=3$.  
ANSWER 6: D  

Problem 7:  
Wire length: 24 cm.  
- Regular hexagon with side 5 cm: Perimeter $6 \times 5 = 30$ cm (not 24).  
- Square with area 36 cm²: Side $6$ cm, perimeter $24$ cm.  
- Right triangle with legs 6 and 8 cm: Hypotenuse 10 cm, perimeter $6+8+10=24$ cm.  
Can make square and triangle.  
Tempting wrong choices: Hexagon (incorrect perimeter).  
ANSWER 7: D  

Problem 8:  
Largest reciprocal means smallest original number.  
$\frac{1}{3} \approx 0.333$, $\frac{2}{5} = 0.4$, $1$, $5$, $1986$.  
Smallest: $\frac{1}{3}$. Reciprocal: 3.  
Others: 2.5, 1, 0.2, 0.0005.  
Tempting wrong choices: $\frac{2}{5}$ (larger than $\frac{1}{3}$).  
ANSWER 8: A  

Problem 9:  
Maximum heart rate: $220 - 26 = 194$.  
Target: $0.80 \times 194 = 155.2 \approx 155$.  
Tempting wrong choices: 134 (70%), 176 (90%), or 243 (incorrect max).  
ANSWER 9: B  

Problem 10:  
Harry: $8 - (2 + 5) = 1$.  
Terry: $8 - 2 + 5 = 11$.  
Difference: $1 - 11 = -10$.  
Tempting wrong choices: 10 (absolute difference) or 0 (if parentheses ignored by both).  
ANSWER 10: A  

Problem 11:  
Sum: $81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99$.  
Average: 90, 10 terms: $90 \times 10 = 900$.  
$2 \times 900 = 1800$.  
Tempting wrong choices: 1600 (min+max=180, times 5=900, times 2=1800, but if miscount terms).  
ANSWER 11: E  

Problem 12:  
Total surveyed: 200.  
Females: 96, so males: $200 - 96 = 104$.  
Total listeners: 136.  
Female listeners: 58, so male listeners: $136 - 58 = 78$.  
Percent male listeners: $\frac{78}{104} = 0.75 = 75\%$.  
Tempting wrong choices: 39 (male non-listeners percent) or 52 (incorrect calculation).  
ANSWER 12: E

### Problem 1:
What is the percentage of yellow tint in the new 35-liter paint mixture, after adding 5 liters of yellow tint to an original 30-liter mixture that was 30% yellow?
First, calculate the original volume of yellow tint: 30% of 30 liters = \(0.3 \times 30 = 9\) liters. Add the 5 new liters of yellow tint to get \(9 + 5 = 14\) liters of yellow in the new mixture. The total new volume of the mixture is \(30 + 5 = 35\) liters. The percentage of yellow is \(\frac{14}{35} = 0.4 = 40\%\). Tempting wrong answers come from forgetting to update the total mixture volume when calculating the new percentage, or miscalculating the original yellow volume.
ANSWER 1: C

---

### Problem 2:
How many ways can Georgie enter through one window and leave through a different window, out of 6 total windows?
Georgie has 6 choices for an entry window. Since he must leave through a different window, he has 5 remaining choices for an exit window. The total number of ways is \(6 \times 5 = 30\). A common wrong answer is 15, which incorrectly treats entry/exit as unordered pairs, when order matters (entering window A and leaving B is different from entering B and leaving A).
ANSWER 2: D

---

### Problem 3:
Simplify \(\frac{1}{10} + \frac{2}{20} + \frac{3}{30}\):
Simplify each fraction first: \(\frac{2}{20} = \frac{1}{10}\) and \(\frac{3}{30} = \frac{1}{10}\). Add them together: \(\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3\). A common wrong answer is 0.1, which incorrectly adds numerators and denominators directly instead of simplifying first.
ANSWER 3: D

---

### Problem 4:
How much money does Susan have left after spending $12 on food and twice that amount on rides, starting with $50?
First calculate the amount spent on rides: twice $12 is \(2 \times 12 = 24\) dollars. Total spent is \(12 + 24 = 36\) dollars. The amount left is \(50 - 36 = 14\) dollars. A common wrong answer is 26, which incorrectly uses $12 instead of $24 for ride costs.
ANSWER 4: B

---

### Problem 5:
Calculate \(200,000 \times 200,000\):
Rewrite 200,000 as \(2 \times 10^5\). Multiplying two copies gives \((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\), which equals 40,000,000,000, or forty billion. Wrong answers come from miscounting the number of zeros in the product.
ANSWER 5: E

---

### Problem 6:
The 3-digit total cost \(1A2\) dollars is divisible by 11 (11 members paid the same integer amount). What is the missing digit A?
The divisibility rule for 11 states that alternating sum of digits must be divisible by 11. For 1A2, the alternating sum is \(1 - A + 2 = 3 - A\). The only value of A (0-9) that makes this sum divisible by 11 is A=3, since \(3-3=0\), which is divisible by 11. Verify: \(132 \div 11 = 12\), a valid integer.
ANSWER 6: D

---

### Problem 7:
Haruki has a 24cm wire. Which of the three shapes can he form? Calculate the perimeter of each shape:
1. Regular hexagon with side 5cm: perimeter = \(6 \times 5 = 30\)cm, which is longer than 24cm, so impossible.
2. Square with area 36cm²: side length = \(\sqrt{36}=6\)cm, perimeter = \(4 \times 6 = 24\)cm, which fits.
3. Right triangle with legs 6cm and 8cm: hypotenuse = 10cm (6-8-10 Pythagorean triple), perimeter = \(6+8+10=24\)cm, which fits.
Only the square and triangle can be formed.
ANSWER 7: D

---

### Problem 8:
Which number has the largest reciprocal? The reciprocal of \(x\) is \(\frac{1}{x}\), so smaller values of \(x\) produce larger reciprocals. List the values of all options:
A. \(\frac{1}{3} \approx 0.333\), B. \(\frac{2}{5}=0.4\), C.1, D.5, E.1986. The smallest value is \(\frac{1}{3}\), whose reciprocal is 3, the largest of all options.
ANSWER 8: A

---

### Problem 9:
Calculate the target heart rate for a 26-year-old athlete: target is 80% of max heart rate, where max = 220 - age.
First calculate max heart rate: \(220 - 26 = 194\) bpm. 80% of 194 is \(0.8 \times 194 = 155.2\), which rounds to 155 bpm. A common wrong answer is 176, which uses 220 instead of the correct max heart rate of 194.
ANSWER 9: B

---

### Problem 10:
Calculate \(H-T\), where H is Harry's correct answer to \(8-(2+5)\), and T is Terry's incorrect answer of \(8-2+5\).
Harry's answer: \(8 - 7 = 1 = H\). Terry's answer: \(6 + 5 = 11 = T\). The difference \(H-T = 1 - 11 = -10\). A common wrong answer is 10, which calculates \(T-H\) instead of \(H-T\).
ANSWER 10: A

---

### Problem 11:
Simplify \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\):
Pair the terms inside the parentheses to simplify addition: \((81+99)+(83+97)+(85+95)+(87+93)+(89+91) = 5 \times 180 = 900\). Multiply by 2: \(2 \times 900 = 1800\).
ANSWER 11: E

---

### Problem 12:
What percentage of males surveyed listen to the radio station? First calculate total number of males: total surveyed = 200, females = 96, so males = \(200 - 96 = 104\). Males who do not listen = 26, so males who listen = \(104 - 26 = 78\). The percentage of males who listen is \(\frac{78}{104} = 0.75 = 75\%\). A common wrong answer is 39%, which divides male listeners by the total surveyed population instead of the total male population.
ANSWER 12: E

### Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.
1.  Calculate original yellow tint volume: 25%? No wait, original is 30% yellow. So 30% of 30 L = $0.3 \times 30 = 9$ liters of yellow.
2.  Add 5 liters of yellow: total yellow = $9 + 5 = 14$ liters.
3.  Total new mixture volume: $30 + 5 = 35$ liters.
4.  Percent yellow: $\frac{14}{35} \times 100 = 40\%$.
ANSWER 1: C
---
### Problem 2:
We count the number of ways to choose an entry window and a different exit window from 6 windows.
1.  Entry window: 6 possible choices.
2.  Exit window: cannot be the entry window, so 5 remaining choices.
3.  Total ways: $6 \times 5 = 30$ (order matters because entry and exit are distinct actions).
ANSWER 2: D
---
### Problem 3:
Simplify and add the fractions:
1.  $\frac{1}{10} = 0.1$, $\frac{2}{20} = \frac{1}{10} = 0.1$, $\frac{3}{30} = \frac{1}{10} = 0.1$.
2.  Sum: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D
---
### Problem 4:
Calculate remaining money after expenses:
1.  Total initial money: $50.
2.  Money spent on rides: twice the food cost = $2 \times 12 = 24$ dollars.
3.  Total spent: $12 + 24 = 36$ dollars.
4.  Remaining: $50 - 36 = 14$ dollars.
ANSWER 4: B
---
### Problem 5:
Compute $200,000 \times 200,000$:
1.  Rewrite as $(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
2.  $10^{10} = 10,000,000,000$, so $4 \times 10^{10} = 40,000,000,000 = 40$ billion.
ANSWER 5: E
---
### Problem 6:
The total payment $1A2$ (a 3-digit number) must be divisible by 11 (since 11 people paid equal integer amounts).
1.  Use the divisibility rule for 11: the alternating sum of digits is divisible by 11. For $1A2$, this is $1 - A + 2 = 3 - A$.
2.  $A$ is a digit (0-9), so $3 - A$ must equal 0 (the only multiple of 11 in range: $3 - A = 11$ gives $A=-8$, $3-A=-11$ gives $A=14$, both invalid).
3.  $3 - A = 0 \implies A=3$. Verify: $132 \div 11 = 12$, which is an integer.
ANSWER 6: D
---
### Problem 7:
Check if the 24 cm wire can form each shape (perimeter of the shape must equal 24 cm):
1.  Regular hexagon (side 5 cm): Perimeter = $6 \times 5 = 30$ cm > 24 cm → cannot make.
2.  Square (area 36 cm²): Side length = $\sqrt{36} = 6$ cm. Perimeter = $4 \times 6 = 24$ cm → can make.
3.  Right triangle (legs 6, 8 cm): Hypotenuse = $\sqrt{6^2 + 8^2} = 10$ cm. Perimeter = $6 + 8 + 10 = 24$ cm → can make.
Only square and triangle can be made.
ANSWER 7: D
---
### Problem 8:
The reciprocal of a positive number $x$ is $\frac{1}{x}$; smaller $x$ gives larger reciprocal.
1.  List values: $\frac{1}{3} \approx 0.33$, $\frac{2}{5}=0.4$, $1$, $5$, $1986$.
2.  Smallest value is $\frac{1}{3}$, whose reciprocal is $3$ (largest among all reciprocals: $\frac{5}{2}=2.5$, $1$, $0.2$, $\frac{1}{1986} \approx 0.0005$).
ANSWER 8: A
---
### Problem 9:
Calculate target heart rate for a 26-year-old:
1.  Maximum heart rate: $220 - 26 = 194$ bpm.
2.  Target heart rate: 80% of 194 = $0.8 \times 194 = 155.2$, which rounds to 155.
ANSWER 9: B
---
### Problem 10:
Compute $H$ (Harry's correct answer) and $T$ (Terry's incorrect answer):
1.  Harry: $8 - (2 + 5) = 8 - 7 = 1$, so $H=1$.
2.  Terry (ignores parentheses, left-to-right): $8 - 2 + 5 = 6 + 5 = 11$, so $T=11$.
3.  Difference: $H - T = 1 - 11 = -10$.
ANSWER 10: A
---
### Problem 11:
Calculate $2 \times (81 + 83 + ... + 99)$:
1.  The sum inside is 10 consecutive odd numbers. Average of first and last term: $\frac{81 + 99}{2} = 90$.
2.  Sum of the 10 terms: $90 \times 10 = 900$.
3.  Multiply by 2: $2 \times 900 = 1800$.
ANSWER 11: E
---
### Problem 12:
Find the percentage of males who listen to the station:
1.  Total males surveyed: $200 - 96 = 104$.
2.  Males who do not listen: 26, so males who listen: $104 - 26 = 78$.
3.  Verify with total listeners: Total listeners = 136, female listeners = 58, so male listeners = $136 - 58 = 78$ (matches).
4.  Percentage: $\frac{78}{104} \times 100 = 75\%$.
ANSWER 12: E

Problem 1:
Original yellow = 30% of 30 = 9 L. Add 5 → 14 L yellow. New total = 35 L. Percent = 14/35 = 40%.

ANSWER 1: C

Problem 2:
Enter: 6 choices, leave: 5 different choices. 6×5 = 30.

ANSWER 2: D

Problem 3:
Each fraction = 0.1. Sum = 0.3.

ANSWER 3: D

Problem 4:
Food 12, rides twice = 24. Spent 36. Left = 50 − 36 = 14.

ANSWER 4: B

Problem 5:
200,000 × 200,000 = 40,000,000,000 = forty billion.

ANSWER 5: E

Problem 6:
1A2 divisible by 11. Alternating sum: 1 − A + 2 = 3 − A must be divisible by 11 → A = 3 (gives 0). Check 132/11 = 12. Yes.

ANSWER 6: D

Problem 7:
Hexagon: 6×5 = 30 > 24, no. Square area 36 → side 6, perimeter 24, yes. Triangle legs 6,8 → hypotenuse 10, perimeter 24, yes. Square and triangle only.

ANSWER 7: D

Problem 8:
Largest reciprocal = smallest number. Smallest is 1/3.

ANSWER 8: A

Problem 9:
Max = 220 − 26 = 194. Target = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

Problem 10:
H = 8 − 7 = 1. T = 8 − 2 + 5 = 11. H − T = 1 − 11 = −10.

ANSWER 10: A

Problem 11:
Sum of 81..99 odd numbers (10 terms): average (81+99)/2 = 90, ×10 = 900. ×2 = 1800.

ANSWER 11: E

Problem 12:
Males total = 200 − 96 = 104. Males who don't listen = 26 → males who listen = 104 − 26 = 78. Percent = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem systematically.

---

**Problem 1:**
Original mixture: 30 liters, 30% yellow = 9 liters yellow. Add 5 liters yellow → new yellow = 14 liters, new total = 35 liters.
Percent yellow = 14/35 = 40%.

ANSWER 1: C

---

**Problem 2:**
Georgie can enter through any of 6 windows, then leave through any of the remaining 5 windows.
6 × 5 = 30.

ANSWER 2: D

---

**Problem 3:**
1/10 = 0.1, 2/20 = 0.1, 3/30 = 0.1. Sum = 0.3.

ANSWER 3: D

---

**Problem 4:**
Food: $12. Rides: 2 × $12 = $24. Total spent: $12 + $24 = $36. Left: $50 − $36 = $14.

ANSWER 4: B

---

**Problem 5:**
200,000 × 200,000 = (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion.

ANSWER 5: E

---

**Problem 6:**
The number 1A2 must be divisible by 11. 1A2 ÷ 11 must be an integer.
11 × 10 = 110, 11 × 11 = 121 (no), 11 × 12 = 132 ✓ → A = 3.
Check: 132/11 = 12 ✓, each paid $12.

ANSWER 6: D

---

**Problem 7:**
Wire = 24 cm.
- Regular hexagon, side 5: perimeter = 6 × 5 = 30 cm > 24. ✗
- Square, area 36: side = 6, perimeter = 24 cm ✓
- Right triangle, legs 6 and 8: hypotenuse = 10, perimeter = 6+8+10 = 24 cm ✓

ANSWER 7: D

---

**Problem 8:**
The reciprocal of x is 1/x. The largest reciprocal comes from the smallest positive number.
Among 1/3, 2/5, 1, 5, 1986 — the smallest is 1/3, and its reciprocal is 3, which is the largest.

ANSWER 8: A

---

**Problem 9:**
Maximum heart rate = 220 − 26 = 194. Target = 80% × 194 = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

---

**Problem 10:**
Harry: 8 − (2 + 5) = 8 − 7 = 1, so H = 1.
Terry: 8 − 2 + 5 = 11, so T = 11.
H − T = 1 − 11 = −10.

ANSWER 10: A

---

**Problem 11:**
The ten odd numbers from 81 to 99: sum = (81+99)/2 × 10 = 90 × 10 = 900.
2 × 900 = 1800.

ANSWER 11: E

---

**Problem 12:**
Total surveyed = 200. Females = 96, so Males = 104.
Total non-listeners = 64. Males who don't listen = 26, so females who don't listen = 64 − 26 = 38.
Males who listen = 104 − 26 = 78.
Percentage = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem step-by-step.

## Problem 1:
**What is asked:** Find the percentage of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.

**Original mixture:**
- Yellow tint: 30% of 30 = 0.30 × 30 = 9 liters
- Total volume: 30 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total volume: 30 + 5 = 35 liters

**Percentage of yellow tint:**
$$\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%$$

ANSWER 1: C

## Problem 2:
**What is asked:** In how many ways can the ghost enter by one window and leave by a different window?

**Solution:**
- Choices for entering: 6 windows
- Choices for leaving: 5 windows (must be different from entry)
- Total ways: 6 × 5 = 30

ANSWER 2: D

## Problem 3:
**What is asked:** Calculate 1/10 + 2/20 + 3/30

**Solution:**
Simplify each fraction:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

ANSWER 3: D

## Problem 4:
**What is asked:** How much money does Susan have left?

**Solution:**
- Started with: $50
- Spent on food: $12
- Spent on rides: 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Left: $50 - $36 = $14

ANSWER 4: B

## Problem 5:
**What is asked:** Calculate 200,000 × 200,000

**Solution:**
$$200,000 × 200,000 = (2 × 10^5) × (2 × 10^5) = 4 × 10^{10} = 40,000,000,000 = 40 \text{ billion}$$

ANSWER 5: E

## Problem 6:
**What is asked:** Find the missing digit A in the 3-digit number 1A2, where 11 club members each paid an equal integer amount that totals 1A2.

**Solution:**
The number 1A2 must be divisible by 11.

Testing divisibility by 11 (alternating sum of digits):
- 1 - A + 2 = 0 (mod 11)
- 3 - A ≡ 0 (mod 11)
- A = 3

Check: 132 ÷ 11 = 12 ✓

ANSWER 6: D

## Problem 7:
**What is asked:** Which shapes can Haruki make with 24 cm of wire?

**Solution:**
Test each shape:

**Regular hexagon with side 5 cm:**
- Perimeter: 6 × 5 = 30 cm
- Need 30 cm, have 24 cm ✗

**Square with area 36 cm²:**
- Side length: √36 = 6 cm
- Perimeter: 4 × 6 = 24 cm ✓

**Right triangle with legs 6 and 8 cm:**
- Hypotenuse: √(6² + 8²) = √(36 + 64) = √100 = 10 cm
- Perimeter: 6 + 8 + 10 = 24 cm ✓

Only square and triangle can be made.

ANSWER 7: D

## Problem 8:
**What is asked:** Which number has the largest reciprocal?

**Solution:**
Find reciprocals:
- 1/(1/3) = 3
- 1/(2/5) = 5/2 = 2.5
- 1/1 = 1
- 1/5 = 0.2
- 1/1986 ≈ 0.0005

The largest reciprocal is 5 (from the number 1/5).

ANSWER 8: D

## Problem 9:
**What is asked:** Find the target heart rate for a 26-year-old athlete.

**Solution:**
- Maximum heart rate: 220 - 26 = 194 beats per minute
- Target heart rate: 80% of 194 = 0.80 × 194 = 155.2 ≈ 155

ANSWER 9: B

## Problem 10:
**What is asked:** Find H - T where H is Harry's answer and T is Terry's answer.

**Solution:**
**Harry's calculation (correct):**
$$H = 8 - (2 + 5) = 8 - 7 = 1$$

**Terry's calculation (ignoring parentheses):**
$$T = 8 - 2 + 5 = 6 + 5 = 11$$

**Difference:**
$$H - T = 1 - 11 = -10$$

ANSWER 10: A

## Problem 11:
**What is asked:** Calculate 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

**Solution:**
This is an arithmetic sequence with 10 terms. Sum = n(first + last)/2
$$81 + 83 + ... + 99 = \frac{10(81 + 99)}{2} = \frac{10 × 180}{2} = 900$$

$$2 × 900 = 1800$$

ANSWER 11: E

## Problem 12:
**What is asked:** What percentage of males surveyed listen to the station?

**Solution:**
- Total surveyed: 200
- Females: 96
- Males: 200 - 96 = 104

- Total non-listeners: 64
- Males who don't listen: 26
- Males who listen: 104 - 26 = 78

**Percentage of males who listen:**
$$\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%$$

ANSWER 12: E

Problem 1:

The original mixture is 30 liters total, with 30% yellow tint.

So the amount of yellow tint at first is:
- \(30\%\) of \(30 = 0.30 \times 30 = 9\) liters

Then 5 liters of yellow tint are added, so the new amount of yellow tint is:
- \(9 + 5 = 14\) liters

The new total mixture is:
- \(30 + 5 = 35\) liters

So the percent of yellow tint is:
- \(\frac{14}{35} = \frac{2}{5} = 0.4 = 40\%\)

Answer choice C matches 40%.

ANSWER 1: C

Problem 2:

Georgie can:
- choose 1 of the 6 windows to enter
- then choose a different window to leave

For the entry, there are 6 choices.

For the exit, there are 5 remaining choices, since it must be a different window.

So the total number of ways is:
- \(6 \times 5 = 30\)

ANSWER 2: D

Problem 3:

Compute each fraction:
- \(\frac{1}{10} = 0.1\)
- \(\frac{2}{20} = \frac{1}{10} = 0.1\)
- \(\frac{3}{30} = \frac{1}{10} = 0.1\)

Add them:
- \(0.1 + 0.1 + 0.1 = 0.3\)

So the value is \(0.3\).

ANSWER 3: D

Problem 4:

Susan had \$50.

She spent:
- \$12 on food
- twice as much on rides, so \(2 \times 12 = 24\)

Total spent:
- \(12 + 24 = 36\)

Money left:
- \(50 - 36 = 14\)

ANSWER 4: B

Problem 5:

Two hundred thousand is:
- \(200{,}000 = 2 \times 10^5\)

Multiply:
- \(200{,}000 \times 200{,}000 = (2 \times 10^5)(2 \times 10^5) = 4 \times 10^{10}\)

And \(4 \times 10^{10} = 40{,}000{,}000{,}000\), which is forty billion.

ANSWER 5: E

Problem 6:

The total is written as \(1A2\), and 11 members each paid the same integer amount.

So \(1A2\) must be divisible by 11.

Test the answer choices:
- 102 ÷ 11 is not an integer
- 112 ÷ 11 is not an integer
- 122 ÷ 11 = 11 remainder 1
- 132 ÷ 11 = 12 exactly
- 142 ÷ 11 is not an integer

So the number is 132, which means \(A = 3\).

ANSWER 6: D

Problem 7:

We check each shape.

### Hexagon
A regular hexagon has 6 equal sides.
If each side is 5 cm, perimeter is:
- \(6 \times 5 = 30\) cm

The wire is only 24 cm, so he cannot make the hexagon.

### Square
A square with area 36 has side length:
- \(\sqrt{36} = 6\) cm

Perimeter:
- \(4 \times 6 = 24\) cm

So he can make the square exactly.

### Right triangle
The legs are 6 and 8, so the hypotenuse is:
- \(\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)

Perimeter:
- \(6 + 8 + 10 = 24\) cm

So he can make the triangle exactly.

Therefore he can make the square and the triangle only.

ANSWER 7: D

Problem 8:

The reciprocal is largest when the original number is smallest, assuming positive numbers.

Compare:
- \(\frac13 \approx 0.333\)
- \(\frac25 = 0.4\)
- \(1\)
- \(5\)
- \(1986\)

The smallest number is \(\frac13\), so it has the largest reciprocal.

Its reciprocal is 3, which is larger than the reciprocals of all the others.

ANSWER 8: A

Problem 9:

Maximum heart rate:
- \(220 - 26 = 194\)

Target heart rate is 80% of that:
- \(0.8 \times 194 = 155.2\)

To the nearest whole number, that is 155.

ANSWER 9: B

Problem 10:

First, Harry uses the parentheses:
- \(8 - (2 + 5) = 8 - 7 = 1\)

So \(H = 1\).

Terry ignores parentheses:
- \(8 - 2 + 5 = 6 + 5 = 11\)

So \(T = 11\).

Difference:
- \(H - T = 1 - 11 = -10\)

ANSWER 10: A

Problem 11:

We need to evaluate:
- \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

These are 10 odd numbers from 81 to 99.

Instead of adding one by one, use the average:
- average of 81 and 99 is \( \frac{81+99}{2} = 90\)

Since there are 10 numbers, their sum is:
- \(10 \times 90 = 900\)

Now multiply by 2:
- \(2 \times 900 = 1800\)

ANSWER 11: E

Problem 12:

We want the percentage of males surveyed who listen.

From the table:
- Total surveyed = 200
- Females = 96
- So males = \(200 - 96 = 104\)

Males who don’t listen = 26, so males who do listen:
- \(104 - 26 = 78\)

Now find the percent of males who listen:
- \(\frac{78}{104} = \frac{3}{4} = 75\%\)

ANSWER 12: E

Problem 1:  
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?  

- Original yellow tint amount = 30 × 0.30 = 9 liters.  
- Add 5 liters of yellow tint → new yellow amount = 9 + 5 = 14 liters.  
- New total volume = 30 + 5 = 35 liters.  
- New percent yellow = (14 / 35) × 100% = (2/5) × 100% = 40%.  

So the correct choice is 40.  

ANSWER 1: C  

---

Problem 2:  
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?  

- Choose an entry window: 6 choices.  
- Choose a different exit window: 5 choices (cannot be the same as entry).  
- Total ways = 6 × 5 = 30.  

ANSWER 2: D  

---

Problem 3:  
Compute: \( \frac{1}{10} + \frac{2}{20} + \frac{3}{30} \).  

Convert each to a common denominator, 60:  
- \( \frac{1}{10} = \frac{6}{60} \)  
- \( \frac{2}{20} = \frac{2}{20} = \frac{6}{60} \) (since multiply top and bottom by 3)  
- \( \frac{3}{30} = \frac{3}{30} = \frac{6}{60} \) (since multiply top and bottom by 2)  

Add:  
\( \frac{6}{60} + \frac{6}{60} + \frac{6}{60} = \frac{18}{60} = \frac{3}{10} = 0.3 \)

So the value is .3.  

ANSWER 3: D  

---

Problem 4:  
Susan had $50 to spend. She spent $12 on food and twice as much on rides. How much left?  

- Food = 12  
- Rides = twice 12 = 24  
- Total spent = 12 + 24 = 36  
- Money left = 50 − 36 = 14  

ANSWER 4: B  

---

Problem 5:  
Two hundred thousand times two hundred thousand equals ?  

\(200{,}000 \times 200{,}000 = (2\times 10^5)(2\times 10^5) = 4 \times 10^{10}\).  
- \(10^{10} = 10{,}000{,}000{,}000\)  
- \(4 \times 10^{10} = 40{,}000{,}000{,}000 = 40\) billion  

ANSWER 5: E  

---

Problem 6:  
Eleven members each paid the same integer amount for a guest speaker. In all, they paid $1A2 (a 3-digit number). Find missing digit A.  

- Total = \(1A2\) means:  
  \(1A2 = 100 + 10A + 2 = 102 + 10A\).  
- This total is divided equally among 11 people:  
  So \(102 + 10A\) must be divisible by 11.  

Check \(102 + 10A \mod 11\):  
- \(102 \div 11 = 9\) remainder 3, so \(102 \equiv 3 \pmod{11}\).  
- \(10A \equiv -A \pmod{11}\) (since 10 ≡ −1 mod 11).  

So:  
\(102 + 10A \equiv 3 - A \equiv 0 \pmod{11}\)  
Thus \(A \equiv 3 \pmod{11}\).  
Since \(A\) is a digit 0–9, \(A=3\).  

ANSWER 6: D  

---

Problem 7:  
Wire length = 24 cm. Which shapes can be made (using all 24 cm of wire)?  

We need the perimeter (total boundary length) of each shape:

**(1) Regular hexagon with side length 5 cm**  
Perimeter = 6 × 5 = 30 cm → too long. Cannot.

**(2) Square with area 36 cm²**  
Side length = √36 = 6 cm  
Perimeter = 4 × 6 = 24 cm → fits. Can.

**(3) Right triangle with legs 6 and 8 cm**  
Right triangle sides are 6 and 8 legs, and hypotenuse:  
\( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)  
Perimeter = 6 + 8 + 10 = 24 cm → fits. Can.

So the shapes possible: **square and triangle only**.  

ANSWER 7: D  

---

Problem 8:  
Which number has the largest reciprocal? Recall: the reciprocal of \(x\) is \(1/x\).  
We compare \(1/3, 2/5, 1, 1/5, 1/1986\).  

Compute roughly:  
- \(1/3 \approx 0.333\)  
- \(2/5 = 0.4\)  
- \(1 = 1\)  
- \(1/5 = 0.2\)  
- \(1/1986\) is tiny  

Largest is 1 (reciprocal of 1).  

ANSWER 8: C  

---

Problem 9:  
Target heart rate = 80% of (maximum heart rate).  
Maximum heart rate = 220 − age.  
Age = 26.

- Maximum = 220 − 26 = 194  
- Target = 0.80 × 194 = 155.2  
Nearest whole number = 155  

ANSWER 9: B  

---

Problem 10:  
Expression: \(8 - (2 + 5)\)  
- Harry computes correctly:  
  \(2 + 5 = 7\), so \(8 - 7 = 1\).  
- Terry ignores parentheses:  
  \(8 - 2 + 5 = 6 + 5 = 11\).  

Difference \(H - T = 1 - 11 = -10\).  

ANSWER 10: A  

---

Problem 11:  
Compute:  
\(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

Notice these are all odd numbers from 81 to 99 stepping by 2.  
There are 10 terms.

Pair from ends:  
- 81 + 99 = 180  
- 83 + 97 = 180  
- 85 + 95 = 180  
- 87 + 93 = 180  
- 89 + 91 = 180  

There are 5 pairs, each sum 180:  
Sum = 5 × 180 = 900  

Now multiply by 2:  
2 × 900 = 1800  

ANSWER 11: E  

---

Problem 12:  
Total surveyed = 200  
Females = 96  
Females who listen = 58  
Males who don’t listen = 26  
Total listeners = 136  

We want: % of males who listen.

Step 1: Find males total.  
Males = 200 − 96 = 104

Step 2: Find males who don’t listen is given = 26.  
So males who listen = 104 − 26 = 78

Step 3: Convert to percent of males who listen:  
Percent = (78 / 104) × 100%  
Simplify 78/104 = 3/4 (divide by 26):  
So percent = (3/4) × 100% = 75%

ANSWER 12: E

Problem 1:
The original mixture is 30 liters. The amount of yellow tint is 30% of 30 liters: $0.30 \times 30 = 9$ liters.
Five liters of yellow tint are added, so the new amount of yellow tint is $9 + 5 = 14$ liters.
The new total volume of the mixture is $30 + 5 = 35$ liters.
The percentage of yellow tint is $\frac{14}{35} \times 100\%$.
Simplifying the fraction: $\frac{14}{35} = \frac{2}{5} = 0.4$.
$0.4 \times 100\% = 40\%$.
ANSWER 1: C

Problem 2:
Georgie has 6 choices for the window he enters. Since he must leave by a different window, he has $6 - 1 = 5$ choices for the window he leaves by.
The total number of ways is $6 \times 5 = 30$.
ANSWER 2: D

Problem 3:
Simplify each fraction:
$1/10 = 0.1$
$2/20 = 1/10 = 0.1$
$3/30 = 1/10 = 0.1$
Summing them: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides, which is $2 \times 12 = 24$.
Total spent: $12 + 24 = 36$.
Amount left: $50 - 36 = 14$.
ANSWER 4: B

Problem 5:
Two hundred thousand is $200,000$ (or $2 \times 10^5$).
$(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
$4 \times 10^{10} = 40,000,000,000$, which is forty billion.
ANSWER 5: E

Problem 6:
The total amount $1A2$ must be divisible by 11.
A number is divisible by 11 if the alternating sum of its digits is divisible by 11.
For $1A2$, the alternating sum is $1 - A + 2 = 3 - A$.
For this to be 0 (the only multiple of 11 possible for a single digit $A$), $3 - A = 0$, so $A = 3$.
Check: $132 / 11 = 12$. This works.
ANSWER 6: D

Problem 7:
Hexagon: Perimeter of a regular hexagon with side 5 is $6 \times 5 = 30$ cm. (Cannot make)
Square: Area is 36, so side is $\sqrt{36} = 6$. Perimeter is $4 \times 6 = 24$ cm. (Can make)
Triangle: Right triangle with legs 6 and 8. Hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. Perimeter is $6 + 8 + 10 = 24$ cm. (Can make)
ANSWER 7: D

Problem 8:
Reciprocals:
A: $1/(1/3) = 3$
B: $1/(2/5) = 2.5$
C: $1/1 = 1$
D: $1/5 = 0.2$
E: $1/1986 \approx 0.0005$
The largest is 3.
ANSWER 8: A

Problem 9:
Max heart rate = $220 - 26 = 194$.
Target heart rate = $80\%$ of $194 = 0.8 \times 194 = 155.2$.
To the nearest whole number, this is 155.
ANSWER 9: B

Problem 10:
$H = 8 - (2 + 5) = 8 - 7 = 1$.
$T = 8 - 2 + 5 = 6 + 5 = 11$.
$H - T = 1 - 11 = -10$.
ANSWER 10: A

Problem 11:
Sum inside parentheses: $81+83+85+87+89+91+93+95+97+99$.
This is an arithmetic progression with 10 terms.
Sum = $\frac{n}{2}(\text{first} + \text{last}) = \frac{10}{2}(81 + 99) = 5 \times 180 = 900$.
The expression is $2 \times 900 = 1800$.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96.
Males = $200 - 96 = 104$.
Males who don't listen = 26.
Males who listen = $104 - 26 = 78$.
Percentage of males who listen = $(78 / 104) \times 100\%$.
$\frac{78}{104} = \frac{39}{52} = \frac{3}{4} = 0.75 = 75\%$.
ANSWER 12: E

Problem 1: The original mixture has volume 30 L, of which 30% is yellow tint, so there are \(0.3 \times 30 = 9\) L of yellow tint. Adding 5 L of yellow tint produces a new total of 14 L of yellow tint and a new mixture volume of 35 L. The new percentage of yellow tint is therefore \((14/35) \times 100\% = 40\%\).

ANSWER 1: C

Problem 2: There are 6 choices for the entry window. For each of those choices there are then exactly 5 remaining choices for a different exit window, giving a total of \(6 \times 5 = 30\) ordered pairs of distinct windows.

ANSWER 2: D

Problem 3: Rewrite each term with a common denominator of 10:  
\(1/10 = 0.1\),  
\(2/20 = 1/10 = 0.1\),  
\(3/30 = 1/10 = 0.1\).  
Their sum is \(0.1 + 0.1 + 0.1 = 0.3\).

ANSWER 3: D

Problem 4: Twice the food cost is \(2 \times 12 = 24\) dollars spent on rides. The total spent is therefore \(12 + 24 = 36\) dollars, leaving \(50 - 36 = 14\) dollars.

ANSWER 4: B

Problem 5: Write the factors in scientific notation:  
\(200000 = 2 \times 10^5\), so  
\((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\),  
which is forty billion.

ANSWER 5: E

Problem 6: The three-digit total \(1A2\) must be divisible by 11. The alternating-sum divisibility rule requires that \(1 - A + 2 = 3 - A\) be a multiple of 11. The only admissible digit \(A\) (0-9) that satisfies the condition is \(A = 3\), giving the total 132, which is exactly \(11 \times 12\).

ANSWER 6: D

Problem 7: A regular hexagon with side 5 cm has perimeter 30 cm, which exceeds the 24 cm length of wire, so it is impossible. A square of area 36 cm² has side length 6 cm and perimeter 24 cm, which matches the wire length. A right triangle with legs 6 cm and 8 cm has hypotenuse 10 cm, so its perimeter is \(6 + 8 + 10 = 24\) cm, which also matches. Consequently only the square and the triangle are possible.

ANSWER 7: D

Problem 8: For positive numbers the reciprocal function is strictly decreasing, so the largest reciprocal belongs to the smallest number in the list, which is \(1/3\).

ANSWER 8: A

Problem 9: The theoretical maximum heart rate is \(220 - 26 = 194\) beats per minute. Eighty percent of that value is \(0.8 \times 194 = 155.2\), which rounds to the nearest whole number 155.

ANSWER 9: B

Problem 10: Harry evaluates the parentheses first and obtains \(8 - (2 + 5) = 1\). Terry omits the parentheses and obtains \(8 - 2 + 5 = 11\). Their difference is therefore \(1 - 11 = -10\).

ANSWER 10: A

Problem 11: The sum inside the parentheses consists of the ten consecutive odd integers from 81 to 99. Their average is \((81 + 99)/2 = 90\), so the sum equals \(10 \times 90 = 900\). Multiplying by the outer factor of 2 yields \(2 \times 900 = 1800\).

ANSWER 11: E

Problem 12: There are 200 people surveyed and 96 females, so there are \(200 - 96 = 104\) males. Of the 96 females, 58 listen, leaving \(96 - 58 = 38\) females who do not listen. Given that 26 males do not listen, the number of males who listen is \(104 - 26 = 78\). The required percentage is therefore \((78/104) \times 100\% = 75\%\).

ANSWER 12: E

Problem 1:
The problem asks us to find the percentage of yellow tint in a new mixture after adding 5 liters of yellow tint to the original 30 liters of paint mixture. The original mixture is 25% red tint, 30% yellow tint, and 45% water.

First, let's calculate the amount of yellow tint in the original mixture. 30% of 30 liters is $0.3 \times 30 = 9$ liters.

After adding 5 liters of yellow tint, the total amount of yellow tint becomes $9 + 5 = 14$ liters.

The total volume of the new mixture is $30 + 5 = 35$ liters.

To find the percentage of yellow tint in the new mixture, we divide the amount of yellow tint by the total volume and multiply by 100: $\frac{14}{35} \times 100 = 40\%$.

Let's examine the choices:
- A. 25: Too low because the percentage of yellow tint increased.
- B. 35: Not the correct calculation.
- C. 40: Matches our calculation.
- D. 45: Too high.
- E. 50: Too high.

The correct percentage is 40%.

ANSWER 1: C

Problem 2:
The problem asks us to determine the number of ways Georgie the Ghost can enter a haunted house through one of six windows and leave by a different window.

To enter, Georgie has 6 choices. After entering, there are 5 remaining windows to choose from for leaving.

The total number of ways to enter and leave is $6 \times 5 = 30$.

Let's examine the choices:
- A. 12: Too low.
- B. 15: Too low.
- C. 18: Too low.
- D. 30: Matches our calculation.
- E. 36: Too high because it would imply Georgie can leave by the same window.

The correct number of ways is 30.

ANSWER 2: D

Problem 3:
The problem requires us to simplify the expression $1⁄10 + 2⁄20 + 3⁄30$.

First, simplify each fraction:
- $1⁄10$ stays the same.
- $2⁄20 = 1⁄10$.
- $3⁄30 = 1⁄10$.

Adding them together: $1⁄10 + 1⁄10 + 1⁄10 = 3⁄10 = 0.3$.

Let's examine the choices:
- A. .1: Too low.
- B. .123: Not the correct sum.
- C. .2: Too low.
- D. .3: Matches our calculation.
- E. .6: Too high.

The correct sum is 0.3.

ANSWER 3: D

Problem 4:
The problem asks us to find out how many dollars Susan had left after spending money on food and rides at the carnival. She started with $50, spent $12 on food, and twice as much on rides.

First, calculate the amount spent on rides: $12 \times 2 = $24.

Total amount spent is $12 + $24 = $36.

The amount left is $50 - $36 = $14.

Let's examine the choices:
- A. 12: Too low.
- B. 14: Matches our calculation.
- C. 26: Too high.
- D. 38: Too high.
- E. 50: Incorrect because she spent money.

The correct amount left is $14.

ANSWER 4: B

Problem 5:
The problem asks us to calculate $200,000 \times 200,000$.

$200,000 = 2 \times 10^5$, so $200,000 \times 200,000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10} = 40,000,000,000$.

Let's examine the choices:
- A. four hundred thousand: Too low.
- B. four million: Too low.
- C. forty thousand: Too low.
- D. four hundred million: Too low.
- E. forty billion: Matches our calculation.

The correct result is forty billion.

ANSWER 5: E

Problem 6:
The problem requires us to find the missing digit A in the 3-digit number $1A2 that represents the total amount paid by 11 members of the Math Club.

The total amount paid is divisible by 11 since there are 11 members who paid the same amount.

To check divisibility by 11, we use the rule: the difference between the sum of digits at odd places and the sum of digits at even places should be divisible by 11.

For $1A2, the sum of digits at odd places is $1 + 2 = 3$, and the sum at even places is $A$. The difference is $3 - A$.

For $3 - A$ to be divisible by 11 (or 0, since $A$ is a single digit), $A = 3$ makes $3 - A = 0$, which is divisible by 11.

Let's examine the choices:
- A. 0: Not correct because $3 - 0 = 3$, not divisible by 11.
- B. 1: Not correct because $3 - 1 = 2$, not divisible by 11.
- C. 2: Not correct because $3 - 2 = 1$, not divisible by 11.
- D. 3: Correct because $3 - 3 = 0$, which is divisible by 11.
- E. 4: Not correct because $3 - 4 = -1$, not divisible by 11.

The missing digit A is 3, but we should verify if 132 is indeed divisible by 11. $132 / 11 = 12$, so it is.

ANSWER 6: D

Problem 7:
The problem asks us to determine which shapes Haruki can form using a 24 cm long wire.

1. A regular hexagon with side length 5 cm: The perimeter is $6 \times 5 = 30$ cm, which is more than 24 cm. So, Haruki cannot form this hexagon.

2. A square of area 36 cm$^2$: The side length is $\sqrt{36} = 6$ cm. The perimeter is $4 \times 6 = 24$ cm, which matches the wire length. So, Haruki can form this square.

3. A right triangle with legs 6 and 8 cm: Using the Pythagorean theorem, the hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm. The perimeter is $6 + 8 + 10 = 24$ cm, which matches the wire length. So, Haruki can form this triangle.

Let's examine the choices:
- A. Triangle only: Incorrect because Haruki can also form the square.
- B. Hexagon and square only: Incorrect because Haruki cannot form the hexagon.
- C. Hexagon and triangle only: Incorrect because Haruki cannot form the hexagon.
- D. Square and triangle only: Matches our findings.
- E. Hexagon, triangle, and square: Incorrect because Haruki cannot form the hexagon.

The correct shapes Haruki can form are the square and the triangle.

ANSWER 7: D

Problem 8:
The problem asks us to find which of the given numbers has the largest reciprocal.

The reciprocal of a number $x$ is $1/x$. The larger $x$ is, the smaller its reciprocal is, and vice versa.

Let's examine the choices:
- A. $1⁄3$: The reciprocal is 3.
- B. $2⁄5$: The reciprocal is $5/2 = 2.5$.
- C. 1: The reciprocal is 1.
- D. 5: The reciprocal is $1/5 = 0.2$.
- E. 1986: The reciprocal is $1/1986$, very small.

Comparing the reciprocals: $3 > 2.5 > 1 > 0.2 > 1/1986$. So, $1/3$ has the largest reciprocal.

Let's examine the choices again:
- A. $1⁄3$: Correct.
- B. $2⁄5$: Not correct.
- C. 1: Not correct.
- D. 5: Not correct.
- E. 1986: Not correct.

The number with the largest reciprocal is $1/3$.

ANSWER 8: A

Problem 9:
The proble

Problem 1:
Original mixture: 30 L. Yellow tint = 30% of 30 = 0.30 × 30 = 9 L.
Add 5 L yellow: new yellow = 9 + 5 = 14 L, new total volume = 30 + 5 = 35 L.
Percent yellow = (14/35) × 100% = 40%.
ANSWER 1: C

Problem 2:
6 choices for entry window, 5 remaining choices for a different exit window.
Total ways = 6 × 5 = 30.
ANSWER 2: D

Problem 3:
1/10 + 2/20 + 3/30 = 1/10 + 1/10 + 1/10 = 3/10 = 0.3.
ANSWER 3: D

Problem 4:
Spent on food: $12. Spent on rides: 2 × $12 = $24.
Total spent = $12 + $24 = $36. Left = $50 − $36 = $14.
ANSWER 4: B

Problem 5:
200,000 × 200,000 = (2×10⁵) × (2×10⁵) = 4×10¹⁰ = 40,000,000,000, which is forty billion.
ANSWER 5: E

Problem 6:
Total paid is $1A2, a multiple of 11. Divisibility by 11: (1 − A + 2) = 3 − A must be a multiple of 11.
With A a digit 0–9, 3 − A = 0 ⇒ A = 3. (Check: 132 ÷ 11 = 12.)
ANSWER 6: D

Problem 7:
Wire length = 24 cm.
- Regular hexagon, side 5 cm: perimeter = 6×5 = 30 cm > 24 cm → cannot make.
- Square, area 36 cm²: side = 6 cm, perimeter = 4×6 = 24 cm → can make.
- Right triangle, legs 6 and 8 cm: hypotenuse = 10 cm, perimeter = 6+8+10 = 24 cm → can make.
So square and triangle only.
ANSWER 7: D

Problem 8:
Reciprocals: 1/(1/3)=3, 1/(2/5)=2.5, 1/1=1, 1/5=0.2, 1/1986≈0.0005. Largest is 3, from 1/3.
ANSWER 8: A

Problem 9:
Maximum heart rate = 220 − 26 = 194 bpm.
Target = 80% of 194 = 0.8 × 194 = 155.2 → 155 to nearest whole number.
ANSWER 9: B

Problem 10:
Harry: H = 8 − (2+5) = 8 − 7 = 1.
Terry: T = 8 − 2 + 5 = 6 + 5 = 11.
H − T = 1 − 11 = −10.
ANSWER 10: A

Problem 11:
Sum inside parentheses: 10 terms, average = (81+99)/2 = 90, sum = 10×90 = 900.
2 × 900 = 1800.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96 ⇒ Males = 200 − 96 = 104.
Total listeners = 136, female listeners = 58 ⇒ male listeners = 136 − 58 = 78.
Percent of males who listen = (78/104) × 100% = 75%.
ANSWER 12: E

Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint.

**Original mixture (30 liters):**
- Red tint: 25% of 30 = 7.5 liters
- Yellow tint: 30% of 30 = 9 liters
- Water: 45% of 30 = 13.5 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total mixture: 30 + 5 = 35 liters

**New percentage of yellow tint:**
14 / 35 = 2/5 = 0.40 = 40%

Tempting wrong answers: Choice B (35) might come from just adding 5 to the original 30%, but we must recalculate based on the new total.

ANSWER 1: C

---

Problem 2:
Georgie enters through one window and leaves through a *different* window.

- Number of choices for the entry window: 6
- Number of choices for the exit window (must be different): 5
- Total ways: 6 × 5 = 30

Tempting wrong answers: Choice E (36) would be 6 × 6, which allows entering and leaving through the same window. Choice A (12) would be 6 + 6, which doesn't account for all combinations.

ANSWER 2: D

---

Problem 3:
Simplify each fraction first:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

Tempting wrong answers: Choice B (.123) might come from concatenating the digits 1, 2, 3 instead of actually computing. Choice C (.2) might come from miscounting the terms.

ANSWER 3: D

---

Problem 4:
Susan starts with $50.
- Spent on food: $12
- Spent on rides: twice the food amount = 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Amount left: $50 − $36 = $14

Tempting wrong answers: Choice C (26) might come from subtracting only the food ($50 − $24 = $26, forgetting food). Choice D (38) comes from subtracting only $12.

ANSWER 4: B

---

Problem 5:
200,000 × 200,000

Write in scientific notation:
(2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰

4 × 10¹⁰ = 40,000,000,000 = forty billion

Tempting wrong answers: Choice A (four hundred thousand) comes from adding instead of multiplying. Choice D (four hundred million) comes from miscounting zeros.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, so the total $1A2 must be divisible by 11.

**Divisibility rule for 11:** The alternating sum of digits must be a multiple of 11.
Alternating sum: 1 − A + 2 = 3 − A

For 3 − A to be divisible by 11 (and A to be a single digit 0–9):
3 − A = 0 → A = 3

**Check:** $132 ÷ 11 = $12 per member. This is a valid integer amount.

Tempting wrong answers: Other values of A don't make 3 − A a multiple of 11 within the digit range.

ANSWER 6: D

---

Problem 7:
Wire length = 24 cm. We check if each shape's perimeter ≤ 24 cm.

**Regular hexagon (side 5 cm):**
Perimeter = 6 × 5 = 30 cm. Since 30 > 24, he **cannot** make this.

**Square (area 36 cm²):**
Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Since 24 = 24, he **can** make this.

**Right triangle (legs 6 and 8 cm):**
Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Perimeter = 6 + 8 + 10 = 24 cm. Since 24 = 24, he **can** make this.

Haruki can make the square and triangle only.

Tempting wrong answers: Choice E includes the hexagon, but its perimeter exceeds the wire length.

ANSWER 7: D

---

Problem 8:
The reciprocal of a number x is 1/x. The *smaller* the positive number, the *larger* its reciprocal.

| Choice | Number | Reciprocal |
|--------|--------|------------|
| A | 1/3 | 3 |
| B | 2/5 | 5/2 = 2.5 |
| C | 1 | 1 |
| D | 5 | 1/5 = 0.2 |
| E | 1986 | 1/1986 ≈ 0.0005 |

The largest reciprocal is 3, which comes from 1/3.

Tempting wrong answers: Choice E (1986) is the largest number, so it has the *smallest* reciprocal.

ANSWER 8: A

---

Problem 9:
Athlete's age = 26 years.

**Step 1:** Maximum heart rate = 220 − 26 = 194 beats per minute.

**Step 2:** Target heart rate = 80% of 194 = 0.80 × 194 = 155.2

**Step 3:** Round to nearest whole number: 155

Tempting wrong answers: Choice C (176) might come from taking 80% of 220 without subtracting the age. Choice D (194) is the maximum heart rate itself.

ANSWER 9: B

---

Problem 10:
**Harry (correct calculation):**
H = 8 − (2 + 5) = 8 − 7 = 1

**Terry (ignores parentheses):**
T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:**
H − T = 1 − 11 = −10

Tempting wrong answers: Choice E (10) is |H − T|, but the question asks for H − T specifically, which is negative since Terry's answer is larger.

ANSWER 10: A

---

Problem 11:
We need: 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

The sum inside is an arithmetic sequence:
- First term: 81, Last term: 99
- Number of terms: 10
- Sum = (number of terms) × (first + last) / 2 = 10 × (81 + 99) / 2 = 10 × 180 / 2 = 10 × 90 = 900

Multiply by 2: 2 × 900 = 1800

Tempting wrong answers: Choice C (1700) might come from miscounting terms or miscalculating the average.

ANSWER 11: E

---

Problem 12:
Let me organize the data from the survey:

- Total surveyed: 200
- Females: 96 → Males: 200 − 96 = 104
- Total listeners: 136
- Females who listen: 58 → Males who listen: 136 − 58 = 78
- Males who don't listen: 26 (Check: 78 + 26 = 104 ✓)

**Percentage of males who listen:**
(78 / 104) × 100% = (3/4) × 100% = 75%

Tempting wrong answers: Choice C (52) might come from computing 104/200 (percentage of total who are male). Choice D (55) might come from 78/136 (percentage of listeners who are male).

ANSWER 12: E

 Problem 1:
We start with 30 liters of paint. The original amount of yellow tint is 30% of 30 liters:
\[
0.30 \times 30 = 9 \text{ liters.}
\]
Five liters of yellow tint are added, so the new amount of yellow tint is:
\[
9 + 5 = 14 \text{ liters.}
\]
The new total volume of the mixture is:
\[
30 + 5 = 35 \text{ liters.}
\]
The percent of yellow tint in the new mixture is:
\[
\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%.
\]
(Choice A is the original red percentage, D is the original water percentage, and E would require 17.5 liters of yellow, which we do not have.)
ANSWER 1: C

Problem 2:
Georgie chooses one of 6 windows to enter. Since he must leave through a different window, there are 5 remaining choices for his exit. By the multiplication principle:
\[
6 \times 5 = 30 \text{ ways.}
\]
(Choice E, 36, would count entering and leaving through the same window; the other choices are too small.)
ANSWER 2: D

Problem 3:
Simplify each fraction:
\[
\frac{2}{20} = \frac{1}{10}, \qquad \frac{3}{30} = \frac{1}{10}.
\]
So the sum is:
\[
\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3.
\]
(Choice A is just one term; C is 0.2; E is 0.6.)
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides:
\[
2 \times 12 = 24 \text{ dollars.}
\]
Total spent:
\[
12 + 24 = 36 \text{ dollars.}
\]
Amount left:
\[
50 - 36 = 14 \text{ dollars.}
\]
(Choice C is the total spent; D is the amount left if she had spent only $12.)
ANSWER 4: B

Problem 5:
\[
200{,}000 \times 200{,}000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}.
\]
\[
4 \times 10^{10} = 40{,}000{,}000{,}000,
\]
which is read as forty billion.
(Choice D is $4 \times 10^8$; B is $4 \times 10^6$.)
ANSWER 5: E

Problem 6:
The total amount paid is the three-digit number $1A2$, which equals:
\[
100 + 10A + 2 = 102 + 10A.
\]
Since 11 members each paid the same integer amount, $102 + 10A$ must be divisible by 11. Testing the choices:
- $A=0$: $102 \div 11 \approx 9.27$ (not an integer)
- $A=1$: $112 \div 11 \approx 10.18$ (not an integer)
- $A=2$: $122 \div 11 \approx 11.09$ (not an integer)
- $A=3$: $132 \div 11 = 12$ (integer!)
- $A=4$: $142 \div 11 \approx 12.9$ (not an integer)
Thus the missing digit is 3.
ANSWER 6: D

Problem 7:
The wire is 24 cm long, so a shape is possible only if its perimeter is at most 24 cm.
- **Regular hexagon, side 5 cm:** Perimeter $= 6 \times 5 = 30$ cm. $30 > 24$, so impossible.
- **Square of area 36 cm²:** Side $= \sqrt{36} = 6$ cm. Perimeter $= 4 \times 6 = 24$ cm. Exactly fits, so possible.
- **Right triangle with legs 6 and 8 cm:** Hypotenuse $= \sqrt{6^2 + 8^2} = \sqrt{100} = 10$ cm. Perimeter $= 6 + 8 + 10 = 24$ cm. Exactly fits, so possible.
Only the square and the triangle can be formed.
ANSWER 7: D

Problem 8:
For positive numbers, the reciprocal function $f(x) = \frac{1}{x}$ is decreasing, so the smallest number has the largest reciprocal. Compare the two smallest choices:
\[
\frac{1}{3} \approx 0.333\ldots, \qquad \frac{2}{5} = 0.4.
\]
Since $\frac{1}{3} < \frac{2}{5} < 1 < 5 < 1986$, the smallest number is $\frac{1}{3}$. Its reciprocal is $3$, which is larger than the reciprocals of all the other choices.
ANSWER 8: A

Problem 9:
First find the theoretical maximum heart rate:
\[
220 - 26 = 194 \text{ beats per minute.}
\]
The target rate is 80% of this:
\[
0.80 \times 194 = 155.2.
\]
Rounded to the nearest whole number, this is 155.
(Choice C is the maximum heart rate itself.)
ANSWER 9: B

Problem 10:
Harry computes correctly:
\[
H = 8 - (2 + 5) = 8 - 7 = 1.
\]
Terry ignores the parentheses:
\[
T = 8 - 2 + 5 = 6 + 5 = 11.
\]
The difference is:
\[
H - T = 1 - 11 = -10.
\]
ANSWER 10: A

Problem 11:
The ten numbers inside the parentheses form an arithmetic sequence: 81, 83, …, 99. The sum of an arithmetic sequence is:
\[
\frac{\text{number of terms}}{2} \times (\text{first} + \text{last}) = \frac{10}{2} \times (81 + 99) = 5 \times 180 = 900.
\]
Now multiply by the factor of 2 outside:
\[
2 \times 900 = 1800.
\]
(One can also pair them: $(81+99)=180$, $(83+97)=180$, etc., giving five pairs that each sum to 180.)
ANSWER 11: E

Problem 12:
Total surveyed = 200, and females surveyed = 96, so:
\[
\text{Males surveyed} = 200 - 96 = 104.
\]
Total listeners = 136, and females who listen = 58, so:
\[
\text{Males who listen} = 136 - 58 = 78.
\]
(We can check: total non-listeners = 64, males who don't listen = 26, so females who don't listen = 38; $58+38=96$ and $78+26=104$, which checks out.)

The percentage of males who listen is:
\[
\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%.
\]
ANSWER 12: E

Problem 1:
A mixture of 30 liters is 25% red, 30% yellow, and 45% water. Five liters of yellow tint are added. What is the percent of yellow tint in the new mixture?

**Setup:** Find the amount of yellow tint originally, add 5 liters, then find the new percentage.

**Original yellow tint:** 30% of 30 = 0.30 × 30 = 9 liters

**After adding 5 liters of yellow tint:**
- New yellow tint: 9 + 5 = 14 liters
- New total volume: 30 + 5 = 35 liters
- New percentage: 14/35 = 2/5 = 40%

**Checking wrong choices:** 25% would be if we just added the original percentages; 35% would be if we just added 5% to 30%; 45% and 50% are too high.

ANSWER 1: C

---

Problem 2:
Six windows. Georgie enters by one and leaves by a different one. How many ways?

**Setup:** Choose an entry window (6 choices), then choose a different exit window (5 choices).

**Calculation:** 6 × 5 = 30 ways

**Checking wrong choices:** 15 would be if we divided by 2 (treating order as irrelevant); 12 would be 6×2; 18 would be 6×3; 36 would be 6×6 (allowing same window).

ANSWER 2: D

---

Problem 3:
1/10 + 2/20 + 3/30 = ?

**Simplify each fraction:**
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

**Sum:** 1/10 + 1/10 + 1/10 = 3/10 = 0.3

**Checking wrong choices:** 0.1 would be just one term; 0.2 would be two terms; 0.6 would be adding numerators and denominators incorrectly.

ANSWER 3: D

---

Problem 4:
Susan had $50. She spent $12 on food and twice as much on rides. How much left?

**Setup:** Food = $12, Rides = 2 × $12 = $24

**Total spent:** $12 + $24 = $36

**Amount left:** $50 - $36 = $14

**Checking wrong choices:** 12 is just the food amount; 26 is 50-24 (forgetting food); 38 is 50-12 (forgetting rides).

ANSWER 4: B

---

Problem 5:
200,000 × 200,000 = ?

**Setup:** Express in scientific notation: 200,000 = 2 × 10⁵

**Calculation:** (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion

**Checking wrong choices:** Four hundred thousand is 4×10⁵; four million is 4×10⁶; forty thousand is 4×10⁴; four hundred million is 4×10⁸.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, totaling $1A2. Find the missing digit A.

**Setup:** The total $1A2 must be divisible by 11.

**Divisibility test for 11:** A number is divisible by 11 if the alternating sum of its digits is divisible by 11.

For 1A2: (1 + 2) - A = 3 - A must be divisible by 11.

Since A is a single digit (0-9), 3 - A can only equal 0 (the only multiple of 11 in range -6 to 3).

So A = 3, giving $132 = 11 × $12 ✓

**Checking:** With A=0: $102 ÷ 11 ≈ 9.27 (not integer); A=1: $112 ÷ 11 ≈ 10.18 (not integer); A=3: $132 ÷ 11 = 12 ✓

ANSWER 6: D

---

Problem 7:
Wire is 24 cm. Which shapes can be made?

**Regular hexagon with side 5 cm:** Perimeter = 6 × 5 = 30 cm. Need 30 cm, only have 24 cm. ❌

**Square with area 36 cm²:** Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Need 24 cm, have 24 cm. ✓

**Right triangle with legs 6 and 8 cm:** Hypotenuse = √(36 + 64) = √100 = 10 cm. Perimeter = 6 + 8 + 10 = 24 cm. Need 24 cm, have 24 cm. ✓

**Conclusion:** Can make the square and triangle, but not the hexagon.

ANSWER 7: D

---

Problem 8:
Which number has the largest reciprocal?

**Find each reciprocal:**
- A. 1/3 → reciprocal = 3
- B. 2/5 → reciprocal = 5/2 = 2.5
- C. 1 → reciprocal = 1
- D. 5 → reciprocal = 1/5 = 0.2
- E. 1986 → reciprocal ≈ 0.0005

**Key insight:** Smaller positive numbers have larger reciprocals. The smallest number listed is 1/3, so it has the largest reciprocal (3).

ANSWER 8: A

---

Problem 9:
Target heart rate = 80% of (220 - age). Find target for 26-year-old.

**Maximum heart rate:** 220 - 26 = 194 bpm

**Target heart rate:** 80% of 194 = 0.80 × 194 = 155.2 ≈ 155 bpm

**Checking wrong choices:** 134 would be about 69%; 176 would be about 91%; 194 is the max (100%); 243 is impossible.

ANSWER 9: B

---

Problem 10:
Harry calculates 8 − (2 + 5) correctly. Terry calculates 8 − 2 + 5. Find H − T.

**Harry's answer:** H = 8 − (2 + 5) = 8 − 7 = 1

**Terry's answer:** T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:** H − T = 1 − 11 = −10

**Checking wrong choices:** −6 would be if Terry got 7; 0 would be if they got the same answer; 6 or 10 would be T − H.

ANSWER 10: A

---

Problem 11:
2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99) = ?

**Setup:** This is 10 consecutive odd numbers. The average of the first and last is (81 + 99)/2 = 90.

**Sum of the 10 numbers:** 10 × 90 = 900

**Multiply by 2:** 2 × 900 = 1800

**Alternative verification:** Pair the numbers: (81+99) + (83+97) + (85+95) + (87+93) + (89+91) = 180 + 180 + 180 + 180 + 180 = 900. Then 2 × 900 = 1800.

ANSWER 11: E

---

Problem 12:
Find the percentage of males who listen to the station.

**Given information:**
- Total surveyed: 200
- Females: 96
- Females who listen: 58
- Males who don't listen: 26
- Total listeners: 136
- Total non-listeners: 64

**Find males:** Males = 200 - 96 = 104

**Find males who listen:** Males who listen = Total listeners - Females who listen = 136 - 58 = 78

**Verification:** Males who don't listen = 104 - 78 = 26 ✓ (matches given)

**Percentage of males who listen:** 78/104 × 100 = 75%

**Checking wrong choices:** 39% would be 78/200 (using total instead of males); 48% would be about 50/104; 52% would be about 54/104; 55% would be about 57/104.

ANSWER 12: E

**Problem 1**

The original mixture is 30 L with 30 % yellow tint → yellow = 0.30·30 = 9 L.  
Add 5 L of yellow → total yellow = 9 + 5 = 14 L.  
Total volume after addition = 30 + 5 = 35 L.  

Percent yellow = (14/35)·100 % = 40 %.

**Answer 1: C**

---

**Problem 2**

Choose an entry window (6 ways) and a different exit window (5 ways).  
Number of ways = 6 × 5 = 30.

**Answer 2: D**

---

**Problem 3**

\[
\frac{1}{10}=0.1,\qquad\frac{2}{20}=0.1,\qquad\frac{3}{30}=0.1
\]
Sum = 0.1 + 0.1 + 0.1 = 0.3.

**Answer 3: D**

---

**Problem 4**

Food cost = \$12.  
Rides cost = twice food = 2·12 = \$24.  
Total spent = 12 + 24 = \$36.  
Money left = 50 − 36 = \$14.

**Answer 4: B**

---

**Problem 5**

\(200{,}000 \times 200{,}000 = (2\times10^{5})^{2}=4\times10^{10}=40{,}000{,}000{,}000\).

That is 40 billion.

**Answer 5: E**

---

**Problem 6**

Total paid = \$1A2 = 100 + 10A + 2 = 102 + 10A.  
Since the 11 members each paid an integer amount, 102 + 10A must be divisible by 11.

\(102\equiv3\pmod{11}\).  
We need \(10A\equiv-3\equiv8\pmod{11}\).  
Because \(10\equiv-1\pmod{11}\), this is \(-A\equiv8\pmod{11}\) → \(A\equiv3\pmod{11}\).  
A digit, so \(A=3\).

**Answer 6: D**

---

**Problem 7**

- Hexagon: perimeter = 6·5 = 30 cm > 24 cm → impossible.  
- Square: side = √36 = 6 cm, perimeter = 4·6 = 24 cm → possible.  
- Right triangle: legs 6 cm, 8 cm, hypotenuse = √(6²+8²)=10 cm, perimeter = 6+8+10 = 24 cm → possible.

Only the square and triangle can be made.

**Answer 7: D**

---

**Problem 8**

Reciprocals:
- \(1/3 → 3\)  
- \(2/5 → 5/2 = 2.5\)  
- \(1 → 1\)  
- \(5 → 1/5 = 0.2\)  
- \(1986 → 1/1986 ≈ 0.0005\)

Largest reciprocal is 3, from \(1/3\).

**Answer 8: A**

---

**Problem 9**

Maximum heart rate = 220 − 26 = 194 bpm.  
Target = 80 % of 194 = 0.8·194 = 155.2 → rounded to 155 bpm.

**Answer 9: B**

---

**Problem 10**

Harry: \(8-(2+5)=8-7=1\).  
Terry (ignoring parentheses): \(8-2+5 = 6+5 = 11\).  
Difference \(H-T = 1-11 = -10\).

**Answer 10: A**

---

**Problem 11**

Sum of the ten terms: average = (81+99)/2 = 90, so sum = 90·10 = 900.  
Multiply by 2: \(2·900 = 1800\).

**Answer 11: E**

---

**Problem 12**

Total surveyed = 200.  
Females = 96 → Males = 200 − 96 = 104.  

Female listeners = 58 → Male listeners = 136 − 58 = 78.  

Percent of males who listen = (78/104)·100 % = (39/52)·100 % = 75 %.

**Answer 12: E**

---

**Final Answers**

ANSWER 1: C  
ANSWER 2: D  
ANSWER 3: D  
ANSWER 4: B  
ANSWER 5: E  
ANSWER 6: D  
ANSWER 7: D  
ANSWER 8: A  
ANSWER 9: B  
ANSWER 10: A  
ANSWER 11: E  
ANSWER 12: E

Problem 1:  
A 30-liter paint mixture is 25% red tint, 30% yellow tint, and 45% water. Five liters of yellow tint are added.  
Original yellow tint: $0.30 \times 30 = 9$ liters.  
After adding 5 liters: $9 + 5 = 14$ liters yellow.  
New total volume: $30 + 5 = 35$ liters.  
Percent yellow: $\frac{14}{35} = 0.40 = 40\%$.  
Tempting wrong choices: 35% (original yellow percent) or 45% (water percent).  
ANSWER 1: D  

Problem 2:  
Georgie enters through one of six windows and exits through a different one.  
Entry choices: 6. Exit choices: 5 (excluding entry window).  
Total ways: $6 \times 5 = 30$.  
Tempting wrong choices: 15 (if order didn’t matter) or 36 (if same window allowed).  
ANSWER 2: D  

Problem 3:  
$\frac{1}{10} + \frac{2}{20} + \frac{3}{30} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3$.  
Tempting wrong choices: 0.1 (one term) or 0.2 (two terms).  
ANSWER 3: D  

Problem 4:  
Susan starts with \$50.  
Food: \$12.  
Rides: $2 \times 12 = \$24$.  
Total spent: $12 + 24 = \$36$.  
Left: $50 - 36 = \$14$.  
Tempting wrong choices: \$26 (if spent \$24 total) or \$38 (if spent \$12 total).  
ANSWER 4: B  

Problem 5:  
$200,000 \times 200,000 = 40,000,000,000 = 40$ billion.  
Tempting wrong choices: 400,000 (added zeros) or 4 million (multiplied digits).  
ANSWER 5: E  

Problem 6:  
Total paid: \$1A2, an integer divisible by 11.  
$1A2 \mod 11 = 0$.  
$1 + 2 - A \equiv 0 \mod 11 \Rightarrow 3 - A \equiv 0 \mod 11 \Rightarrow A = 3$.  
Tempting wrong choices: 0, 1, 2, or 4 (incorrect divisibility).  
ANSWER 6: E (Wait, let's recheck: 1A2 must be divisible by 11. 132 ÷ 11 = 12, so A=3. But 132 is 11×12, so correct. However, the answer choices are A=0,B=1,C=2,D=3,E=4. So D is 3. So correct answer is D, not E. I made a mistake in the initial answer. Correcting:)  
**Correction**: $1A2 = 132$ when $A=3$, and $132 \div 11 = 12$, so $A=3$.  
ANSWER 6: D  

Problem 7:  
Wire length: 24 cm.  
- Regular hexagon with side 5 cm: Perimeter $6 \times 5 = 30$ cm (not 24).  
- Square with area 36 cm²: Side $6$ cm, perimeter $24$ cm.  
- Right triangle with legs 6 and 8 cm: Hypotenuse 10 cm, perimeter $6+8+10=24$ cm.  
Can make square and triangle.  
Tempting wrong choices: Hexagon (incorrect perimeter).  
ANSWER 7: D  

Problem 8:  
Largest reciprocal means smallest original number.  
$\frac{1}{3} \approx 0.333$, $\frac{2}{5} = 0.4$, $1$, $5$, $1986$.  
Smallest: $\frac{1}{3}$. Reciprocal: 3.  
Others: 2.5, 1, 0.2, 0.0005.  
Tempting wrong choices: $\frac{2}{5}$ (larger than $\frac{1}{3}$).  
ANSWER 8: A  

Problem 9:  
Maximum heart rate: $220 - 26 = 194$.  
Target: $0.80 \times 194 = 155.2 \approx 155$.  
Tempting wrong choices: 134 (70%), 176 (90%), or 243 (incorrect max).  
ANSWER 9: B  

Problem 10:  
Harry: $8 - (2 + 5) = 1$.  
Terry: $8 - 2 + 5 = 11$.  
Difference: $1 - 11 = -10$.  
Tempting wrong choices: 10 (absolute difference) or 0 (if parentheses ignored by both).  
ANSWER 10: A  

Problem 11:  
Sum: $81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99$.  
Average: 90, 10 terms: $90 \times 10 = 900$.  
$2 \times 900 = 1800$.  
Tempting wrong choices: 1600 (min+max=180, times 5=900, times 2=1800, but if miscount terms).  
ANSWER 11: E  

Problem 12:  
Total surveyed: 200.  
Females: 96, so males: $200 - 96 = 104$.  
Total listeners: 136.  
Female listeners: 58, so male listeners: $136 - 58 = 78$.  
Percent male listeners: $\frac{78}{104} = 0.75 = 75\%$.  
Tempting wrong choices: 39 (male non-listeners percent) or 52 (incorrect calculation).  
ANSWER 12: E

### Problem 1:
What is the percentage of yellow tint in the new 35-liter paint mixture, after adding 5 liters of yellow tint to an original 30-liter mixture that was 30% yellow?
First, calculate the original volume of yellow tint: 30% of 30 liters = \(0.3 \times 30 = 9\) liters. Add the 5 new liters of yellow tint to get \(9 + 5 = 14\) liters of yellow in the new mixture. The total new volume of the mixture is \(30 + 5 = 35\) liters. The percentage of yellow is \(\frac{14}{35} = 0.4 = 40\%\). Tempting wrong answers come from forgetting to update the total mixture volume when calculating the new percentage, or miscalculating the original yellow volume.
ANSWER 1: C

---

### Problem 2:
How many ways can Georgie enter through one window and leave through a different window, out of 6 total windows?
Georgie has 6 choices for an entry window. Since he must leave through a different window, he has 5 remaining choices for an exit window. The total number of ways is \(6 \times 5 = 30\). A common wrong answer is 15, which incorrectly treats entry/exit as unordered pairs, when order matters (entering window A and leaving B is different from entering B and leaving A).
ANSWER 2: D

---

### Problem 3:
Simplify \(\frac{1}{10} + \frac{2}{20} + \frac{3}{30}\):
Simplify each fraction first: \(\frac{2}{20} = \frac{1}{10}\) and \(\frac{3}{30} = \frac{1}{10}\). Add them together: \(\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3\). A common wrong answer is 0.1, which incorrectly adds numerators and denominators directly instead of simplifying first.
ANSWER 3: D

---

### Problem 4:
How much money does Susan have left after spending $12 on food and twice that amount on rides, starting with $50?
First calculate the amount spent on rides: twice $12 is \(2 \times 12 = 24\) dollars. Total spent is \(12 + 24 = 36\) dollars. The amount left is \(50 - 36 = 14\) dollars. A common wrong answer is 26, which incorrectly uses $12 instead of $24 for ride costs.
ANSWER 4: B

---

### Problem 5:
Calculate \(200,000 \times 200,000\):
Rewrite 200,000 as \(2 \times 10^5\). Multiplying two copies gives \((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\), which equals 40,000,000,000, or forty billion. Wrong answers come from miscounting the number of zeros in the product.
ANSWER 5: E

---

### Problem 6:
The 3-digit total cost \(1A2\) dollars is divisible by 11 (11 members paid the same integer amount). What is the missing digit A?
The divisibility rule for 11 states that alternating sum of digits must be divisible by 11. For 1A2, the alternating sum is \(1 - A + 2 = 3 - A\). The only value of A (0-9) that makes this sum divisible by 11 is A=3, since \(3-3=0\), which is divisible by 11. Verify: \(132 \div 11 = 12\), a valid integer.
ANSWER 6: D

---

### Problem 7:
Haruki has a 24cm wire. Which of the three shapes can he form? Calculate the perimeter of each shape:
1. Regular hexagon with side 5cm: perimeter = \(6 \times 5 = 30\)cm, which is longer than 24cm, so impossible.
2. Square with area 36cm²: side length = \(\sqrt{36}=6\)cm, perimeter = \(4 \times 6 = 24\)cm, which fits.
3. Right triangle with legs 6cm and 8cm: hypotenuse = 10cm (6-8-10 Pythagorean triple), perimeter = \(6+8+10=24\)cm, which fits.
Only the square and triangle can be formed.
ANSWER 7: D

---

### Problem 8:
Which number has the largest reciprocal? The reciprocal of \(x\) is \(\frac{1}{x}\), so smaller values of \(x\) produce larger reciprocals. List the values of all options:
A. \(\frac{1}{3} \approx 0.333\), B. \(\frac{2}{5}=0.4\), C.1, D.5, E.1986. The smallest value is \(\frac{1}{3}\), whose reciprocal is 3, the largest of all options.
ANSWER 8: A

---

### Problem 9:
Calculate the target heart rate for a 26-year-old athlete: target is 80% of max heart rate, where max = 220 - age.
First calculate max heart rate: \(220 - 26 = 194\) bpm. 80% of 194 is \(0.8 \times 194 = 155.2\), which rounds to 155 bpm. A common wrong answer is 176, which uses 220 instead of the correct max heart rate of 194.
ANSWER 9: B

---

### Problem 10:
Calculate \(H-T\), where H is Harry's correct answer to \(8-(2+5)\), and T is Terry's incorrect answer of \(8-2+5\).
Harry's answer: \(8 - 7 = 1 = H\). Terry's answer: \(6 + 5 = 11 = T\). The difference \(H-T = 1 - 11 = -10\). A common wrong answer is 10, which calculates \(T-H\) instead of \(H-T\).
ANSWER 10: A

---

### Problem 11:
Simplify \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\):
Pair the terms inside the parentheses to simplify addition: \((81+99)+(83+97)+(85+95)+(87+93)+(89+91) = 5 \times 180 = 900\). Multiply by 2: \(2 \times 900 = 1800\).
ANSWER 11: E

---

### Problem 12:
What percentage of males surveyed listen to the radio station? First calculate total number of males: total surveyed = 200, females = 96, so males = \(200 - 96 = 104\). Males who do not listen = 26, so males who listen = \(104 - 26 = 78\). The percentage of males who listen is \(\frac{78}{104} = 0.75 = 75\%\). A common wrong answer is 39%, which divides male listeners by the total surveyed population instead of the total male population.
ANSWER 12: E

### Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.
1.  Calculate original yellow tint volume: 25%? No wait, original is 30% yellow. So 30% of 30 L = $0.3 \times 30 = 9$ liters of yellow.
2.  Add 5 liters of yellow: total yellow = $9 + 5 = 14$ liters.
3.  Total new mixture volume: $30 + 5 = 35$ liters.
4.  Percent yellow: $\frac{14}{35} \times 100 = 40\%$.
ANSWER 1: C
---
### Problem 2:
We count the number of ways to choose an entry window and a different exit window from 6 windows.
1.  Entry window: 6 possible choices.
2.  Exit window: cannot be the entry window, so 5 remaining choices.
3.  Total ways: $6 \times 5 = 30$ (order matters because entry and exit are distinct actions).
ANSWER 2: D
---
### Problem 3:
Simplify and add the fractions:
1.  $\frac{1}{10} = 0.1$, $\frac{2}{20} = \frac{1}{10} = 0.1$, $\frac{3}{30} = \frac{1}{10} = 0.1$.
2.  Sum: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D
---
### Problem 4:
Calculate remaining money after expenses:
1.  Total initial money: $50.
2.  Money spent on rides: twice the food cost = $2 \times 12 = 24$ dollars.
3.  Total spent: $12 + 24 = 36$ dollars.
4.  Remaining: $50 - 36 = 14$ dollars.
ANSWER 4: B
---
### Problem 5:
Compute $200,000 \times 200,000$:
1.  Rewrite as $(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
2.  $10^{10} = 10,000,000,000$, so $4 \times 10^{10} = 40,000,000,000 = 40$ billion.
ANSWER 5: E
---
### Problem 6:
The total payment $1A2$ (a 3-digit number) must be divisible by 11 (since 11 people paid equal integer amounts).
1.  Use the divisibility rule for 11: the alternating sum of digits is divisible by 11. For $1A2$, this is $1 - A + 2 = 3 - A$.
2.  $A$ is a digit (0-9), so $3 - A$ must equal 0 (the only multiple of 11 in range: $3 - A = 11$ gives $A=-8$, $3-A=-11$ gives $A=14$, both invalid).
3.  $3 - A = 0 \implies A=3$. Verify: $132 \div 11 = 12$, which is an integer.
ANSWER 6: D
---
### Problem 7:
Check if the 24 cm wire can form each shape (perimeter of the shape must equal 24 cm):
1.  Regular hexagon (side 5 cm): Perimeter = $6 \times 5 = 30$ cm > 24 cm → cannot make.
2.  Square (area 36 cm²): Side length = $\sqrt{36} = 6$ cm. Perimeter = $4 \times 6 = 24$ cm → can make.
3.  Right triangle (legs 6, 8 cm): Hypotenuse = $\sqrt{6^2 + 8^2} = 10$ cm. Perimeter = $6 + 8 + 10 = 24$ cm → can make.
Only square and triangle can be made.
ANSWER 7: D
---
### Problem 8:
The reciprocal of a positive number $x$ is $\frac{1}{x}$; smaller $x$ gives larger reciprocal.
1.  List values: $\frac{1}{3} \approx 0.33$, $\frac{2}{5}=0.4$, $1$, $5$, $1986$.
2.  Smallest value is $\frac{1}{3}$, whose reciprocal is $3$ (largest among all reciprocals: $\frac{5}{2}=2.5$, $1$, $0.2$, $\frac{1}{1986} \approx 0.0005$).
ANSWER 8: A
---
### Problem 9:
Calculate target heart rate for a 26-year-old:
1.  Maximum heart rate: $220 - 26 = 194$ bpm.
2.  Target heart rate: 80% of 194 = $0.8 \times 194 = 155.2$, which rounds to 155.
ANSWER 9: B
---
### Problem 10:
Compute $H$ (Harry's correct answer) and $T$ (Terry's incorrect answer):
1.  Harry: $8 - (2 + 5) = 8 - 7 = 1$, so $H=1$.
2.  Terry (ignores parentheses, left-to-right): $8 - 2 + 5 = 6 + 5 = 11$, so $T=11$.
3.  Difference: $H - T = 1 - 11 = -10$.
ANSWER 10: A
---
### Problem 11:
Calculate $2 \times (81 + 83 + ... + 99)$:
1.  The sum inside is 10 consecutive odd numbers. Average of first and last term: $\frac{81 + 99}{2} = 90$.
2.  Sum of the 10 terms: $90 \times 10 = 900$.
3.  Multiply by 2: $2 \times 900 = 1800$.
ANSWER 11: E
---
### Problem 12:
Find the percentage of males who listen to the station:
1.  Total males surveyed: $200 - 96 = 104$.
2.  Males who do not listen: 26, so males who listen: $104 - 26 = 78$.
3.  Verify with total listeners: Total listeners = 136, female listeners = 58, so male listeners = $136 - 58 = 78$ (matches).
4.  Percentage: $\frac{78}{104} \times 100 = 75\%$.
ANSWER 12: E

Problem 1:
Original yellow = 30% of 30 = 9 L. Add 5 → 14 L yellow. New total = 35 L. Percent = 14/35 = 40%.

ANSWER 1: C

Problem 2:
Enter: 6 choices, leave: 5 different choices. 6×5 = 30.

ANSWER 2: D

Problem 3:
Each fraction = 0.1. Sum = 0.3.

ANSWER 3: D

Problem 4:
Food 12, rides twice = 24. Spent 36. Left = 50 − 36 = 14.

ANSWER 4: B

Problem 5:
200,000 × 200,000 = 40,000,000,000 = forty billion.

ANSWER 5: E

Problem 6:
1A2 divisible by 11. Alternating sum: 1 − A + 2 = 3 − A must be divisible by 11 → A = 3 (gives 0). Check 132/11 = 12. Yes.

ANSWER 6: D

Problem 7:
Hexagon: 6×5 = 30 > 24, no. Square area 36 → side 6, perimeter 24, yes. Triangle legs 6,8 → hypotenuse 10, perimeter 24, yes. Square and triangle only.

ANSWER 7: D

Problem 8:
Largest reciprocal = smallest number. Smallest is 1/3.

ANSWER 8: A

Problem 9:
Max = 220 − 26 = 194. Target = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

Problem 10:
H = 8 − 7 = 1. T = 8 − 2 + 5 = 11. H − T = 1 − 11 = −10.

ANSWER 10: A

Problem 11:
Sum of 81..99 odd numbers (10 terms): average (81+99)/2 = 90, ×10 = 900. ×2 = 1800.

ANSWER 11: E

Problem 12:
Males total = 200 − 96 = 104. Males who don't listen = 26 → males who listen = 104 − 26 = 78. Percent = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem systematically.

---

**Problem 1:**
Original mixture: 30 liters, 30% yellow = 9 liters yellow. Add 5 liters yellow → new yellow = 14 liters, new total = 35 liters.
Percent yellow = 14/35 = 40%.

ANSWER 1: C

---

**Problem 2:**
Georgie can enter through any of 6 windows, then leave through any of the remaining 5 windows.
6 × 5 = 30.

ANSWER 2: D

---

**Problem 3:**
1/10 = 0.1, 2/20 = 0.1, 3/30 = 0.1. Sum = 0.3.

ANSWER 3: D

---

**Problem 4:**
Food: $12. Rides: 2 × $12 = $24. Total spent: $12 + $24 = $36. Left: $50 − $36 = $14.

ANSWER 4: B

---

**Problem 5:**
200,000 × 200,000 = (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion.

ANSWER 5: E

---

**Problem 6:**
The number 1A2 must be divisible by 11. 1A2 ÷ 11 must be an integer.
11 × 10 = 110, 11 × 11 = 121 (no), 11 × 12 = 132 ✓ → A = 3.
Check: 132/11 = 12 ✓, each paid $12.

ANSWER 6: D

---

**Problem 7:**
Wire = 24 cm.
- Regular hexagon, side 5: perimeter = 6 × 5 = 30 cm > 24. ✗
- Square, area 36: side = 6, perimeter = 24 cm ✓
- Right triangle, legs 6 and 8: hypotenuse = 10, perimeter = 6+8+10 = 24 cm ✓

ANSWER 7: D

---

**Problem 8:**
The reciprocal of x is 1/x. The largest reciprocal comes from the smallest positive number.
Among 1/3, 2/5, 1, 5, 1986 — the smallest is 1/3, and its reciprocal is 3, which is the largest.

ANSWER 8: A

---

**Problem 9:**
Maximum heart rate = 220 − 26 = 194. Target = 80% × 194 = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

---

**Problem 10:**
Harry: 8 − (2 + 5) = 8 − 7 = 1, so H = 1.
Terry: 8 − 2 + 5 = 11, so T = 11.
H − T = 1 − 11 = −10.

ANSWER 10: A

---

**Problem 11:**
The ten odd numbers from 81 to 99: sum = (81+99)/2 × 10 = 90 × 10 = 900.
2 × 900 = 1800.

ANSWER 11: E

---

**Problem 12:**
Total surveyed = 200. Females = 96, so Males = 104.
Total non-listeners = 64. Males who don't listen = 26, so females who don't listen = 64 − 26 = 38.
Males who listen = 104 − 26 = 78.
Percentage = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem step-by-step.

## Problem 1:
**What is asked:** Find the percentage of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.

**Original mixture:**
- Yellow tint: 30% of 30 = 0.30 × 30 = 9 liters
- Total volume: 30 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total volume: 30 + 5 = 35 liters

**Percentage of yellow tint:**
$$\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%$$

ANSWER 1: C

## Problem 2:
**What is asked:** In how many ways can the ghost enter by one window and leave by a different window?

**Solution:**
- Choices for entering: 6 windows
- Choices for leaving: 5 windows (must be different from entry)
- Total ways: 6 × 5 = 30

ANSWER 2: D

## Problem 3:
**What is asked:** Calculate 1/10 + 2/20 + 3/30

**Solution:**
Simplify each fraction:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

ANSWER 3: D

## Problem 4:
**What is asked:** How much money does Susan have left?

**Solution:**
- Started with: $50
- Spent on food: $12
- Spent on rides: 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Left: $50 - $36 = $14

ANSWER 4: B

## Problem 5:
**What is asked:** Calculate 200,000 × 200,000

**Solution:**
$$200,000 × 200,000 = (2 × 10^5) × (2 × 10^5) = 4 × 10^{10} = 40,000,000,000 = 40 \text{ billion}$$

ANSWER 5: E

## Problem 6:
**What is asked:** Find the missing digit A in the 3-digit number 1A2, where 11 club members each paid an equal integer amount that totals 1A2.

**Solution:**
The number 1A2 must be divisible by 11.

Testing divisibility by 11 (alternating sum of digits):
- 1 - A + 2 = 0 (mod 11)
- 3 - A ≡ 0 (mod 11)
- A = 3

Check: 132 ÷ 11 = 12 ✓

ANSWER 6: D

## Problem 7:
**What is asked:** Which shapes can Haruki make with 24 cm of wire?

**Solution:**
Test each shape:

**Regular hexagon with side 5 cm:**
- Perimeter: 6 × 5 = 30 cm
- Need 30 cm, have 24 cm ✗

**Square with area 36 cm²:**
- Side length: √36 = 6 cm
- Perimeter: 4 × 6 = 24 cm ✓

**Right triangle with legs 6 and 8 cm:**
- Hypotenuse: √(6² + 8²) = √(36 + 64) = √100 = 10 cm
- Perimeter: 6 + 8 + 10 = 24 cm ✓

Only square and triangle can be made.

ANSWER 7: D

## Problem 8:
**What is asked:** Which number has the largest reciprocal?

**Solution:**
Find reciprocals:
- 1/(1/3) = 3
- 1/(2/5) = 5/2 = 2.5
- 1/1 = 1
- 1/5 = 0.2
- 1/1986 ≈ 0.0005

The largest reciprocal is 5 (from the number 1/5).

ANSWER 8: D

## Problem 9:
**What is asked:** Find the target heart rate for a 26-year-old athlete.

**Solution:**
- Maximum heart rate: 220 - 26 = 194 beats per minute
- Target heart rate: 80% of 194 = 0.80 × 194 = 155.2 ≈ 155

ANSWER 9: B

## Problem 10:
**What is asked:** Find H - T where H is Harry's answer and T is Terry's answer.

**Solution:**
**Harry's calculation (correct):**
$$H = 8 - (2 + 5) = 8 - 7 = 1$$

**Terry's calculation (ignoring parentheses):**
$$T = 8 - 2 + 5 = 6 + 5 = 11$$

**Difference:**
$$H - T = 1 - 11 = -10$$

ANSWER 10: A

## Problem 11:
**What is asked:** Calculate 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

**Solution:**
This is an arithmetic sequence with 10 terms. Sum = n(first + last)/2
$$81 + 83 + ... + 99 = \frac{10(81 + 99)}{2} = \frac{10 × 180}{2} = 900$$

$$2 × 900 = 1800$$

ANSWER 11: E

## Problem 12:
**What is asked:** What percentage of males surveyed listen to the station?

**Solution:**
- Total surveyed: 200
- Females: 96
- Males: 200 - 96 = 104

- Total non-listeners: 64
- Males who don't listen: 26
- Males who listen: 104 - 26 = 78

**Percentage of males who listen:**
$$\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%$$

ANSWER 12: E

Problem 1:

The original mixture is 30 liters total, with 30% yellow tint.

So the amount of yellow tint at first is:
- \(30\%\) of \(30 = 0.30 \times 30 = 9\) liters

Then 5 liters of yellow tint are added, so the new amount of yellow tint is:
- \(9 + 5 = 14\) liters

The new total mixture is:
- \(30 + 5 = 35\) liters

So the percent of yellow tint is:
- \(\frac{14}{35} = \frac{2}{5} = 0.4 = 40\%\)

Answer choice C matches 40%.

ANSWER 1: C

Problem 2:

Georgie can:
- choose 1 of the 6 windows to enter
- then choose a different window to leave

For the entry, there are 6 choices.

For the exit, there are 5 remaining choices, since it must be a different window.

So the total number of ways is:
- \(6 \times 5 = 30\)

ANSWER 2: D

Problem 3:

Compute each fraction:
- \(\frac{1}{10} = 0.1\)
- \(\frac{2}{20} = \frac{1}{10} = 0.1\)
- \(\frac{3}{30} = \frac{1}{10} = 0.1\)

Add them:
- \(0.1 + 0.1 + 0.1 = 0.3\)

So the value is \(0.3\).

ANSWER 3: D

Problem 4:

Susan had \$50.

She spent:
- \$12 on food
- twice as much on rides, so \(2 \times 12 = 24\)

Total spent:
- \(12 + 24 = 36\)

Money left:
- \(50 - 36 = 14\)

ANSWER 4: B

Problem 5:

Two hundred thousand is:
- \(200{,}000 = 2 \times 10^5\)

Multiply:
- \(200{,}000 \times 200{,}000 = (2 \times 10^5)(2 \times 10^5) = 4 \times 10^{10}\)

And \(4 \times 10^{10} = 40{,}000{,}000{,}000\), which is forty billion.

ANSWER 5: E

Problem 6:

The total is written as \(1A2\), and 11 members each paid the same integer amount.

So \(1A2\) must be divisible by 11.

Test the answer choices:
- 102 ÷ 11 is not an integer
- 112 ÷ 11 is not an integer
- 122 ÷ 11 = 11 remainder 1
- 132 ÷ 11 = 12 exactly
- 142 ÷ 11 is not an integer

So the number is 132, which means \(A = 3\).

ANSWER 6: D

Problem 7:

We check each shape.

### Hexagon
A regular hexagon has 6 equal sides.
If each side is 5 cm, perimeter is:
- \(6 \times 5 = 30\) cm

The wire is only 24 cm, so he cannot make the hexagon.

### Square
A square with area 36 has side length:
- \(\sqrt{36} = 6\) cm

Perimeter:
- \(4 \times 6 = 24\) cm

So he can make the square exactly.

### Right triangle
The legs are 6 and 8, so the hypotenuse is:
- \(\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)

Perimeter:
- \(6 + 8 + 10 = 24\) cm

So he can make the triangle exactly.

Therefore he can make the square and the triangle only.

ANSWER 7: D

Problem 8:

The reciprocal is largest when the original number is smallest, assuming positive numbers.

Compare:
- \(\frac13 \approx 0.333\)
- \(\frac25 = 0.4\)
- \(1\)
- \(5\)
- \(1986\)

The smallest number is \(\frac13\), so it has the largest reciprocal.

Its reciprocal is 3, which is larger than the reciprocals of all the others.

ANSWER 8: A

Problem 9:

Maximum heart rate:
- \(220 - 26 = 194\)

Target heart rate is 80% of that:
- \(0.8 \times 194 = 155.2\)

To the nearest whole number, that is 155.

ANSWER 9: B

Problem 10:

First, Harry uses the parentheses:
- \(8 - (2 + 5) = 8 - 7 = 1\)

So \(H = 1\).

Terry ignores parentheses:
- \(8 - 2 + 5 = 6 + 5 = 11\)

So \(T = 11\).

Difference:
- \(H - T = 1 - 11 = -10\)

ANSWER 10: A

Problem 11:

We need to evaluate:
- \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

These are 10 odd numbers from 81 to 99.

Instead of adding one by one, use the average:
- average of 81 and 99 is \( \frac{81+99}{2} = 90\)

Since there are 10 numbers, their sum is:
- \(10 \times 90 = 900\)

Now multiply by 2:
- \(2 \times 900 = 1800\)

ANSWER 11: E

Problem 12:

We want the percentage of males surveyed who listen.

From the table:
- Total surveyed = 200
- Females = 96
- So males = \(200 - 96 = 104\)

Males who don’t listen = 26, so males who do listen:
- \(104 - 26 = 78\)

Now find the percent of males who listen:
- \(\frac{78}{104} = \frac{3}{4} = 75\%\)

ANSWER 12: E

Problem 1:  
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?  

- Original yellow tint amount = 30 × 0.30 = 9 liters.  
- Add 5 liters of yellow tint → new yellow amount = 9 + 5 = 14 liters.  
- New total volume = 30 + 5 = 35 liters.  
- New percent yellow = (14 / 35) × 100% = (2/5) × 100% = 40%.  

So the correct choice is 40.  

ANSWER 1: C  

---

Problem 2:  
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?  

- Choose an entry window: 6 choices.  
- Choose a different exit window: 5 choices (cannot be the same as entry).  
- Total ways = 6 × 5 = 30.  

ANSWER 2: D  

---

Problem 3:  
Compute: \( \frac{1}{10} + \frac{2}{20} + \frac{3}{30} \).  

Convert each to a common denominator, 60:  
- \( \frac{1}{10} = \frac{6}{60} \)  
- \( \frac{2}{20} = \frac{2}{20} = \frac{6}{60} \) (since multiply top and bottom by 3)  
- \( \frac{3}{30} = \frac{3}{30} = \frac{6}{60} \) (since multiply top and bottom by 2)  

Add:  
\( \frac{6}{60} + \frac{6}{60} + \frac{6}{60} = \frac{18}{60} = \frac{3}{10} = 0.3 \)

So the value is .3.  

ANSWER 3: D  

---

Problem 4:  
Susan had $50 to spend. She spent $12 on food and twice as much on rides. How much left?  

- Food = 12  
- Rides = twice 12 = 24  
- Total spent = 12 + 24 = 36  
- Money left = 50 − 36 = 14  

ANSWER 4: B  

---

Problem 5:  
Two hundred thousand times two hundred thousand equals ?  

\(200{,}000 \times 200{,}000 = (2\times 10^5)(2\times 10^5) = 4 \times 10^{10}\).  
- \(10^{10} = 10{,}000{,}000{,}000\)  
- \(4 \times 10^{10} = 40{,}000{,}000{,}000 = 40\) billion  

ANSWER 5: E  

---

Problem 6:  
Eleven members each paid the same integer amount for a guest speaker. In all, they paid $1A2 (a 3-digit number). Find missing digit A.  

- Total = \(1A2\) means:  
  \(1A2 = 100 + 10A + 2 = 102 + 10A\).  
- This total is divided equally among 11 people:  
  So \(102 + 10A\) must be divisible by 11.  

Check \(102 + 10A \mod 11\):  
- \(102 \div 11 = 9\) remainder 3, so \(102 \equiv 3 \pmod{11}\).  
- \(10A \equiv -A \pmod{11}\) (since 10 ≡ −1 mod 11).  

So:  
\(102 + 10A \equiv 3 - A \equiv 0 \pmod{11}\)  
Thus \(A \equiv 3 \pmod{11}\).  
Since \(A\) is a digit 0–9, \(A=3\).  

ANSWER 6: D  

---

Problem 7:  
Wire length = 24 cm. Which shapes can be made (using all 24 cm of wire)?  

We need the perimeter (total boundary length) of each shape:

**(1) Regular hexagon with side length 5 cm**  
Perimeter = 6 × 5 = 30 cm → too long. Cannot.

**(2) Square with area 36 cm²**  
Side length = √36 = 6 cm  
Perimeter = 4 × 6 = 24 cm → fits. Can.

**(3) Right triangle with legs 6 and 8 cm**  
Right triangle sides are 6 and 8 legs, and hypotenuse:  
\( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)  
Perimeter = 6 + 8 + 10 = 24 cm → fits. Can.

So the shapes possible: **square and triangle only**.  

ANSWER 7: D  

---

Problem 8:  
Which number has the largest reciprocal? Recall: the reciprocal of \(x\) is \(1/x\).  
We compare \(1/3, 2/5, 1, 1/5, 1/1986\).  

Compute roughly:  
- \(1/3 \approx 0.333\)  
- \(2/5 = 0.4\)  
- \(1 = 1\)  
- \(1/5 = 0.2\)  
- \(1/1986\) is tiny  

Largest is 1 (reciprocal of 1).  

ANSWER 8: C  

---

Problem 9:  
Target heart rate = 80% of (maximum heart rate).  
Maximum heart rate = 220 − age.  
Age = 26.

- Maximum = 220 − 26 = 194  
- Target = 0.80 × 194 = 155.2  
Nearest whole number = 155  

ANSWER 9: B  

---

Problem 10:  
Expression: \(8 - (2 + 5)\)  
- Harry computes correctly:  
  \(2 + 5 = 7\), so \(8 - 7 = 1\).  
- Terry ignores parentheses:  
  \(8 - 2 + 5 = 6 + 5 = 11\).  

Difference \(H - T = 1 - 11 = -10\).  

ANSWER 10: A  

---

Problem 11:  
Compute:  
\(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

Notice these are all odd numbers from 81 to 99 stepping by 2.  
There are 10 terms.

Pair from ends:  
- 81 + 99 = 180  
- 83 + 97 = 180  
- 85 + 95 = 180  
- 87 + 93 = 180  
- 89 + 91 = 180  

There are 5 pairs, each sum 180:  
Sum = 5 × 180 = 900  

Now multiply by 2:  
2 × 900 = 1800  

ANSWER 11: E  

---

Problem 12:  
Total surveyed = 200  
Females = 96  
Females who listen = 58  
Males who don’t listen = 26  
Total listeners = 136  

We want: % of males who listen.

Step 1: Find males total.  
Males = 200 − 96 = 104

Step 2: Find males who don’t listen is given = 26.  
So males who listen = 104 − 26 = 78

Step 3: Convert to percent of males who listen:  
Percent = (78 / 104) × 100%  
Simplify 78/104 = 3/4 (divide by 26):  
So percent = (3/4) × 100% = 75%

ANSWER 12: E

Problem 1:
The original mixture is 30 liters. The amount of yellow tint is 30% of 30 liters: $0.30 \times 30 = 9$ liters.
Five liters of yellow tint are added, so the new amount of yellow tint is $9 + 5 = 14$ liters.
The new total volume of the mixture is $30 + 5 = 35$ liters.
The percentage of yellow tint is $\frac{14}{35} \times 100\%$.
Simplifying the fraction: $\frac{14}{35} = \frac{2}{5} = 0.4$.
$0.4 \times 100\% = 40\%$.
ANSWER 1: C

Problem 2:
Georgie has 6 choices for the window he enters. Since he must leave by a different window, he has $6 - 1 = 5$ choices for the window he leaves by.
The total number of ways is $6 \times 5 = 30$.
ANSWER 2: D

Problem 3:
Simplify each fraction:
$1/10 = 0.1$
$2/20 = 1/10 = 0.1$
$3/30 = 1/10 = 0.1$
Summing them: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides, which is $2 \times 12 = 24$.
Total spent: $12 + 24 = 36$.
Amount left: $50 - 36 = 14$.
ANSWER 4: B

Problem 5:
Two hundred thousand is $200,000$ (or $2 \times 10^5$).
$(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
$4 \times 10^{10} = 40,000,000,000$, which is forty billion.
ANSWER 5: E

Problem 6:
The total amount $1A2$ must be divisible by 11.
A number is divisible by 11 if the alternating sum of its digits is divisible by 11.
For $1A2$, the alternating sum is $1 - A + 2 = 3 - A$.
For this to be 0 (the only multiple of 11 possible for a single digit $A$), $3 - A = 0$, so $A = 3$.
Check: $132 / 11 = 12$. This works.
ANSWER 6: D

Problem 7:
Hexagon: Perimeter of a regular hexagon with side 5 is $6 \times 5 = 30$ cm. (Cannot make)
Square: Area is 36, so side is $\sqrt{36} = 6$. Perimeter is $4 \times 6 = 24$ cm. (Can make)
Triangle: Right triangle with legs 6 and 8. Hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. Perimeter is $6 + 8 + 10 = 24$ cm. (Can make)
ANSWER 7: D

Problem 8:
Reciprocals:
A: $1/(1/3) = 3$
B: $1/(2/5) = 2.5$
C: $1/1 = 1$
D: $1/5 = 0.2$
E: $1/1986 \approx 0.0005$
The largest is 3.
ANSWER 8: A

Problem 9:
Max heart rate = $220 - 26 = 194$.
Target heart rate = $80\%$ of $194 = 0.8 \times 194 = 155.2$.
To the nearest whole number, this is 155.
ANSWER 9: B

Problem 10:
$H = 8 - (2 + 5) = 8 - 7 = 1$.
$T = 8 - 2 + 5 = 6 + 5 = 11$.
$H - T = 1 - 11 = -10$.
ANSWER 10: A

Problem 11:
Sum inside parentheses: $81+83+85+87+89+91+93+95+97+99$.
This is an arithmetic progression with 10 terms.
Sum = $\frac{n}{2}(\text{first} + \text{last}) = \frac{10}{2}(81 + 99) = 5 \times 180 = 900$.
The expression is $2 \times 900 = 1800$.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96.
Males = $200 - 96 = 104$.
Males who don't listen = 26.
Males who listen = $104 - 26 = 78$.
Percentage of males who listen = $(78 / 104) \times 100\%$.
$\frac{78}{104} = \frac{39}{52} = \frac{3}{4} = 0.75 = 75\%$.
ANSWER 12: E

Problem 1: The original mixture has volume 30 L, of which 30% is yellow tint, so there are \(0.3 \times 30 = 9\) L of yellow tint. Adding 5 L of yellow tint produces a new total of 14 L of yellow tint and a new mixture volume of 35 L. The new percentage of yellow tint is therefore \((14/35) \times 100\% = 40\%\).

ANSWER 1: C

Problem 2: There are 6 choices for the entry window. For each of those choices there are then exactly 5 remaining choices for a different exit window, giving a total of \(6 \times 5 = 30\) ordered pairs of distinct windows.

ANSWER 2: D

Problem 3: Rewrite each term with a common denominator of 10:  
\(1/10 = 0.1\),  
\(2/20 = 1/10 = 0.1\),  
\(3/30 = 1/10 = 0.1\).  
Their sum is \(0.1 + 0.1 + 0.1 = 0.3\).

ANSWER 3: D

Problem 4: Twice the food cost is \(2 \times 12 = 24\) dollars spent on rides. The total spent is therefore \(12 + 24 = 36\) dollars, leaving \(50 - 36 = 14\) dollars.

ANSWER 4: B

Problem 5: Write the factors in scientific notation:  
\(200000 = 2 \times 10^5\), so  
\((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\),  
which is forty billion.

ANSWER 5: E

Problem 6: The three-digit total \(1A2\) must be divisible by 11. The alternating-sum divisibility rule requires that \(1 - A + 2 = 3 - A\) be a multiple of 11. The only admissible digit \(A\) (0-9) that satisfies the condition is \(A = 3\), giving the total 132, which is exactly \(11 \times 12\).

ANSWER 6: D

Problem 7: A regular hexagon with side 5 cm has perimeter 30 cm, which exceeds the 24 cm length of wire, so it is impossible. A square of area 36 cm² has side length 6 cm and perimeter 24 cm, which matches the wire length. A right triangle with legs 6 cm and 8 cm has hypotenuse 10 cm, so its perimeter is \(6 + 8 + 10 = 24\) cm, which also matches. Consequently only the square and the triangle are possible.

ANSWER 7: D

Problem 8: For positive numbers the reciprocal function is strictly decreasing, so the largest reciprocal belongs to the smallest number in the list, which is \(1/3\).

ANSWER 8: A

Problem 9: The theoretical maximum heart rate is \(220 - 26 = 194\) beats per minute. Eighty percent of that value is \(0.8 \times 194 = 155.2\), which rounds to the nearest whole number 155.

ANSWER 9: B

Problem 10: Harry evaluates the parentheses first and obtains \(8 - (2 + 5) = 1\). Terry omits the parentheses and obtains \(8 - 2 + 5 = 11\). Their difference is therefore \(1 - 11 = -10\).

ANSWER 10: A

Problem 11: The sum inside the parentheses consists of the ten consecutive odd integers from 81 to 99. Their average is \((81 + 99)/2 = 90\), so the sum equals \(10 \times 90 = 900\). Multiplying by the outer factor of 2 yields \(2 \times 900 = 1800\).

ANSWER 11: E

Problem 12: There are 200 people surveyed and 96 females, so there are \(200 - 96 = 104\) males. Of the 96 females, 58 listen, leaving \(96 - 58 = 38\) females who do not listen. Given that 26 males do not listen, the number of males who listen is \(104 - 26 = 78\). The required percentage is therefore \((78/104) \times 100\% = 75\%\).

ANSWER 12: E

Problem 1:
The problem asks us to find the percentage of yellow tint in a new mixture after adding 5 liters of yellow tint to the original 30 liters of paint mixture. The original mixture is 25% red tint, 30% yellow tint, and 45% water.

First, let's calculate the amount of yellow tint in the original mixture. 30% of 30 liters is $0.3 \times 30 = 9$ liters.

After adding 5 liters of yellow tint, the total amount of yellow tint becomes $9 + 5 = 14$ liters.

The total volume of the new mixture is $30 + 5 = 35$ liters.

To find the percentage of yellow tint in the new mixture, we divide the amount of yellow tint by the total volume and multiply by 100: $\frac{14}{35} \times 100 = 40\%$.

Let's examine the choices:
- A. 25: Too low because the percentage of yellow tint increased.
- B. 35: Not the correct calculation.
- C. 40: Matches our calculation.
- D. 45: Too high.
- E. 50: Too high.

The correct percentage is 40%.

ANSWER 1: C

Problem 2:
The problem asks us to determine the number of ways Georgie the Ghost can enter a haunted house through one of six windows and leave by a different window.

To enter, Georgie has 6 choices. After entering, there are 5 remaining windows to choose from for leaving.

The total number of ways to enter and leave is $6 \times 5 = 30$.

Let's examine the choices:
- A. 12: Too low.
- B. 15: Too low.
- C. 18: Too low.
- D. 30: Matches our calculation.
- E. 36: Too high because it would imply Georgie can leave by the same window.

The correct number of ways is 30.

ANSWER 2: D

Problem 3:
The problem requires us to simplify the expression $1⁄10 + 2⁄20 + 3⁄30$.

First, simplify each fraction:
- $1⁄10$ stays the same.
- $2⁄20 = 1⁄10$.
- $3⁄30 = 1⁄10$.

Adding them together: $1⁄10 + 1⁄10 + 1⁄10 = 3⁄10 = 0.3$.

Let's examine the choices:
- A. .1: Too low.
- B. .123: Not the correct sum.
- C. .2: Too low.
- D. .3: Matches our calculation.
- E. .6: Too high.

The correct sum is 0.3.

ANSWER 3: D

Problem 4:
The problem asks us to find out how many dollars Susan had left after spending money on food and rides at the carnival. She started with $50, spent $12 on food, and twice as much on rides.

First, calculate the amount spent on rides: $12 \times 2 = $24.

Total amount spent is $12 + $24 = $36.

The amount left is $50 - $36 = $14.

Let's examine the choices:
- A. 12: Too low.
- B. 14: Matches our calculation.
- C. 26: Too high.
- D. 38: Too high.
- E. 50: Incorrect because she spent money.

The correct amount left is $14.

ANSWER 4: B

Problem 5:
The problem asks us to calculate $200,000 \times 200,000$.

$200,000 = 2 \times 10^5$, so $200,000 \times 200,000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10} = 40,000,000,000$.

Let's examine the choices:
- A. four hundred thousand: Too low.
- B. four million: Too low.
- C. forty thousand: Too low.
- D. four hundred million: Too low.
- E. forty billion: Matches our calculation.

The correct result is forty billion.

ANSWER 5: E

Problem 6:
The problem requires us to find the missing digit A in the 3-digit number $1A2 that represents the total amount paid by 11 members of the Math Club.

The total amount paid is divisible by 11 since there are 11 members who paid the same amount.

To check divisibility by 11, we use the rule: the difference between the sum of digits at odd places and the sum of digits at even places should be divisible by 11.

For $1A2, the sum of digits at odd places is $1 + 2 = 3$, and the sum at even places is $A$. The difference is $3 - A$.

For $3 - A$ to be divisible by 11 (or 0, since $A$ is a single digit), $A = 3$ makes $3 - A = 0$, which is divisible by 11.

Let's examine the choices:
- A. 0: Not correct because $3 - 0 = 3$, not divisible by 11.
- B. 1: Not correct because $3 - 1 = 2$, not divisible by 11.
- C. 2: Not correct because $3 - 2 = 1$, not divisible by 11.
- D. 3: Correct because $3 - 3 = 0$, which is divisible by 11.
- E. 4: Not correct because $3 - 4 = -1$, not divisible by 11.

The missing digit A is 3, but we should verify if 132 is indeed divisible by 11. $132 / 11 = 12$, so it is.

ANSWER 6: D

Problem 7:
The problem asks us to determine which shapes Haruki can form using a 24 cm long wire.

1. A regular hexagon with side length 5 cm: The perimeter is $6 \times 5 = 30$ cm, which is more than 24 cm. So, Haruki cannot form this hexagon.

2. A square of area 36 cm$^2$: The side length is $\sqrt{36} = 6$ cm. The perimeter is $4 \times 6 = 24$ cm, which matches the wire length. So, Haruki can form this square.

3. A right triangle with legs 6 and 8 cm: Using the Pythagorean theorem, the hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm. The perimeter is $6 + 8 + 10 = 24$ cm, which matches the wire length. So, Haruki can form this triangle.

Let's examine the choices:
- A. Triangle only: Incorrect because Haruki can also form the square.
- B. Hexagon and square only: Incorrect because Haruki cannot form the hexagon.
- C. Hexagon and triangle only: Incorrect because Haruki cannot form the hexagon.
- D. Square and triangle only: Matches our findings.
- E. Hexagon, triangle, and square: Incorrect because Haruki cannot form the hexagon.

The correct shapes Haruki can form are the square and the triangle.

ANSWER 7: D

Problem 8:
The problem asks us to find which of the given numbers has the largest reciprocal.

The reciprocal of a number $x$ is $1/x$. The larger $x$ is, the smaller its reciprocal is, and vice versa.

Let's examine the choices:
- A. $1⁄3$: The reciprocal is 3.
- B. $2⁄5$: The reciprocal is $5/2 = 2.5$.
- C. 1: The reciprocal is 1.
- D. 5: The reciprocal is $1/5 = 0.2$.
- E. 1986: The reciprocal is $1/1986$, very small.

Comparing the reciprocals: $3 > 2.5 > 1 > 0.2 > 1/1986$. So, $1/3$ has the largest reciprocal.

Let's examine the choices again:
- A. $1⁄3$: Correct.
- B. $2⁄5$: Not correct.
- C. 1: Not correct.
- D. 5: Not correct.
- E. 1986: Not correct.

The number with the largest reciprocal is $1/3$.

ANSWER 8: A

Problem 9:
The proble

Problem 1:
Original mixture: 30 L. Yellow tint = 30% of 30 = 0.30 × 30 = 9 L.
Add 5 L yellow: new yellow = 9 + 5 = 14 L, new total volume = 30 + 5 = 35 L.
Percent yellow = (14/35) × 100% = 40%.
ANSWER 1: C

Problem 2:
6 choices for entry window, 5 remaining choices for a different exit window.
Total ways = 6 × 5 = 30.
ANSWER 2: D

Problem 3:
1/10 + 2/20 + 3/30 = 1/10 + 1/10 + 1/10 = 3/10 = 0.3.
ANSWER 3: D

Problem 4:
Spent on food: $12. Spent on rides: 2 × $12 = $24.
Total spent = $12 + $24 = $36. Left = $50 − $36 = $14.
ANSWER 4: B

Problem 5:
200,000 × 200,000 = (2×10⁵) × (2×10⁵) = 4×10¹⁰ = 40,000,000,000, which is forty billion.
ANSWER 5: E

Problem 6:
Total paid is $1A2, a multiple of 11. Divisibility by 11: (1 − A + 2) = 3 − A must be a multiple of 11.
With A a digit 0–9, 3 − A = 0 ⇒ A = 3. (Check: 132 ÷ 11 = 12.)
ANSWER 6: D

Problem 7:
Wire length = 24 cm.
- Regular hexagon, side 5 cm: perimeter = 6×5 = 30 cm > 24 cm → cannot make.
- Square, area 36 cm²: side = 6 cm, perimeter = 4×6 = 24 cm → can make.
- Right triangle, legs 6 and 8 cm: hypotenuse = 10 cm, perimeter = 6+8+10 = 24 cm → can make.
So square and triangle only.
ANSWER 7: D

Problem 8:
Reciprocals: 1/(1/3)=3, 1/(2/5)=2.5, 1/1=1, 1/5=0.2, 1/1986≈0.0005. Largest is 3, from 1/3.
ANSWER 8: A

Problem 9:
Maximum heart rate = 220 − 26 = 194 bpm.
Target = 80% of 194 = 0.8 × 194 = 155.2 → 155 to nearest whole number.
ANSWER 9: B

Problem 10:
Harry: H = 8 − (2+5) = 8 − 7 = 1.
Terry: T = 8 − 2 + 5 = 6 + 5 = 11.
H − T = 1 − 11 = −10.
ANSWER 10: A

Problem 11:
Sum inside parentheses: 10 terms, average = (81+99)/2 = 90, sum = 10×90 = 900.
2 × 900 = 1800.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96 ⇒ Males = 200 − 96 = 104.
Total listeners = 136, female listeners = 58 ⇒ male listeners = 136 − 58 = 78.
Percent of males who listen = (78/104) × 100% = 75%.
ANSWER 12: E

Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint.

**Original mixture (30 liters):**
- Red tint: 25% of 30 = 7.5 liters
- Yellow tint: 30% of 30 = 9 liters
- Water: 45% of 30 = 13.5 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total mixture: 30 + 5 = 35 liters

**New percentage of yellow tint:**
14 / 35 = 2/5 = 0.40 = 40%

Tempting wrong answers: Choice B (35) might come from just adding 5 to the original 30%, but we must recalculate based on the new total.

ANSWER 1: C

---

Problem 2:
Georgie enters through one window and leaves through a *different* window.

- Number of choices for the entry window: 6
- Number of choices for the exit window (must be different): 5
- Total ways: 6 × 5 = 30

Tempting wrong answers: Choice E (36) would be 6 × 6, which allows entering and leaving through the same window. Choice A (12) would be 6 + 6, which doesn't account for all combinations.

ANSWER 2: D

---

Problem 3:
Simplify each fraction first:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

Tempting wrong answers: Choice B (.123) might come from concatenating the digits 1, 2, 3 instead of actually computing. Choice C (.2) might come from miscounting the terms.

ANSWER 3: D

---

Problem 4:
Susan starts with $50.
- Spent on food: $12
- Spent on rides: twice the food amount = 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Amount left: $50 − $36 = $14

Tempting wrong answers: Choice C (26) might come from subtracting only the food ($50 − $24 = $26, forgetting food). Choice D (38) comes from subtracting only $12.

ANSWER 4: B

---

Problem 5:
200,000 × 200,000

Write in scientific notation:
(2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰

4 × 10¹⁰ = 40,000,000,000 = forty billion

Tempting wrong answers: Choice A (four hundred thousand) comes from adding instead of multiplying. Choice D (four hundred million) comes from miscounting zeros.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, so the total $1A2 must be divisible by 11.

**Divisibility rule for 11:** The alternating sum of digits must be a multiple of 11.
Alternating sum: 1 − A + 2 = 3 − A

For 3 − A to be divisible by 11 (and A to be a single digit 0–9):
3 − A = 0 → A = 3

**Check:** $132 ÷ 11 = $12 per member. This is a valid integer amount.

Tempting wrong answers: Other values of A don't make 3 − A a multiple of 11 within the digit range.

ANSWER 6: D

---

Problem 7:
Wire length = 24 cm. We check if each shape's perimeter ≤ 24 cm.

**Regular hexagon (side 5 cm):**
Perimeter = 6 × 5 = 30 cm. Since 30 > 24, he **cannot** make this.

**Square (area 36 cm²):**
Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Since 24 = 24, he **can** make this.

**Right triangle (legs 6 and 8 cm):**
Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Perimeter = 6 + 8 + 10 = 24 cm. Since 24 = 24, he **can** make this.

Haruki can make the square and triangle only.

Tempting wrong answers: Choice E includes the hexagon, but its perimeter exceeds the wire length.

ANSWER 7: D

---

Problem 8:
The reciprocal of a number x is 1/x. The *smaller* the positive number, the *larger* its reciprocal.

| Choice | Number | Reciprocal |
|--------|--------|------------|
| A | 1/3 | 3 |
| B | 2/5 | 5/2 = 2.5 |
| C | 1 | 1 |
| D | 5 | 1/5 = 0.2 |
| E | 1986 | 1/1986 ≈ 0.0005 |

The largest reciprocal is 3, which comes from 1/3.

Tempting wrong answers: Choice E (1986) is the largest number, so it has the *smallest* reciprocal.

ANSWER 8: A

---

Problem 9:
Athlete's age = 26 years.

**Step 1:** Maximum heart rate = 220 − 26 = 194 beats per minute.

**Step 2:** Target heart rate = 80% of 194 = 0.80 × 194 = 155.2

**Step 3:** Round to nearest whole number: 155

Tempting wrong answers: Choice C (176) might come from taking 80% of 220 without subtracting the age. Choice D (194) is the maximum heart rate itself.

ANSWER 9: B

---

Problem 10:
**Harry (correct calculation):**
H = 8 − (2 + 5) = 8 − 7 = 1

**Terry (ignores parentheses):**
T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:**
H − T = 1 − 11 = −10

Tempting wrong answers: Choice E (10) is |H − T|, but the question asks for H − T specifically, which is negative since Terry's answer is larger.

ANSWER 10: A

---

Problem 11:
We need: 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

The sum inside is an arithmetic sequence:
- First term: 81, Last term: 99
- Number of terms: 10
- Sum = (number of terms) × (first + last) / 2 = 10 × (81 + 99) / 2 = 10 × 180 / 2 = 10 × 90 = 900

Multiply by 2: 2 × 900 = 1800

Tempting wrong answers: Choice C (1700) might come from miscounting terms or miscalculating the average.

ANSWER 11: E

---

Problem 12:
Let me organize the data from the survey:

- Total surveyed: 200
- Females: 96 → Males: 200 − 96 = 104
- Total listeners: 136
- Females who listen: 58 → Males who listen: 136 − 58 = 78
- Males who don't listen: 26 (Check: 78 + 26 = 104 ✓)

**Percentage of males who listen:**
(78 / 104) × 100% = (3/4) × 100% = 75%

Tempting wrong answers: Choice C (52) might come from computing 104/200 (percentage of total who are male). Choice D (55) might come from 78/136 (percentage of listeners who are male).

ANSWER 12: E

 Problem 1:
We start with 30 liters of paint. The original amount of yellow tint is 30% of 30 liters:
\[
0.30 \times 30 = 9 \text{ liters.}
\]
Five liters of yellow tint are added, so the new amount of yellow tint is:
\[
9 + 5 = 14 \text{ liters.}
\]
The new total volume of the mixture is:
\[
30 + 5 = 35 \text{ liters.}
\]
The percent of yellow tint in the new mixture is:
\[
\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%.
\]
(Choice A is the original red percentage, D is the original water percentage, and E would require 17.5 liters of yellow, which we do not have.)
ANSWER 1: C

Problem 2:
Georgie chooses one of 6 windows to enter. Since he must leave through a different window, there are 5 remaining choices for his exit. By the multiplication principle:
\[
6 \times 5 = 30 \text{ ways.}
\]
(Choice E, 36, would count entering and leaving through the same window; the other choices are too small.)
ANSWER 2: D

Problem 3:
Simplify each fraction:
\[
\frac{2}{20} = \frac{1}{10}, \qquad \frac{3}{30} = \frac{1}{10}.
\]
So the sum is:
\[
\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3.
\]
(Choice A is just one term; C is 0.2; E is 0.6.)
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides:
\[
2 \times 12 = 24 \text{ dollars.}
\]
Total spent:
\[
12 + 24 = 36 \text{ dollars.}
\]
Amount left:
\[
50 - 36 = 14 \text{ dollars.}
\]
(Choice C is the total spent; D is the amount left if she had spent only $12.)
ANSWER 4: B

Problem 5:
\[
200{,}000 \times 200{,}000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}.
\]
\[
4 \times 10^{10} = 40{,}000{,}000{,}000,
\]
which is read as forty billion.
(Choice D is $4 \times 10^8$; B is $4 \times 10^6$.)
ANSWER 5: E

Problem 6:
The total amount paid is the three-digit number $1A2$, which equals:
\[
100 + 10A + 2 = 102 + 10A.
\]
Since 11 members each paid the same integer amount, $102 + 10A$ must be divisible by 11. Testing the choices:
- $A=0$: $102 \div 11 \approx 9.27$ (not an integer)
- $A=1$: $112 \div 11 \approx 10.18$ (not an integer)
- $A=2$: $122 \div 11 \approx 11.09$ (not an integer)
- $A=3$: $132 \div 11 = 12$ (integer!)
- $A=4$: $142 \div 11 \approx 12.9$ (not an integer)
Thus the missing digit is 3.
ANSWER 6: D

Problem 7:
The wire is 24 cm long, so a shape is possible only if its perimeter is at most 24 cm.
- **Regular hexagon, side 5 cm:** Perimeter $= 6 \times 5 = 30$ cm. $30 > 24$, so impossible.
- **Square of area 36 cm²:** Side $= \sqrt{36} = 6$ cm. Perimeter $= 4 \times 6 = 24$ cm. Exactly fits, so possible.
- **Right triangle with legs 6 and 8 cm:** Hypotenuse $= \sqrt{6^2 + 8^2} = \sqrt{100} = 10$ cm. Perimeter $= 6 + 8 + 10 = 24$ cm. Exactly fits, so possible.
Only the square and the triangle can be formed.
ANSWER 7: D

Problem 8:
For positive numbers, the reciprocal function $f(x) = \frac{1}{x}$ is decreasing, so the smallest number has the largest reciprocal. Compare the two smallest choices:
\[
\frac{1}{3} \approx 0.333\ldots, \qquad \frac{2}{5} = 0.4.
\]
Since $\frac{1}{3} < \frac{2}{5} < 1 < 5 < 1986$, the smallest number is $\frac{1}{3}$. Its reciprocal is $3$, which is larger than the reciprocals of all the other choices.
ANSWER 8: A

Problem 9:
First find the theoretical maximum heart rate:
\[
220 - 26 = 194 \text{ beats per minute.}
\]
The target rate is 80% of this:
\[
0.80 \times 194 = 155.2.
\]
Rounded to the nearest whole number, this is 155.
(Choice C is the maximum heart rate itself.)
ANSWER 9: B

Problem 10:
Harry computes correctly:
\[
H = 8 - (2 + 5) = 8 - 7 = 1.
\]
Terry ignores the parentheses:
\[
T = 8 - 2 + 5 = 6 + 5 = 11.
\]
The difference is:
\[
H - T = 1 - 11 = -10.
\]
ANSWER 10: A

Problem 11:
The ten numbers inside the parentheses form an arithmetic sequence: 81, 83, …, 99. The sum of an arithmetic sequence is:
\[
\frac{\text{number of terms}}{2} \times (\text{first} + \text{last}) = \frac{10}{2} \times (81 + 99) = 5 \times 180 = 900.
\]
Now multiply by the factor of 2 outside:
\[
2 \times 900 = 1800.
\]
(One can also pair them: $(81+99)=180$, $(83+97)=180$, etc., giving five pairs that each sum to 180.)
ANSWER 11: E

Problem 12:
Total surveyed = 200, and females surveyed = 96, so:
\[
\text{Males surveyed} = 200 - 96 = 104.
\]
Total listeners = 136, and females who listen = 58, so:
\[
\text{Males who listen} = 136 - 58 = 78.
\]
(We can check: total non-listeners = 64, males who don't listen = 26, so females who don't listen = 38; $58+38=96$ and $78+26=104$, which checks out.)

The percentage of males who listen is:
\[
\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%.
\]
ANSWER 12: E

Problem 1:
A mixture of 30 liters is 25% red, 30% yellow, and 45% water. Five liters of yellow tint are added. What is the percent of yellow tint in the new mixture?

**Setup:** Find the amount of yellow tint originally, add 5 liters, then find the new percentage.

**Original yellow tint:** 30% of 30 = 0.30 × 30 = 9 liters

**After adding 5 liters of yellow tint:**
- New yellow tint: 9 + 5 = 14 liters
- New total volume: 30 + 5 = 35 liters
- New percentage: 14/35 = 2/5 = 40%

**Checking wrong choices:** 25% would be if we just added the original percentages; 35% would be if we just added 5% to 30%; 45% and 50% are too high.

ANSWER 1: C

---

Problem 2:
Six windows. Georgie enters by one and leaves by a different one. How many ways?

**Setup:** Choose an entry window (6 choices), then choose a different exit window (5 choices).

**Calculation:** 6 × 5 = 30 ways

**Checking wrong choices:** 15 would be if we divided by 2 (treating order as irrelevant); 12 would be 6×2; 18 would be 6×3; 36 would be 6×6 (allowing same window).

ANSWER 2: D

---

Problem 3:
1/10 + 2/20 + 3/30 = ?

**Simplify each fraction:**
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

**Sum:** 1/10 + 1/10 + 1/10 = 3/10 = 0.3

**Checking wrong choices:** 0.1 would be just one term; 0.2 would be two terms; 0.6 would be adding numerators and denominators incorrectly.

ANSWER 3: D

---

Problem 4:
Susan had $50. She spent $12 on food and twice as much on rides. How much left?

**Setup:** Food = $12, Rides = 2 × $12 = $24

**Total spent:** $12 + $24 = $36

**Amount left:** $50 - $36 = $14

**Checking wrong choices:** 12 is just the food amount; 26 is 50-24 (forgetting food); 38 is 50-12 (forgetting rides).

ANSWER 4: B

---

Problem 5:
200,000 × 200,000 = ?

**Setup:** Express in scientific notation: 200,000 = 2 × 10⁵

**Calculation:** (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion

**Checking wrong choices:** Four hundred thousand is 4×10⁵; four million is 4×10⁶; forty thousand is 4×10⁴; four hundred million is 4×10⁸.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, totaling $1A2. Find the missing digit A.

**Setup:** The total $1A2 must be divisible by 11.

**Divisibility test for 11:** A number is divisible by 11 if the alternating sum of its digits is divisible by 11.

For 1A2: (1 + 2) - A = 3 - A must be divisible by 11.

Since A is a single digit (0-9), 3 - A can only equal 0 (the only multiple of 11 in range -6 to 3).

So A = 3, giving $132 = 11 × $12 ✓

**Checking:** With A=0: $102 ÷ 11 ≈ 9.27 (not integer); A=1: $112 ÷ 11 ≈ 10.18 (not integer); A=3: $132 ÷ 11 = 12 ✓

ANSWER 6: D

---

Problem 7:
Wire is 24 cm. Which shapes can be made?

**Regular hexagon with side 5 cm:** Perimeter = 6 × 5 = 30 cm. Need 30 cm, only have 24 cm. ❌

**Square with area 36 cm²:** Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Need 24 cm, have 24 cm. ✓

**Right triangle with legs 6 and 8 cm:** Hypotenuse = √(36 + 64) = √100 = 10 cm. Perimeter = 6 + 8 + 10 = 24 cm. Need 24 cm, have 24 cm. ✓

**Conclusion:** Can make the square and triangle, but not the hexagon.

ANSWER 7: D

---

Problem 8:
Which number has the largest reciprocal?

**Find each reciprocal:**
- A. 1/3 → reciprocal = 3
- B. 2/5 → reciprocal = 5/2 = 2.5
- C. 1 → reciprocal = 1
- D. 5 → reciprocal = 1/5 = 0.2
- E. 1986 → reciprocal ≈ 0.0005

**Key insight:** Smaller positive numbers have larger reciprocals. The smallest number listed is 1/3, so it has the largest reciprocal (3).

ANSWER 8: A

---

Problem 9:
Target heart rate = 80% of (220 - age). Find target for 26-year-old.

**Maximum heart rate:** 220 - 26 = 194 bpm

**Target heart rate:** 80% of 194 = 0.80 × 194 = 155.2 ≈ 155 bpm

**Checking wrong choices:** 134 would be about 69%; 176 would be about 91%; 194 is the max (100%); 243 is impossible.

ANSWER 9: B

---

Problem 10:
Harry calculates 8 − (2 + 5) correctly. Terry calculates 8 − 2 + 5. Find H − T.

**Harry's answer:** H = 8 − (2 + 5) = 8 − 7 = 1

**Terry's answer:** T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:** H − T = 1 − 11 = −10

**Checking wrong choices:** −6 would be if Terry got 7; 0 would be if they got the same answer; 6 or 10 would be T − H.

ANSWER 10: A

---

Problem 11:
2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99) = ?

**Setup:** This is 10 consecutive odd numbers. The average of the first and last is (81 + 99)/2 = 90.

**Sum of the 10 numbers:** 10 × 90 = 900

**Multiply by 2:** 2 × 900 = 1800

**Alternative verification:** Pair the numbers: (81+99) + (83+97) + (85+95) + (87+93) + (89+91) = 180 + 180 + 180 + 180 + 180 = 900. Then 2 × 900 = 1800.

ANSWER 11: E

---

Problem 12:
Find the percentage of males who listen to the station.

**Given information:**
- Total surveyed: 200
- Females: 96
- Females who listen: 58
- Males who don't listen: 26
- Total listeners: 136
- Total non-listeners: 64

**Find males:** Males = 200 - 96 = 104

**Find males who listen:** Males who listen = Total listeners - Females who listen = 136 - 58 = 78

**Verification:** Males who don't listen = 104 - 78 = 26 ✓ (matches given)

**Percentage of males who listen:** 78/104 × 100 = 75%

**Checking wrong choices:** 39% would be 78/200 (using total instead of males); 48% would be about 50/104; 52% would be about 54/104; 55% would be about 57/104.

ANSWER 12: E

**Problem 1**

The original mixture is 30 L with 30 % yellow tint → yellow = 0.30·30 = 9 L.  
Add 5 L of yellow → total yellow = 9 + 5 = 14 L.  
Total volume after addition = 30 + 5 = 35 L.  

Percent yellow = (14/35)·100 % = 40 %.

**Answer 1: C**

---

**Problem 2**

Choose an entry window (6 ways) and a different exit window (5 ways).  
Number of ways = 6 × 5 = 30.

**Answer 2: D**

---

**Problem 3**

\[
\frac{1}{10}=0.1,\qquad\frac{2}{20}=0.1,\qquad\frac{3}{30}=0.1
\]
Sum = 0.1 + 0.1 + 0.1 = 0.3.

**Answer 3: D**

---

**Problem 4**

Food cost = \$12.  
Rides cost = twice food = 2·12 = \$24.  
Total spent = 12 + 24 = \$36.  
Money left = 50 − 36 = \$14.

**Answer 4: B**

---

**Problem 5**

\(200{,}000 \times 200{,}000 = (2\times10^{5})^{2}=4\times10^{10}=40{,}000{,}000{,}000\).

That is 40 billion.

**Answer 5: E**

---

**Problem 6**

Total paid = \$1A2 = 100 + 10A + 2 = 102 + 10A.  
Since the 11 members each paid an integer amount, 102 + 10A must be divisible by 11.

\(102\equiv3\pmod{11}\).  
We need \(10A\equiv-3\equiv8\pmod{11}\).  
Because \(10\equiv-1\pmod{11}\), this is \(-A\equiv8\pmod{11}\) → \(A\equiv3\pmod{11}\).  
A digit, so \(A=3\).

**Answer 6: D**

---

**Problem 7**

- Hexagon: perimeter = 6·5 = 30 cm > 24 cm → impossible.  
- Square: side = √36 = 6 cm, perimeter = 4·6 = 24 cm → possible.  
- Right triangle: legs 6 cm, 8 cm, hypotenuse = √(6²+8²)=10 cm, perimeter = 6+8+10 = 24 cm → possible.

Only the square and triangle can be made.

**Answer 7: D**

---

**Problem 8**

Reciprocals:
- \(1/3 → 3\)  
- \(2/5 → 5/2 = 2.5\)  
- \(1 → 1\)  
- \(5 → 1/5 = 0.2\)  
- \(1986 → 1/1986 ≈ 0.0005\)

Largest reciprocal is 3, from \(1/3\).

**Answer 8: A**

---

**Problem 9**

Maximum heart rate = 220 − 26 = 194 bpm.  
Target = 80 % of 194 = 0.8·194 = 155.2 → rounded to 155 bpm.

**Answer 9: B**

---

**Problem 10**

Harry: \(8-(2+5)=8-7=1\).  
Terry (ignoring parentheses): \(8-2+5 = 6+5 = 11\).  
Difference \(H-T = 1-11 = -10\).

**Answer 10: A**

---

**Problem 11**

Sum of the ten terms: average = (81+99)/2 = 90, so sum = 90·10 = 900.  
Multiply by 2: \(2·900 = 1800\).

**Answer 11: E**

---

**Problem 12**

Total surveyed = 200.  
Females = 96 → Males = 200 − 96 = 104.  

Female listeners = 58 → Male listeners = 136 − 58 = 78.  

Percent of males who listen = (78/104)·100 % = (39/52)·100 % = 75 %.

**Answer 12: E**

---

**Final Answers**

ANSWER 1: C  
ANSWER 2: D  
ANSWER 3: D  
ANSWER 4: B  
ANSWER 5: E  
ANSWER 6: D  
ANSWER 7: D  
ANSWER 8: A  
ANSWER 9: B  
ANSWER 10: A  
ANSWER 11: E  
ANSWER 12: E

Problem 1:  
A 30-liter paint mixture is 25% red tint, 30% yellow tint, and 45% water. Five liters of yellow tint are added.  
Original yellow tint: $0.30 \times 30 = 9$ liters.  
After adding 5 liters: $9 + 5 = 14$ liters yellow.  
New total volume: $30 + 5 = 35$ liters.  
Percent yellow: $\frac{14}{35} = 0.40 = 40\%$.  
Tempting wrong choices: 35% (original yellow percent) or 45% (water percent).  
ANSWER 1: D  

Problem 2:  
Georgie enters through one of six windows and exits through a different one.  
Entry choices: 6. Exit choices: 5 (excluding entry window).  
Total ways: $6 \times 5 = 30$.  
Tempting wrong choices: 15 (if order didn’t matter) or 36 (if same window allowed).  
ANSWER 2: D  

Problem 3:  
$\frac{1}{10} + \frac{2}{20} + \frac{3}{30} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3$.  
Tempting wrong choices: 0.1 (one term) or 0.2 (two terms).  
ANSWER 3: D  

Problem 4:  
Susan starts with \$50.  
Food: \$12.  
Rides: $2 \times 12 = \$24$.  
Total spent: $12 + 24 = \$36$.  
Left: $50 - 36 = \$14$.  
Tempting wrong choices: \$26 (if spent \$24 total) or \$38 (if spent \$12 total).  
ANSWER 4: B  

Problem 5:  
$200,000 \times 200,000 = 40,000,000,000 = 40$ billion.  
Tempting wrong choices: 400,000 (added zeros) or 4 million (multiplied digits).  
ANSWER 5: E  

Problem 6:  
Total paid: \$1A2, an integer divisible by 11.  
$1A2 \mod 11 = 0$.  
$1 + 2 - A \equiv 0 \mod 11 \Rightarrow 3 - A \equiv 0 \mod 11 \Rightarrow A = 3$.  
Tempting wrong choices: 0, 1, 2, or 4 (incorrect divisibility).  
ANSWER 6: E (Wait, let's recheck: 1A2 must be divisible by 11. 132 ÷ 11 = 12, so A=3. But 132 is 11×12, so correct. However, the answer choices are A=0,B=1,C=2,D=3,E=4. So D is 3. So correct answer is D, not E. I made a mistake in the initial answer. Correcting:)  
**Correction**: $1A2 = 132$ when $A=3$, and $132 \div 11 = 12$, so $A=3$.  
ANSWER 6: D  

Problem 7:  
Wire length: 24 cm.  
- Regular hexagon with side 5 cm: Perimeter $6 \times 5 = 30$ cm (not 24).  
- Square with area 36 cm²: Side $6$ cm, perimeter $24$ cm.  
- Right triangle with legs 6 and 8 cm: Hypotenuse 10 cm, perimeter $6+8+10=24$ cm.  
Can make square and triangle.  
Tempting wrong choices: Hexagon (incorrect perimeter).  
ANSWER 7: D  

Problem 8:  
Largest reciprocal means smallest original number.  
$\frac{1}{3} \approx 0.333$, $\frac{2}{5} = 0.4$, $1$, $5$, $1986$.  
Smallest: $\frac{1}{3}$. Reciprocal: 3.  
Others: 2.5, 1, 0.2, 0.0005.  
Tempting wrong choices: $\frac{2}{5}$ (larger than $\frac{1}{3}$).  
ANSWER 8: A  

Problem 9:  
Maximum heart rate: $220 - 26 = 194$.  
Target: $0.80 \times 194 = 155.2 \approx 155$.  
Tempting wrong choices: 134 (70%), 176 (90%), or 243 (incorrect max).  
ANSWER 9: B  

Problem 10:  
Harry: $8 - (2 + 5) = 1$.  
Terry: $8 - 2 + 5 = 11$.  
Difference: $1 - 11 = -10$.  
Tempting wrong choices: 10 (absolute difference) or 0 (if parentheses ignored by both).  
ANSWER 10: A  

Problem 11:  
Sum: $81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99$.  
Average: 90, 10 terms: $90 \times 10 = 900$.  
$2 \times 900 = 1800$.  
Tempting wrong choices: 1600 (min+max=180, times 5=900, times 2=1800, but if miscount terms).  
ANSWER 11: E  

Problem 12:  
Total surveyed: 200.  
Females: 96, so males: $200 - 96 = 104$.  
Total listeners: 136.  
Female listeners: 58, so male listeners: $136 - 58 = 78$.  
Percent male listeners: $\frac{78}{104} = 0.75 = 75\%$.  
Tempting wrong choices: 39 (male non-listeners percent) or 52 (incorrect calculation).  
ANSWER 12: E

### Problem 1:
What is the percentage of yellow tint in the new 35-liter paint mixture, after adding 5 liters of yellow tint to an original 30-liter mixture that was 30% yellow?
First, calculate the original volume of yellow tint: 30% of 30 liters = \(0.3 \times 30 = 9\) liters. Add the 5 new liters of yellow tint to get \(9 + 5 = 14\) liters of yellow in the new mixture. The total new volume of the mixture is \(30 + 5 = 35\) liters. The percentage of yellow is \(\frac{14}{35} = 0.4 = 40\%\). Tempting wrong answers come from forgetting to update the total mixture volume when calculating the new percentage, or miscalculating the original yellow volume.
ANSWER 1: C

---

### Problem 2:
How many ways can Georgie enter through one window and leave through a different window, out of 6 total windows?
Georgie has 6 choices for an entry window. Since he must leave through a different window, he has 5 remaining choices for an exit window. The total number of ways is \(6 \times 5 = 30\). A common wrong answer is 15, which incorrectly treats entry/exit as unordered pairs, when order matters (entering window A and leaving B is different from entering B and leaving A).
ANSWER 2: D

---

### Problem 3:
Simplify \(\frac{1}{10} + \frac{2}{20} + \frac{3}{30}\):
Simplify each fraction first: \(\frac{2}{20} = \frac{1}{10}\) and \(\frac{3}{30} = \frac{1}{10}\). Add them together: \(\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3\). A common wrong answer is 0.1, which incorrectly adds numerators and denominators directly instead of simplifying first.
ANSWER 3: D

---

### Problem 4:
How much money does Susan have left after spending $12 on food and twice that amount on rides, starting with $50?
First calculate the amount spent on rides: twice $12 is \(2 \times 12 = 24\) dollars. Total spent is \(12 + 24 = 36\) dollars. The amount left is \(50 - 36 = 14\) dollars. A common wrong answer is 26, which incorrectly uses $12 instead of $24 for ride costs.
ANSWER 4: B

---

### Problem 5:
Calculate \(200,000 \times 200,000\):
Rewrite 200,000 as \(2 \times 10^5\). Multiplying two copies gives \((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\), which equals 40,000,000,000, or forty billion. Wrong answers come from miscounting the number of zeros in the product.
ANSWER 5: E

---

### Problem 6:
The 3-digit total cost \(1A2\) dollars is divisible by 11 (11 members paid the same integer amount). What is the missing digit A?
The divisibility rule for 11 states that alternating sum of digits must be divisible by 11. For 1A2, the alternating sum is \(1 - A + 2 = 3 - A\). The only value of A (0-9) that makes this sum divisible by 11 is A=3, since \(3-3=0\), which is divisible by 11. Verify: \(132 \div 11 = 12\), a valid integer.
ANSWER 6: D

---

### Problem 7:
Haruki has a 24cm wire. Which of the three shapes can he form? Calculate the perimeter of each shape:
1. Regular hexagon with side 5cm: perimeter = \(6 \times 5 = 30\)cm, which is longer than 24cm, so impossible.
2. Square with area 36cm²: side length = \(\sqrt{36}=6\)cm, perimeter = \(4 \times 6 = 24\)cm, which fits.
3. Right triangle with legs 6cm and 8cm: hypotenuse = 10cm (6-8-10 Pythagorean triple), perimeter = \(6+8+10=24\)cm, which fits.
Only the square and triangle can be formed.
ANSWER 7: D

---

### Problem 8:
Which number has the largest reciprocal? The reciprocal of \(x\) is \(\frac{1}{x}\), so smaller values of \(x\) produce larger reciprocals. List the values of all options:
A. \(\frac{1}{3} \approx 0.333\), B. \(\frac{2}{5}=0.4\), C.1, D.5, E.1986. The smallest value is \(\frac{1}{3}\), whose reciprocal is 3, the largest of all options.
ANSWER 8: A

---

### Problem 9:
Calculate the target heart rate for a 26-year-old athlete: target is 80% of max heart rate, where max = 220 - age.
First calculate max heart rate: \(220 - 26 = 194\) bpm. 80% of 194 is \(0.8 \times 194 = 155.2\), which rounds to 155 bpm. A common wrong answer is 176, which uses 220 instead of the correct max heart rate of 194.
ANSWER 9: B

---

### Problem 10:
Calculate \(H-T\), where H is Harry's correct answer to \(8-(2+5)\), and T is Terry's incorrect answer of \(8-2+5\).
Harry's answer: \(8 - 7 = 1 = H\). Terry's answer: \(6 + 5 = 11 = T\). The difference \(H-T = 1 - 11 = -10\). A common wrong answer is 10, which calculates \(T-H\) instead of \(H-T\).
ANSWER 10: A

---

### Problem 11:
Simplify \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\):
Pair the terms inside the parentheses to simplify addition: \((81+99)+(83+97)+(85+95)+(87+93)+(89+91) = 5 \times 180 = 900\). Multiply by 2: \(2 \times 900 = 1800\).
ANSWER 11: E

---

### Problem 12:
What percentage of males surveyed listen to the radio station? First calculate total number of males: total surveyed = 200, females = 96, so males = \(200 - 96 = 104\). Males who do not listen = 26, so males who listen = \(104 - 26 = 78\). The percentage of males who listen is \(\frac{78}{104} = 0.75 = 75\%\). A common wrong answer is 39%, which divides male listeners by the total surveyed population instead of the total male population.
ANSWER 12: E

### Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.
1.  Calculate original yellow tint volume: 25%? No wait, original is 30% yellow. So 30% of 30 L = $0.3 \times 30 = 9$ liters of yellow.
2.  Add 5 liters of yellow: total yellow = $9 + 5 = 14$ liters.
3.  Total new mixture volume: $30 + 5 = 35$ liters.
4.  Percent yellow: $\frac{14}{35} \times 100 = 40\%$.
ANSWER 1: C
---
### Problem 2:
We count the number of ways to choose an entry window and a different exit window from 6 windows.
1.  Entry window: 6 possible choices.
2.  Exit window: cannot be the entry window, so 5 remaining choices.
3.  Total ways: $6 \times 5 = 30$ (order matters because entry and exit are distinct actions).
ANSWER 2: D
---
### Problem 3:
Simplify and add the fractions:
1.  $\frac{1}{10} = 0.1$, $\frac{2}{20} = \frac{1}{10} = 0.1$, $\frac{3}{30} = \frac{1}{10} = 0.1$.
2.  Sum: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D
---
### Problem 4:
Calculate remaining money after expenses:
1.  Total initial money: $50.
2.  Money spent on rides: twice the food cost = $2 \times 12 = 24$ dollars.
3.  Total spent: $12 + 24 = 36$ dollars.
4.  Remaining: $50 - 36 = 14$ dollars.
ANSWER 4: B
---
### Problem 5:
Compute $200,000 \times 200,000$:
1.  Rewrite as $(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
2.  $10^{10} = 10,000,000,000$, so $4 \times 10^{10} = 40,000,000,000 = 40$ billion.
ANSWER 5: E
---
### Problem 6:
The total payment $1A2$ (a 3-digit number) must be divisible by 11 (since 11 people paid equal integer amounts).
1.  Use the divisibility rule for 11: the alternating sum of digits is divisible by 11. For $1A2$, this is $1 - A + 2 = 3 - A$.
2.  $A$ is a digit (0-9), so $3 - A$ must equal 0 (the only multiple of 11 in range: $3 - A = 11$ gives $A=-8$, $3-A=-11$ gives $A=14$, both invalid).
3.  $3 - A = 0 \implies A=3$. Verify: $132 \div 11 = 12$, which is an integer.
ANSWER 6: D
---
### Problem 7:
Check if the 24 cm wire can form each shape (perimeter of the shape must equal 24 cm):
1.  Regular hexagon (side 5 cm): Perimeter = $6 \times 5 = 30$ cm > 24 cm → cannot make.
2.  Square (area 36 cm²): Side length = $\sqrt{36} = 6$ cm. Perimeter = $4 \times 6 = 24$ cm → can make.
3.  Right triangle (legs 6, 8 cm): Hypotenuse = $\sqrt{6^2 + 8^2} = 10$ cm. Perimeter = $6 + 8 + 10 = 24$ cm → can make.
Only square and triangle can be made.
ANSWER 7: D
---
### Problem 8:
The reciprocal of a positive number $x$ is $\frac{1}{x}$; smaller $x$ gives larger reciprocal.
1.  List values: $\frac{1}{3} \approx 0.33$, $\frac{2}{5}=0.4$, $1$, $5$, $1986$.
2.  Smallest value is $\frac{1}{3}$, whose reciprocal is $3$ (largest among all reciprocals: $\frac{5}{2}=2.5$, $1$, $0.2$, $\frac{1}{1986} \approx 0.0005$).
ANSWER 8: A
---
### Problem 9:
Calculate target heart rate for a 26-year-old:
1.  Maximum heart rate: $220 - 26 = 194$ bpm.
2.  Target heart rate: 80% of 194 = $0.8 \times 194 = 155.2$, which rounds to 155.
ANSWER 9: B
---
### Problem 10:
Compute $H$ (Harry's correct answer) and $T$ (Terry's incorrect answer):
1.  Harry: $8 - (2 + 5) = 8 - 7 = 1$, so $H=1$.
2.  Terry (ignores parentheses, left-to-right): $8 - 2 + 5 = 6 + 5 = 11$, so $T=11$.
3.  Difference: $H - T = 1 - 11 = -10$.
ANSWER 10: A
---
### Problem 11:
Calculate $2 \times (81 + 83 + ... + 99)$:
1.  The sum inside is 10 consecutive odd numbers. Average of first and last term: $\frac{81 + 99}{2} = 90$.
2.  Sum of the 10 terms: $90 \times 10 = 900$.
3.  Multiply by 2: $2 \times 900 = 1800$.
ANSWER 11: E
---
### Problem 12:
Find the percentage of males who listen to the station:
1.  Total males surveyed: $200 - 96 = 104$.
2.  Males who do not listen: 26, so males who listen: $104 - 26 = 78$.
3.  Verify with total listeners: Total listeners = 136, female listeners = 58, so male listeners = $136 - 58 = 78$ (matches).
4.  Percentage: $\frac{78}{104} \times 100 = 75\%$.
ANSWER 12: E

Problem 1:
Original yellow = 30% of 30 = 9 L. Add 5 → 14 L yellow. New total = 35 L. Percent = 14/35 = 40%.

ANSWER 1: C

Problem 2:
Enter: 6 choices, leave: 5 different choices. 6×5 = 30.

ANSWER 2: D

Problem 3:
Each fraction = 0.1. Sum = 0.3.

ANSWER 3: D

Problem 4:
Food 12, rides twice = 24. Spent 36. Left = 50 − 36 = 14.

ANSWER 4: B

Problem 5:
200,000 × 200,000 = 40,000,000,000 = forty billion.

ANSWER 5: E

Problem 6:
1A2 divisible by 11. Alternating sum: 1 − A + 2 = 3 − A must be divisible by 11 → A = 3 (gives 0). Check 132/11 = 12. Yes.

ANSWER 6: D

Problem 7:
Hexagon: 6×5 = 30 > 24, no. Square area 36 → side 6, perimeter 24, yes. Triangle legs 6,8 → hypotenuse 10, perimeter 24, yes. Square and triangle only.

ANSWER 7: D

Problem 8:
Largest reciprocal = smallest number. Smallest is 1/3.

ANSWER 8: A

Problem 9:
Max = 220 − 26 = 194. Target = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

Problem 10:
H = 8 − 7 = 1. T = 8 − 2 + 5 = 11. H − T = 1 − 11 = −10.

ANSWER 10: A

Problem 11:
Sum of 81..99 odd numbers (10 terms): average (81+99)/2 = 90, ×10 = 900. ×2 = 1800.

ANSWER 11: E

Problem 12:
Males total = 200 − 96 = 104. Males who don't listen = 26 → males who listen = 104 − 26 = 78. Percent = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem systematically.

---

**Problem 1:**
Original mixture: 30 liters, 30% yellow = 9 liters yellow. Add 5 liters yellow → new yellow = 14 liters, new total = 35 liters.
Percent yellow = 14/35 = 40%.

ANSWER 1: C

---

**Problem 2:**
Georgie can enter through any of 6 windows, then leave through any of the remaining 5 windows.
6 × 5 = 30.

ANSWER 2: D

---

**Problem 3:**
1/10 = 0.1, 2/20 = 0.1, 3/30 = 0.1. Sum = 0.3.

ANSWER 3: D

---

**Problem 4:**
Food: $12. Rides: 2 × $12 = $24. Total spent: $12 + $24 = $36. Left: $50 − $36 = $14.

ANSWER 4: B

---

**Problem 5:**
200,000 × 200,000 = (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion.

ANSWER 5: E

---

**Problem 6:**
The number 1A2 must be divisible by 11. 1A2 ÷ 11 must be an integer.
11 × 10 = 110, 11 × 11 = 121 (no), 11 × 12 = 132 ✓ → A = 3.
Check: 132/11 = 12 ✓, each paid $12.

ANSWER 6: D

---

**Problem 7:**
Wire = 24 cm.
- Regular hexagon, side 5: perimeter = 6 × 5 = 30 cm > 24. ✗
- Square, area 36: side = 6, perimeter = 24 cm ✓
- Right triangle, legs 6 and 8: hypotenuse = 10, perimeter = 6+8+10 = 24 cm ✓

ANSWER 7: D

---

**Problem 8:**
The reciprocal of x is 1/x. The largest reciprocal comes from the smallest positive number.
Among 1/3, 2/5, 1, 5, 1986 — the smallest is 1/3, and its reciprocal is 3, which is the largest.

ANSWER 8: A

---

**Problem 9:**
Maximum heart rate = 220 − 26 = 194. Target = 80% × 194 = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

---

**Problem 10:**
Harry: 8 − (2 + 5) = 8 − 7 = 1, so H = 1.
Terry: 8 − 2 + 5 = 11, so T = 11.
H − T = 1 − 11 = −10.

ANSWER 10: A

---

**Problem 11:**
The ten odd numbers from 81 to 99: sum = (81+99)/2 × 10 = 90 × 10 = 900.
2 × 900 = 1800.

ANSWER 11: E

---

**Problem 12:**
Total surveyed = 200. Females = 96, so Males = 104.
Total non-listeners = 64. Males who don't listen = 26, so females who don't listen = 64 − 26 = 38.
Males who listen = 104 − 26 = 78.
Percentage = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem step-by-step.

## Problem 1:
**What is asked:** Find the percentage of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.

**Original mixture:**
- Yellow tint: 30% of 30 = 0.30 × 30 = 9 liters
- Total volume: 30 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total volume: 30 + 5 = 35 liters

**Percentage of yellow tint:**
$$\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%$$

ANSWER 1: C

## Problem 2:
**What is asked:** In how many ways can the ghost enter by one window and leave by a different window?

**Solution:**
- Choices for entering: 6 windows
- Choices for leaving: 5 windows (must be different from entry)
- Total ways: 6 × 5 = 30

ANSWER 2: D

## Problem 3:
**What is asked:** Calculate 1/10 + 2/20 + 3/30

**Solution:**
Simplify each fraction:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

ANSWER 3: D

## Problem 4:
**What is asked:** How much money does Susan have left?

**Solution:**
- Started with: $50
- Spent on food: $12
- Spent on rides: 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Left: $50 - $36 = $14

ANSWER 4: B

## Problem 5:
**What is asked:** Calculate 200,000 × 200,000

**Solution:**
$$200,000 × 200,000 = (2 × 10^5) × (2 × 10^5) = 4 × 10^{10} = 40,000,000,000 = 40 \text{ billion}$$

ANSWER 5: E

## Problem 6:
**What is asked:** Find the missing digit A in the 3-digit number 1A2, where 11 club members each paid an equal integer amount that totals 1A2.

**Solution:**
The number 1A2 must be divisible by 11.

Testing divisibility by 11 (alternating sum of digits):
- 1 - A + 2 = 0 (mod 11)
- 3 - A ≡ 0 (mod 11)
- A = 3

Check: 132 ÷ 11 = 12 ✓

ANSWER 6: D

## Problem 7:
**What is asked:** Which shapes can Haruki make with 24 cm of wire?

**Solution:**
Test each shape:

**Regular hexagon with side 5 cm:**
- Perimeter: 6 × 5 = 30 cm
- Need 30 cm, have 24 cm ✗

**Square with area 36 cm²:**
- Side length: √36 = 6 cm
- Perimeter: 4 × 6 = 24 cm ✓

**Right triangle with legs 6 and 8 cm:**
- Hypotenuse: √(6² + 8²) = √(36 + 64) = √100 = 10 cm
- Perimeter: 6 + 8 + 10 = 24 cm ✓

Only square and triangle can be made.

ANSWER 7: D

## Problem 8:
**What is asked:** Which number has the largest reciprocal?

**Solution:**
Find reciprocals:
- 1/(1/3) = 3
- 1/(2/5) = 5/2 = 2.5
- 1/1 = 1
- 1/5 = 0.2
- 1/1986 ≈ 0.0005

The largest reciprocal is 5 (from the number 1/5).

ANSWER 8: D

## Problem 9:
**What is asked:** Find the target heart rate for a 26-year-old athlete.

**Solution:**
- Maximum heart rate: 220 - 26 = 194 beats per minute
- Target heart rate: 80% of 194 = 0.80 × 194 = 155.2 ≈ 155

ANSWER 9: B

## Problem 10:
**What is asked:** Find H - T where H is Harry's answer and T is Terry's answer.

**Solution:**
**Harry's calculation (correct):**
$$H = 8 - (2 + 5) = 8 - 7 = 1$$

**Terry's calculation (ignoring parentheses):**
$$T = 8 - 2 + 5 = 6 + 5 = 11$$

**Difference:**
$$H - T = 1 - 11 = -10$$

ANSWER 10: A

## Problem 11:
**What is asked:** Calculate 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

**Solution:**
This is an arithmetic sequence with 10 terms. Sum = n(first + last)/2
$$81 + 83 + ... + 99 = \frac{10(81 + 99)}{2} = \frac{10 × 180}{2} = 900$$

$$2 × 900 = 1800$$

ANSWER 11: E

## Problem 12:
**What is asked:** What percentage of males surveyed listen to the station?

**Solution:**
- Total surveyed: 200
- Females: 96
- Males: 200 - 96 = 104

- Total non-listeners: 64
- Males who don't listen: 26
- Males who listen: 104 - 26 = 78

**Percentage of males who listen:**
$$\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%$$

ANSWER 12: E

Problem 1:

The original mixture is 30 liters total, with 30% yellow tint.

So the amount of yellow tint at first is:
- \(30\%\) of \(30 = 0.30 \times 30 = 9\) liters

Then 5 liters of yellow tint are added, so the new amount of yellow tint is:
- \(9 + 5 = 14\) liters

The new total mixture is:
- \(30 + 5 = 35\) liters

So the percent of yellow tint is:
- \(\frac{14}{35} = \frac{2}{5} = 0.4 = 40\%\)

Answer choice C matches 40%.

ANSWER 1: C

Problem 2:

Georgie can:
- choose 1 of the 6 windows to enter
- then choose a different window to leave

For the entry, there are 6 choices.

For the exit, there are 5 remaining choices, since it must be a different window.

So the total number of ways is:
- \(6 \times 5 = 30\)

ANSWER 2: D

Problem 3:

Compute each fraction:
- \(\frac{1}{10} = 0.1\)
- \(\frac{2}{20} = \frac{1}{10} = 0.1\)
- \(\frac{3}{30} = \frac{1}{10} = 0.1\)

Add them:
- \(0.1 + 0.1 + 0.1 = 0.3\)

So the value is \(0.3\).

ANSWER 3: D

Problem 4:

Susan had \$50.

She spent:
- \$12 on food
- twice as much on rides, so \(2 \times 12 = 24\)

Total spent:
- \(12 + 24 = 36\)

Money left:
- \(50 - 36 = 14\)

ANSWER 4: B

Problem 5:

Two hundred thousand is:
- \(200{,}000 = 2 \times 10^5\)

Multiply:
- \(200{,}000 \times 200{,}000 = (2 \times 10^5)(2 \times 10^5) = 4 \times 10^{10}\)

And \(4 \times 10^{10} = 40{,}000{,}000{,}000\), which is forty billion.

ANSWER 5: E

Problem 6:

The total is written as \(1A2\), and 11 members each paid the same integer amount.

So \(1A2\) must be divisible by 11.

Test the answer choices:
- 102 ÷ 11 is not an integer
- 112 ÷ 11 is not an integer
- 122 ÷ 11 = 11 remainder 1
- 132 ÷ 11 = 12 exactly
- 142 ÷ 11 is not an integer

So the number is 132, which means \(A = 3\).

ANSWER 6: D

Problem 7:

We check each shape.

### Hexagon
A regular hexagon has 6 equal sides.
If each side is 5 cm, perimeter is:
- \(6 \times 5 = 30\) cm

The wire is only 24 cm, so he cannot make the hexagon.

### Square
A square with area 36 has side length:
- \(\sqrt{36} = 6\) cm

Perimeter:
- \(4 \times 6 = 24\) cm

So he can make the square exactly.

### Right triangle
The legs are 6 and 8, so the hypotenuse is:
- \(\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)

Perimeter:
- \(6 + 8 + 10 = 24\) cm

So he can make the triangle exactly.

Therefore he can make the square and the triangle only.

ANSWER 7: D

Problem 8:

The reciprocal is largest when the original number is smallest, assuming positive numbers.

Compare:
- \(\frac13 \approx 0.333\)
- \(\frac25 = 0.4\)
- \(1\)
- \(5\)
- \(1986\)

The smallest number is \(\frac13\), so it has the largest reciprocal.

Its reciprocal is 3, which is larger than the reciprocals of all the others.

ANSWER 8: A

Problem 9:

Maximum heart rate:
- \(220 - 26 = 194\)

Target heart rate is 80% of that:
- \(0.8 \times 194 = 155.2\)

To the nearest whole number, that is 155.

ANSWER 9: B

Problem 10:

First, Harry uses the parentheses:
- \(8 - (2 + 5) = 8 - 7 = 1\)

So \(H = 1\).

Terry ignores parentheses:
- \(8 - 2 + 5 = 6 + 5 = 11\)

So \(T = 11\).

Difference:
- \(H - T = 1 - 11 = -10\)

ANSWER 10: A

Problem 11:

We need to evaluate:
- \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

These are 10 odd numbers from 81 to 99.

Instead of adding one by one, use the average:
- average of 81 and 99 is \( \frac{81+99}{2} = 90\)

Since there are 10 numbers, their sum is:
- \(10 \times 90 = 900\)

Now multiply by 2:
- \(2 \times 900 = 1800\)

ANSWER 11: E

Problem 12:

We want the percentage of males surveyed who listen.

From the table:
- Total surveyed = 200
- Females = 96
- So males = \(200 - 96 = 104\)

Males who don’t listen = 26, so males who do listen:
- \(104 - 26 = 78\)

Now find the percent of males who listen:
- \(\frac{78}{104} = \frac{3}{4} = 75\%\)

ANSWER 12: E

Problem 1:  
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?  

- Original yellow tint amount = 30 × 0.30 = 9 liters.  
- Add 5 liters of yellow tint → new yellow amount = 9 + 5 = 14 liters.  
- New total volume = 30 + 5 = 35 liters.  
- New percent yellow = (14 / 35) × 100% = (2/5) × 100% = 40%.  

So the correct choice is 40.  

ANSWER 1: C  

---

Problem 2:  
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?  

- Choose an entry window: 6 choices.  
- Choose a different exit window: 5 choices (cannot be the same as entry).  
- Total ways = 6 × 5 = 30.  

ANSWER 2: D  

---

Problem 3:  
Compute: \( \frac{1}{10} + \frac{2}{20} + \frac{3}{30} \).  

Convert each to a common denominator, 60:  
- \( \frac{1}{10} = \frac{6}{60} \)  
- \( \frac{2}{20} = \frac{2}{20} = \frac{6}{60} \) (since multiply top and bottom by 3)  
- \( \frac{3}{30} = \frac{3}{30} = \frac{6}{60} \) (since multiply top and bottom by 2)  

Add:  
\( \frac{6}{60} + \frac{6}{60} + \frac{6}{60} = \frac{18}{60} = \frac{3}{10} = 0.3 \)

So the value is .3.  

ANSWER 3: D  

---

Problem 4:  
Susan had $50 to spend. She spent $12 on food and twice as much on rides. How much left?  

- Food = 12  
- Rides = twice 12 = 24  
- Total spent = 12 + 24 = 36  
- Money left = 50 − 36 = 14  

ANSWER 4: B  

---

Problem 5:  
Two hundred thousand times two hundred thousand equals ?  

\(200{,}000 \times 200{,}000 = (2\times 10^5)(2\times 10^5) = 4 \times 10^{10}\).  
- \(10^{10} = 10{,}000{,}000{,}000\)  
- \(4 \times 10^{10} = 40{,}000{,}000{,}000 = 40\) billion  

ANSWER 5: E  

---

Problem 6:  
Eleven members each paid the same integer amount for a guest speaker. In all, they paid $1A2 (a 3-digit number). Find missing digit A.  

- Total = \(1A2\) means:  
  \(1A2 = 100 + 10A + 2 = 102 + 10A\).  
- This total is divided equally among 11 people:  
  So \(102 + 10A\) must be divisible by 11.  

Check \(102 + 10A \mod 11\):  
- \(102 \div 11 = 9\) remainder 3, so \(102 \equiv 3 \pmod{11}\).  
- \(10A \equiv -A \pmod{11}\) (since 10 ≡ −1 mod 11).  

So:  
\(102 + 10A \equiv 3 - A \equiv 0 \pmod{11}\)  
Thus \(A \equiv 3 \pmod{11}\).  
Since \(A\) is a digit 0–9, \(A=3\).  

ANSWER 6: D  

---

Problem 7:  
Wire length = 24 cm. Which shapes can be made (using all 24 cm of wire)?  

We need the perimeter (total boundary length) of each shape:

**(1) Regular hexagon with side length 5 cm**  
Perimeter = 6 × 5 = 30 cm → too long. Cannot.

**(2) Square with area 36 cm²**  
Side length = √36 = 6 cm  
Perimeter = 4 × 6 = 24 cm → fits. Can.

**(3) Right triangle with legs 6 and 8 cm**  
Right triangle sides are 6 and 8 legs, and hypotenuse:  
\( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)  
Perimeter = 6 + 8 + 10 = 24 cm → fits. Can.

So the shapes possible: **square and triangle only**.  

ANSWER 7: D  

---

Problem 8:  
Which number has the largest reciprocal? Recall: the reciprocal of \(x\) is \(1/x\).  
We compare \(1/3, 2/5, 1, 1/5, 1/1986\).  

Compute roughly:  
- \(1/3 \approx 0.333\)  
- \(2/5 = 0.4\)  
- \(1 = 1\)  
- \(1/5 = 0.2\)  
- \(1/1986\) is tiny  

Largest is 1 (reciprocal of 1).  

ANSWER 8: C  

---

Problem 9:  
Target heart rate = 80% of (maximum heart rate).  
Maximum heart rate = 220 − age.  
Age = 26.

- Maximum = 220 − 26 = 194  
- Target = 0.80 × 194 = 155.2  
Nearest whole number = 155  

ANSWER 9: B  

---

Problem 10:  
Expression: \(8 - (2 + 5)\)  
- Harry computes correctly:  
  \(2 + 5 = 7\), so \(8 - 7 = 1\).  
- Terry ignores parentheses:  
  \(8 - 2 + 5 = 6 + 5 = 11\).  

Difference \(H - T = 1 - 11 = -10\).  

ANSWER 10: A  

---

Problem 11:  
Compute:  
\(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

Notice these are all odd numbers from 81 to 99 stepping by 2.  
There are 10 terms.

Pair from ends:  
- 81 + 99 = 180  
- 83 + 97 = 180  
- 85 + 95 = 180  
- 87 + 93 = 180  
- 89 + 91 = 180  

There are 5 pairs, each sum 180:  
Sum = 5 × 180 = 900  

Now multiply by 2:  
2 × 900 = 1800  

ANSWER 11: E  

---

Problem 12:  
Total surveyed = 200  
Females = 96  
Females who listen = 58  
Males who don’t listen = 26  
Total listeners = 136  

We want: % of males who listen.

Step 1: Find males total.  
Males = 200 − 96 = 104

Step 2: Find males who don’t listen is given = 26.  
So males who listen = 104 − 26 = 78

Step 3: Convert to percent of males who listen:  
Percent = (78 / 104) × 100%  
Simplify 78/104 = 3/4 (divide by 26):  
So percent = (3/4) × 100% = 75%

ANSWER 12: E

Problem 1:
The original mixture is 30 liters. The amount of yellow tint is 30% of 30 liters: $0.30 \times 30 = 9$ liters.
Five liters of yellow tint are added, so the new amount of yellow tint is $9 + 5 = 14$ liters.
The new total volume of the mixture is $30 + 5 = 35$ liters.
The percentage of yellow tint is $\frac{14}{35} \times 100\%$.
Simplifying the fraction: $\frac{14}{35} = \frac{2}{5} = 0.4$.
$0.4 \times 100\% = 40\%$.
ANSWER 1: C

Problem 2:
Georgie has 6 choices for the window he enters. Since he must leave by a different window, he has $6 - 1 = 5$ choices for the window he leaves by.
The total number of ways is $6 \times 5 = 30$.
ANSWER 2: D

Problem 3:
Simplify each fraction:
$1/10 = 0.1$
$2/20 = 1/10 = 0.1$
$3/30 = 1/10 = 0.1$
Summing them: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides, which is $2 \times 12 = 24$.
Total spent: $12 + 24 = 36$.
Amount left: $50 - 36 = 14$.
ANSWER 4: B

Problem 5:
Two hundred thousand is $200,000$ (or $2 \times 10^5$).
$(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
$4 \times 10^{10} = 40,000,000,000$, which is forty billion.
ANSWER 5: E

Problem 6:
The total amount $1A2$ must be divisible by 11.
A number is divisible by 11 if the alternating sum of its digits is divisible by 11.
For $1A2$, the alternating sum is $1 - A + 2 = 3 - A$.
For this to be 0 (the only multiple of 11 possible for a single digit $A$), $3 - A = 0$, so $A = 3$.
Check: $132 / 11 = 12$. This works.
ANSWER 6: D

Problem 7:
Hexagon: Perimeter of a regular hexagon with side 5 is $6 \times 5 = 30$ cm. (Cannot make)
Square: Area is 36, so side is $\sqrt{36} = 6$. Perimeter is $4 \times 6 = 24$ cm. (Can make)
Triangle: Right triangle with legs 6 and 8. Hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. Perimeter is $6 + 8 + 10 = 24$ cm. (Can make)
ANSWER 7: D

Problem 8:
Reciprocals:
A: $1/(1/3) = 3$
B: $1/(2/5) = 2.5$
C: $1/1 = 1$
D: $1/5 = 0.2$
E: $1/1986 \approx 0.0005$
The largest is 3.
ANSWER 8: A

Problem 9:
Max heart rate = $220 - 26 = 194$.
Target heart rate = $80\%$ of $194 = 0.8 \times 194 = 155.2$.
To the nearest whole number, this is 155.
ANSWER 9: B

Problem 10:
$H = 8 - (2 + 5) = 8 - 7 = 1$.
$T = 8 - 2 + 5 = 6 + 5 = 11$.
$H - T = 1 - 11 = -10$.
ANSWER 10: A

Problem 11:
Sum inside parentheses: $81+83+85+87+89+91+93+95+97+99$.
This is an arithmetic progression with 10 terms.
Sum = $\frac{n}{2}(\text{first} + \text{last}) = \frac{10}{2}(81 + 99) = 5 \times 180 = 900$.
The expression is $2 \times 900 = 1800$.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96.
Males = $200 - 96 = 104$.
Males who don't listen = 26.
Males who listen = $104 - 26 = 78$.
Percentage of males who listen = $(78 / 104) \times 100\%$.
$\frac{78}{104} = \frac{39}{52} = \frac{3}{4} = 0.75 = 75\%$.
ANSWER 12: E

Problem 1: The original mixture has volume 30 L, of which 30% is yellow tint, so there are \(0.3 \times 30 = 9\) L of yellow tint. Adding 5 L of yellow tint produces a new total of 14 L of yellow tint and a new mixture volume of 35 L. The new percentage of yellow tint is therefore \((14/35) \times 100\% = 40\%\).

ANSWER 1: C

Problem 2: There are 6 choices for the entry window. For each of those choices there are then exactly 5 remaining choices for a different exit window, giving a total of \(6 \times 5 = 30\) ordered pairs of distinct windows.

ANSWER 2: D

Problem 3: Rewrite each term with a common denominator of 10:  
\(1/10 = 0.1\),  
\(2/20 = 1/10 = 0.1\),  
\(3/30 = 1/10 = 0.1\).  
Their sum is \(0.1 + 0.1 + 0.1 = 0.3\).

ANSWER 3: D

Problem 4: Twice the food cost is \(2 \times 12 = 24\) dollars spent on rides. The total spent is therefore \(12 + 24 = 36\) dollars, leaving \(50 - 36 = 14\) dollars.

ANSWER 4: B

Problem 5: Write the factors in scientific notation:  
\(200000 = 2 \times 10^5\), so  
\((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\),  
which is forty billion.

ANSWER 5: E

Problem 6: The three-digit total \(1A2\) must be divisible by 11. The alternating-sum divisibility rule requires that \(1 - A + 2 = 3 - A\) be a multiple of 11. The only admissible digit \(A\) (0-9) that satisfies the condition is \(A = 3\), giving the total 132, which is exactly \(11 \times 12\).

ANSWER 6: D

Problem 7: A regular hexagon with side 5 cm has perimeter 30 cm, which exceeds the 24 cm length of wire, so it is impossible. A square of area 36 cm² has side length 6 cm and perimeter 24 cm, which matches the wire length. A right triangle with legs 6 cm and 8 cm has hypotenuse 10 cm, so its perimeter is \(6 + 8 + 10 = 24\) cm, which also matches. Consequently only the square and the triangle are possible.

ANSWER 7: D

Problem 8: For positive numbers the reciprocal function is strictly decreasing, so the largest reciprocal belongs to the smallest number in the list, which is \(1/3\).

ANSWER 8: A

Problem 9: The theoretical maximum heart rate is \(220 - 26 = 194\) beats per minute. Eighty percent of that value is \(0.8 \times 194 = 155.2\), which rounds to the nearest whole number 155.

ANSWER 9: B

Problem 10: Harry evaluates the parentheses first and obtains \(8 - (2 + 5) = 1\). Terry omits the parentheses and obtains \(8 - 2 + 5 = 11\). Their difference is therefore \(1 - 11 = -10\).

ANSWER 10: A

Problem 11: The sum inside the parentheses consists of the ten consecutive odd integers from 81 to 99. Their average is \((81 + 99)/2 = 90\), so the sum equals \(10 \times 90 = 900\). Multiplying by the outer factor of 2 yields \(2 \times 900 = 1800\).

ANSWER 11: E

Problem 12: There are 200 people surveyed and 96 females, so there are \(200 - 96 = 104\) males. Of the 96 females, 58 listen, leaving \(96 - 58 = 38\) females who do not listen. Given that 26 males do not listen, the number of males who listen is \(104 - 26 = 78\). The required percentage is therefore \((78/104) \times 100\% = 75\%\).

ANSWER 12: E

Problem 1:
The problem asks us to find the percentage of yellow tint in a new mixture after adding 5 liters of yellow tint to the original 30 liters of paint mixture. The original mixture is 25% red tint, 30% yellow tint, and 45% water.

First, let's calculate the amount of yellow tint in the original mixture. 30% of 30 liters is $0.3 \times 30 = 9$ liters.

After adding 5 liters of yellow tint, the total amount of yellow tint becomes $9 + 5 = 14$ liters.

The total volume of the new mixture is $30 + 5 = 35$ liters.

To find the percentage of yellow tint in the new mixture, we divide the amount of yellow tint by the total volume and multiply by 100: $\frac{14}{35} \times 100 = 40\%$.

Let's examine the choices:
- A. 25: Too low because the percentage of yellow tint increased.
- B. 35: Not the correct calculation.
- C. 40: Matches our calculation.
- D. 45: Too high.
- E. 50: Too high.

The correct percentage is 40%.

ANSWER 1: C

Problem 2:
The problem asks us to determine the number of ways Georgie the Ghost can enter a haunted house through one of six windows and leave by a different window.

To enter, Georgie has 6 choices. After entering, there are 5 remaining windows to choose from for leaving.

The total number of ways to enter and leave is $6 \times 5 = 30$.

Let's examine the choices:
- A. 12: Too low.
- B. 15: Too low.
- C. 18: Too low.
- D. 30: Matches our calculation.
- E. 36: Too high because it would imply Georgie can leave by the same window.

The correct number of ways is 30.

ANSWER 2: D

Problem 3:
The problem requires us to simplify the expression $1⁄10 + 2⁄20 + 3⁄30$.

First, simplify each fraction:
- $1⁄10$ stays the same.
- $2⁄20 = 1⁄10$.
- $3⁄30 = 1⁄10$.

Adding them together: $1⁄10 + 1⁄10 + 1⁄10 = 3⁄10 = 0.3$.

Let's examine the choices:
- A. .1: Too low.
- B. .123: Not the correct sum.
- C. .2: Too low.
- D. .3: Matches our calculation.
- E. .6: Too high.

The correct sum is 0.3.

ANSWER 3: D

Problem 4:
The problem asks us to find out how many dollars Susan had left after spending money on food and rides at the carnival. She started with $50, spent $12 on food, and twice as much on rides.

First, calculate the amount spent on rides: $12 \times 2 = $24.

Total amount spent is $12 + $24 = $36.

The amount left is $50 - $36 = $14.

Let's examine the choices:
- A. 12: Too low.
- B. 14: Matches our calculation.
- C. 26: Too high.
- D. 38: Too high.
- E. 50: Incorrect because she spent money.

The correct amount left is $14.

ANSWER 4: B

Problem 5:
The problem asks us to calculate $200,000 \times 200,000$.

$200,000 = 2 \times 10^5$, so $200,000 \times 200,000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10} = 40,000,000,000$.

Let's examine the choices:
- A. four hundred thousand: Too low.
- B. four million: Too low.
- C. forty thousand: Too low.
- D. four hundred million: Too low.
- E. forty billion: Matches our calculation.

The correct result is forty billion.

ANSWER 5: E

Problem 6:
The problem requires us to find the missing digit A in the 3-digit number $1A2 that represents the total amount paid by 11 members of the Math Club.

The total amount paid is divisible by 11 since there are 11 members who paid the same amount.

To check divisibility by 11, we use the rule: the difference between the sum of digits at odd places and the sum of digits at even places should be divisible by 11.

For $1A2, the sum of digits at odd places is $1 + 2 = 3$, and the sum at even places is $A$. The difference is $3 - A$.

For $3 - A$ to be divisible by 11 (or 0, since $A$ is a single digit), $A = 3$ makes $3 - A = 0$, which is divisible by 11.

Let's examine the choices:
- A. 0: Not correct because $3 - 0 = 3$, not divisible by 11.
- B. 1: Not correct because $3 - 1 = 2$, not divisible by 11.
- C. 2: Not correct because $3 - 2 = 1$, not divisible by 11.
- D. 3: Correct because $3 - 3 = 0$, which is divisible by 11.
- E. 4: Not correct because $3 - 4 = -1$, not divisible by 11.

The missing digit A is 3, but we should verify if 132 is indeed divisible by 11. $132 / 11 = 12$, so it is.

ANSWER 6: D

Problem 7:
The problem asks us to determine which shapes Haruki can form using a 24 cm long wire.

1. A regular hexagon with side length 5 cm: The perimeter is $6 \times 5 = 30$ cm, which is more than 24 cm. So, Haruki cannot form this hexagon.

2. A square of area 36 cm$^2$: The side length is $\sqrt{36} = 6$ cm. The perimeter is $4 \times 6 = 24$ cm, which matches the wire length. So, Haruki can form this square.

3. A right triangle with legs 6 and 8 cm: Using the Pythagorean theorem, the hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm. The perimeter is $6 + 8 + 10 = 24$ cm, which matches the wire length. So, Haruki can form this triangle.

Let's examine the choices:
- A. Triangle only: Incorrect because Haruki can also form the square.
- B. Hexagon and square only: Incorrect because Haruki cannot form the hexagon.
- C. Hexagon and triangle only: Incorrect because Haruki cannot form the hexagon.
- D. Square and triangle only: Matches our findings.
- E. Hexagon, triangle, and square: Incorrect because Haruki cannot form the hexagon.

The correct shapes Haruki can form are the square and the triangle.

ANSWER 7: D

Problem 8:
The problem asks us to find which of the given numbers has the largest reciprocal.

The reciprocal of a number $x$ is $1/x$. The larger $x$ is, the smaller its reciprocal is, and vice versa.

Let's examine the choices:
- A. $1⁄3$: The reciprocal is 3.
- B. $2⁄5$: The reciprocal is $5/2 = 2.5$.
- C. 1: The reciprocal is 1.
- D. 5: The reciprocal is $1/5 = 0.2$.
- E. 1986: The reciprocal is $1/1986$, very small.

Comparing the reciprocals: $3 > 2.5 > 1 > 0.2 > 1/1986$. So, $1/3$ has the largest reciprocal.

Let's examine the choices again:
- A. $1⁄3$: Correct.
- B. $2⁄5$: Not correct.
- C. 1: Not correct.
- D. 5: Not correct.
- E. 1986: Not correct.

The number with the largest reciprocal is $1/3$.

ANSWER 8: A

Problem 9:
The proble

Problem 1:
Original mixture: 30 L. Yellow tint = 30% of 30 = 0.30 × 30 = 9 L.
Add 5 L yellow: new yellow = 9 + 5 = 14 L, new total volume = 30 + 5 = 35 L.
Percent yellow = (14/35) × 100% = 40%.
ANSWER 1: C

Problem 2:
6 choices for entry window, 5 remaining choices for a different exit window.
Total ways = 6 × 5 = 30.
ANSWER 2: D

Problem 3:
1/10 + 2/20 + 3/30 = 1/10 + 1/10 + 1/10 = 3/10 = 0.3.
ANSWER 3: D

Problem 4:
Spent on food: $12. Spent on rides: 2 × $12 = $24.
Total spent = $12 + $24 = $36. Left = $50 − $36 = $14.
ANSWER 4: B

Problem 5:
200,000 × 200,000 = (2×10⁵) × (2×10⁵) = 4×10¹⁰ = 40,000,000,000, which is forty billion.
ANSWER 5: E

Problem 6:
Total paid is $1A2, a multiple of 11. Divisibility by 11: (1 − A + 2) = 3 − A must be a multiple of 11.
With A a digit 0–9, 3 − A = 0 ⇒ A = 3. (Check: 132 ÷ 11 = 12.)
ANSWER 6: D

Problem 7:
Wire length = 24 cm.
- Regular hexagon, side 5 cm: perimeter = 6×5 = 30 cm > 24 cm → cannot make.
- Square, area 36 cm²: side = 6 cm, perimeter = 4×6 = 24 cm → can make.
- Right triangle, legs 6 and 8 cm: hypotenuse = 10 cm, perimeter = 6+8+10 = 24 cm → can make.
So square and triangle only.
ANSWER 7: D

Problem 8:
Reciprocals: 1/(1/3)=3, 1/(2/5)=2.5, 1/1=1, 1/5=0.2, 1/1986≈0.0005. Largest is 3, from 1/3.
ANSWER 8: A

Problem 9:
Maximum heart rate = 220 − 26 = 194 bpm.
Target = 80% of 194 = 0.8 × 194 = 155.2 → 155 to nearest whole number.
ANSWER 9: B

Problem 10:
Harry: H = 8 − (2+5) = 8 − 7 = 1.
Terry: T = 8 − 2 + 5 = 6 + 5 = 11.
H − T = 1 − 11 = −10.
ANSWER 10: A

Problem 11:
Sum inside parentheses: 10 terms, average = (81+99)/2 = 90, sum = 10×90 = 900.
2 × 900 = 1800.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96 ⇒ Males = 200 − 96 = 104.
Total listeners = 136, female listeners = 58 ⇒ male listeners = 136 − 58 = 78.
Percent of males who listen = (78/104) × 100% = 75%.
ANSWER 12: E

Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint.

**Original mixture (30 liters):**
- Red tint: 25% of 30 = 7.5 liters
- Yellow tint: 30% of 30 = 9 liters
- Water: 45% of 30 = 13.5 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total mixture: 30 + 5 = 35 liters

**New percentage of yellow tint:**
14 / 35 = 2/5 = 0.40 = 40%

Tempting wrong answers: Choice B (35) might come from just adding 5 to the original 30%, but we must recalculate based on the new total.

ANSWER 1: C

---

Problem 2:
Georgie enters through one window and leaves through a *different* window.

- Number of choices for the entry window: 6
- Number of choices for the exit window (must be different): 5
- Total ways: 6 × 5 = 30

Tempting wrong answers: Choice E (36) would be 6 × 6, which allows entering and leaving through the same window. Choice A (12) would be 6 + 6, which doesn't account for all combinations.

ANSWER 2: D

---

Problem 3:
Simplify each fraction first:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

Tempting wrong answers: Choice B (.123) might come from concatenating the digits 1, 2, 3 instead of actually computing. Choice C (.2) might come from miscounting the terms.

ANSWER 3: D

---

Problem 4:
Susan starts with $50.
- Spent on food: $12
- Spent on rides: twice the food amount = 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Amount left: $50 − $36 = $14

Tempting wrong answers: Choice C (26) might come from subtracting only the food ($50 − $24 = $26, forgetting food). Choice D (38) comes from subtracting only $12.

ANSWER 4: B

---

Problem 5:
200,000 × 200,000

Write in scientific notation:
(2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰

4 × 10¹⁰ = 40,000,000,000 = forty billion

Tempting wrong answers: Choice A (four hundred thousand) comes from adding instead of multiplying. Choice D (four hundred million) comes from miscounting zeros.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, so the total $1A2 must be divisible by 11.

**Divisibility rule for 11:** The alternating sum of digits must be a multiple of 11.
Alternating sum: 1 − A + 2 = 3 − A

For 3 − A to be divisible by 11 (and A to be a single digit 0–9):
3 − A = 0 → A = 3

**Check:** $132 ÷ 11 = $12 per member. This is a valid integer amount.

Tempting wrong answers: Other values of A don't make 3 − A a multiple of 11 within the digit range.

ANSWER 6: D

---

Problem 7:
Wire length = 24 cm. We check if each shape's perimeter ≤ 24 cm.

**Regular hexagon (side 5 cm):**
Perimeter = 6 × 5 = 30 cm. Since 30 > 24, he **cannot** make this.

**Square (area 36 cm²):**
Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Since 24 = 24, he **can** make this.

**Right triangle (legs 6 and 8 cm):**
Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Perimeter = 6 + 8 + 10 = 24 cm. Since 24 = 24, he **can** make this.

Haruki can make the square and triangle only.

Tempting wrong answers: Choice E includes the hexagon, but its perimeter exceeds the wire length.

ANSWER 7: D

---

Problem 8:
The reciprocal of a number x is 1/x. The *smaller* the positive number, the *larger* its reciprocal.

| Choice | Number | Reciprocal |
|--------|--------|------------|
| A | 1/3 | 3 |
| B | 2/5 | 5/2 = 2.5 |
| C | 1 | 1 |
| D | 5 | 1/5 = 0.2 |
| E | 1986 | 1/1986 ≈ 0.0005 |

The largest reciprocal is 3, which comes from 1/3.

Tempting wrong answers: Choice E (1986) is the largest number, so it has the *smallest* reciprocal.

ANSWER 8: A

---

Problem 9:
Athlete's age = 26 years.

**Step 1:** Maximum heart rate = 220 − 26 = 194 beats per minute.

**Step 2:** Target heart rate = 80% of 194 = 0.80 × 194 = 155.2

**Step 3:** Round to nearest whole number: 155

Tempting wrong answers: Choice C (176) might come from taking 80% of 220 without subtracting the age. Choice D (194) is the maximum heart rate itself.

ANSWER 9: B

---

Problem 10:
**Harry (correct calculation):**
H = 8 − (2 + 5) = 8 − 7 = 1

**Terry (ignores parentheses):**
T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:**
H − T = 1 − 11 = −10

Tempting wrong answers: Choice E (10) is |H − T|, but the question asks for H − T specifically, which is negative since Terry's answer is larger.

ANSWER 10: A

---

Problem 11:
We need: 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

The sum inside is an arithmetic sequence:
- First term: 81, Last term: 99
- Number of terms: 10
- Sum = (number of terms) × (first + last) / 2 = 10 × (81 + 99) / 2 = 10 × 180 / 2 = 10 × 90 = 900

Multiply by 2: 2 × 900 = 1800

Tempting wrong answers: Choice C (1700) might come from miscounting terms or miscalculating the average.

ANSWER 11: E

---

Problem 12:
Let me organize the data from the survey:

- Total surveyed: 200
- Females: 96 → Males: 200 − 96 = 104
- Total listeners: 136
- Females who listen: 58 → Males who listen: 136 − 58 = 78
- Males who don't listen: 26 (Check: 78 + 26 = 104 ✓)

**Percentage of males who listen:**
(78 / 104) × 100% = (3/4) × 100% = 75%

Tempting wrong answers: Choice C (52) might come from computing 104/200 (percentage of total who are male). Choice D (55) might come from 78/136 (percentage of listeners who are male).

ANSWER 12: E

 Problem 1:
We start with 30 liters of paint. The original amount of yellow tint is 30% of 30 liters:
\[
0.30 \times 30 = 9 \text{ liters.}
\]
Five liters of yellow tint are added, so the new amount of yellow tint is:
\[
9 + 5 = 14 \text{ liters.}
\]
The new total volume of the mixture is:
\[
30 + 5 = 35 \text{ liters.}
\]
The percent of yellow tint in the new mixture is:
\[
\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%.
\]
(Choice A is the original red percentage, D is the original water percentage, and E would require 17.5 liters of yellow, which we do not have.)
ANSWER 1: C

Problem 2:
Georgie chooses one of 6 windows to enter. Since he must leave through a different window, there are 5 remaining choices for his exit. By the multiplication principle:
\[
6 \times 5 = 30 \text{ ways.}
\]
(Choice E, 36, would count entering and leaving through the same window; the other choices are too small.)
ANSWER 2: D

Problem 3:
Simplify each fraction:
\[
\frac{2}{20} = \frac{1}{10}, \qquad \frac{3}{30} = \frac{1}{10}.
\]
So the sum is:
\[
\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3.
\]
(Choice A is just one term; C is 0.2; E is 0.6.)
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides:
\[
2 \times 12 = 24 \text{ dollars.}
\]
Total spent:
\[
12 + 24 = 36 \text{ dollars.}
\]
Amount left:
\[
50 - 36 = 14 \text{ dollars.}
\]
(Choice C is the total spent; D is the amount left if she had spent only $12.)
ANSWER 4: B

Problem 5:
\[
200{,}000 \times 200{,}000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}.
\]
\[
4 \times 10^{10} = 40{,}000{,}000{,}000,
\]
which is read as forty billion.
(Choice D is $4 \times 10^8$; B is $4 \times 10^6$.)
ANSWER 5: E

Problem 6:
The total amount paid is the three-digit number $1A2$, which equals:
\[
100 + 10A + 2 = 102 + 10A.
\]
Since 11 members each paid the same integer amount, $102 + 10A$ must be divisible by 11. Testing the choices:
- $A=0$: $102 \div 11 \approx 9.27$ (not an integer)
- $A=1$: $112 \div 11 \approx 10.18$ (not an integer)
- $A=2$: $122 \div 11 \approx 11.09$ (not an integer)
- $A=3$: $132 \div 11 = 12$ (integer!)
- $A=4$: $142 \div 11 \approx 12.9$ (not an integer)
Thus the missing digit is 3.
ANSWER 6: D

Problem 7:
The wire is 24 cm long, so a shape is possible only if its perimeter is at most 24 cm.
- **Regular hexagon, side 5 cm:** Perimeter $= 6 \times 5 = 30$ cm. $30 > 24$, so impossible.
- **Square of area 36 cm²:** Side $= \sqrt{36} = 6$ cm. Perimeter $= 4 \times 6 = 24$ cm. Exactly fits, so possible.
- **Right triangle with legs 6 and 8 cm:** Hypotenuse $= \sqrt{6^2 + 8^2} = \sqrt{100} = 10$ cm. Perimeter $= 6 + 8 + 10 = 24$ cm. Exactly fits, so possible.
Only the square and the triangle can be formed.
ANSWER 7: D

Problem 8:
For positive numbers, the reciprocal function $f(x) = \frac{1}{x}$ is decreasing, so the smallest number has the largest reciprocal. Compare the two smallest choices:
\[
\frac{1}{3} \approx 0.333\ldots, \qquad \frac{2}{5} = 0.4.
\]
Since $\frac{1}{3} < \frac{2}{5} < 1 < 5 < 1986$, the smallest number is $\frac{1}{3}$. Its reciprocal is $3$, which is larger than the reciprocals of all the other choices.
ANSWER 8: A

Problem 9:
First find the theoretical maximum heart rate:
\[
220 - 26 = 194 \text{ beats per minute.}
\]
The target rate is 80% of this:
\[
0.80 \times 194 = 155.2.
\]
Rounded to the nearest whole number, this is 155.
(Choice C is the maximum heart rate itself.)
ANSWER 9: B

Problem 10:
Harry computes correctly:
\[
H = 8 - (2 + 5) = 8 - 7 = 1.
\]
Terry ignores the parentheses:
\[
T = 8 - 2 + 5 = 6 + 5 = 11.
\]
The difference is:
\[
H - T = 1 - 11 = -10.
\]
ANSWER 10: A

Problem 11:
The ten numbers inside the parentheses form an arithmetic sequence: 81, 83, …, 99. The sum of an arithmetic sequence is:
\[
\frac{\text{number of terms}}{2} \times (\text{first} + \text{last}) = \frac{10}{2} \times (81 + 99) = 5 \times 180 = 900.
\]
Now multiply by the factor of 2 outside:
\[
2 \times 900 = 1800.
\]
(One can also pair them: $(81+99)=180$, $(83+97)=180$, etc., giving five pairs that each sum to 180.)
ANSWER 11: E

Problem 12:
Total surveyed = 200, and females surveyed = 96, so:
\[
\text{Males surveyed} = 200 - 96 = 104.
\]
Total listeners = 136, and females who listen = 58, so:
\[
\text{Males who listen} = 136 - 58 = 78.
\]
(We can check: total non-listeners = 64, males who don't listen = 26, so females who don't listen = 38; $58+38=96$ and $78+26=104$, which checks out.)

The percentage of males who listen is:
\[
\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%.
\]
ANSWER 12: E

Problem 1:
A mixture of 30 liters is 25% red, 30% yellow, and 45% water. Five liters of yellow tint are added. What is the percent of yellow tint in the new mixture?

**Setup:** Find the amount of yellow tint originally, add 5 liters, then find the new percentage.

**Original yellow tint:** 30% of 30 = 0.30 × 30 = 9 liters

**After adding 5 liters of yellow tint:**
- New yellow tint: 9 + 5 = 14 liters
- New total volume: 30 + 5 = 35 liters
- New percentage: 14/35 = 2/5 = 40%

**Checking wrong choices:** 25% would be if we just added the original percentages; 35% would be if we just added 5% to 30%; 45% and 50% are too high.

ANSWER 1: C

---

Problem 2:
Six windows. Georgie enters by one and leaves by a different one. How many ways?

**Setup:** Choose an entry window (6 choices), then choose a different exit window (5 choices).

**Calculation:** 6 × 5 = 30 ways

**Checking wrong choices:** 15 would be if we divided by 2 (treating order as irrelevant); 12 would be 6×2; 18 would be 6×3; 36 would be 6×6 (allowing same window).

ANSWER 2: D

---

Problem 3:
1/10 + 2/20 + 3/30 = ?

**Simplify each fraction:**
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

**Sum:** 1/10 + 1/10 + 1/10 = 3/10 = 0.3

**Checking wrong choices:** 0.1 would be just one term; 0.2 would be two terms; 0.6 would be adding numerators and denominators incorrectly.

ANSWER 3: D

---

Problem 4:
Susan had $50. She spent $12 on food and twice as much on rides. How much left?

**Setup:** Food = $12, Rides = 2 × $12 = $24

**Total spent:** $12 + $24 = $36

**Amount left:** $50 - $36 = $14

**Checking wrong choices:** 12 is just the food amount; 26 is 50-24 (forgetting food); 38 is 50-12 (forgetting rides).

ANSWER 4: B

---

Problem 5:
200,000 × 200,000 = ?

**Setup:** Express in scientific notation: 200,000 = 2 × 10⁵

**Calculation:** (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion

**Checking wrong choices:** Four hundred thousand is 4×10⁵; four million is 4×10⁶; forty thousand is 4×10⁴; four hundred million is 4×10⁸.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, totaling $1A2. Find the missing digit A.

**Setup:** The total $1A2 must be divisible by 11.

**Divisibility test for 11:** A number is divisible by 11 if the alternating sum of its digits is divisible by 11.

For 1A2: (1 + 2) - A = 3 - A must be divisible by 11.

Since A is a single digit (0-9), 3 - A can only equal 0 (the only multiple of 11 in range -6 to 3).

So A = 3, giving $132 = 11 × $12 ✓

**Checking:** With A=0: $102 ÷ 11 ≈ 9.27 (not integer); A=1: $112 ÷ 11 ≈ 10.18 (not integer); A=3: $132 ÷ 11 = 12 ✓

ANSWER 6: D

---

Problem 7:
Wire is 24 cm. Which shapes can be made?

**Regular hexagon with side 5 cm:** Perimeter = 6 × 5 = 30 cm. Need 30 cm, only have 24 cm. ❌

**Square with area 36 cm²:** Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Need 24 cm, have 24 cm. ✓

**Right triangle with legs 6 and 8 cm:** Hypotenuse = √(36 + 64) = √100 = 10 cm. Perimeter = 6 + 8 + 10 = 24 cm. Need 24 cm, have 24 cm. ✓

**Conclusion:** Can make the square and triangle, but not the hexagon.

ANSWER 7: D

---

Problem 8:
Which number has the largest reciprocal?

**Find each reciprocal:**
- A. 1/3 → reciprocal = 3
- B. 2/5 → reciprocal = 5/2 = 2.5
- C. 1 → reciprocal = 1
- D. 5 → reciprocal = 1/5 = 0.2
- E. 1986 → reciprocal ≈ 0.0005

**Key insight:** Smaller positive numbers have larger reciprocals. The smallest number listed is 1/3, so it has the largest reciprocal (3).

ANSWER 8: A

---

Problem 9:
Target heart rate = 80% of (220 - age). Find target for 26-year-old.

**Maximum heart rate:** 220 - 26 = 194 bpm

**Target heart rate:** 80% of 194 = 0.80 × 194 = 155.2 ≈ 155 bpm

**Checking wrong choices:** 134 would be about 69%; 176 would be about 91%; 194 is the max (100%); 243 is impossible.

ANSWER 9: B

---

Problem 10:
Harry calculates 8 − (2 + 5) correctly. Terry calculates 8 − 2 + 5. Find H − T.

**Harry's answer:** H = 8 − (2 + 5) = 8 − 7 = 1

**Terry's answer:** T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:** H − T = 1 − 11 = −10

**Checking wrong choices:** −6 would be if Terry got 7; 0 would be if they got the same answer; 6 or 10 would be T − H.

ANSWER 10: A

---

Problem 11:
2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99) = ?

**Setup:** This is 10 consecutive odd numbers. The average of the first and last is (81 + 99)/2 = 90.

**Sum of the 10 numbers:** 10 × 90 = 900

**Multiply by 2:** 2 × 900 = 1800

**Alternative verification:** Pair the numbers: (81+99) + (83+97) + (85+95) + (87+93) + (89+91) = 180 + 180 + 180 + 180 + 180 = 900. Then 2 × 900 = 1800.

ANSWER 11: E

---

Problem 12:
Find the percentage of males who listen to the station.

**Given information:**
- Total surveyed: 200
- Females: 96
- Females who listen: 58
- Males who don't listen: 26
- Total listeners: 136
- Total non-listeners: 64

**Find males:** Males = 200 - 96 = 104

**Find males who listen:** Males who listen = Total listeners - Females who listen = 136 - 58 = 78

**Verification:** Males who don't listen = 104 - 78 = 26 ✓ (matches given)

**Percentage of males who listen:** 78/104 × 100 = 75%

**Checking wrong choices:** 39% would be 78/200 (using total instead of males); 48% would be about 50/104; 52% would be about 54/104; 55% would be about 57/104.

ANSWER 12: E

**Problem 1**

The original mixture is 30 L with 30 % yellow tint → yellow = 0.30·30 = 9 L.  
Add 5 L of yellow → total yellow = 9 + 5 = 14 L.  
Total volume after addition = 30 + 5 = 35 L.  

Percent yellow = (14/35)·100 % = 40 %.

**Answer 1: C**

---

**Problem 2**

Choose an entry window (6 ways) and a different exit window (5 ways).  
Number of ways = 6 × 5 = 30.

**Answer 2: D**

---

**Problem 3**

\[
\frac{1}{10}=0.1,\qquad\frac{2}{20}=0.1,\qquad\frac{3}{30}=0.1
\]
Sum = 0.1 + 0.1 + 0.1 = 0.3.

**Answer 3: D**

---

**Problem 4**

Food cost = \$12.  
Rides cost = twice food = 2·12 = \$24.  
Total spent = 12 + 24 = \$36.  
Money left = 50 − 36 = \$14.

**Answer 4: B**

---

**Problem 5**

\(200{,}000 \times 200{,}000 = (2\times10^{5})^{2}=4\times10^{10}=40{,}000{,}000{,}000\).

That is 40 billion.

**Answer 5: E**

---

**Problem 6**

Total paid = \$1A2 = 100 + 10A + 2 = 102 + 10A.  
Since the 11 members each paid an integer amount, 102 + 10A must be divisible by 11.

\(102\equiv3\pmod{11}\).  
We need \(10A\equiv-3\equiv8\pmod{11}\).  
Because \(10\equiv-1\pmod{11}\), this is \(-A\equiv8\pmod{11}\) → \(A\equiv3\pmod{11}\).  
A digit, so \(A=3\).

**Answer 6: D**

---

**Problem 7**

- Hexagon: perimeter = 6·5 = 30 cm > 24 cm → impossible.  
- Square: side = √36 = 6 cm, perimeter = 4·6 = 24 cm → possible.  
- Right triangle: legs 6 cm, 8 cm, hypotenuse = √(6²+8²)=10 cm, perimeter = 6+8+10 = 24 cm → possible.

Only the square and triangle can be made.

**Answer 7: D**

---

**Problem 8**

Reciprocals:
- \(1/3 → 3\)  
- \(2/5 → 5/2 = 2.5\)  
- \(1 → 1\)  
- \(5 → 1/5 = 0.2\)  
- \(1986 → 1/1986 ≈ 0.0005\)

Largest reciprocal is 3, from \(1/3\).

**Answer 8: A**

---

**Problem 9**

Maximum heart rate = 220 − 26 = 194 bpm.  
Target = 80 % of 194 = 0.8·194 = 155.2 → rounded to 155 bpm.

**Answer 9: B**

---

**Problem 10**

Harry: \(8-(2+5)=8-7=1\).  
Terry (ignoring parentheses): \(8-2+5 = 6+5 = 11\).  
Difference \(H-T = 1-11 = -10\).

**Answer 10: A**

---

**Problem 11**

Sum of the ten terms: average = (81+99)/2 = 90, so sum = 90·10 = 900.  
Multiply by 2: \(2·900 = 1800\).

**Answer 11: E**

---

**Problem 12**

Total surveyed = 200.  
Females = 96 → Males = 200 − 96 = 104.  

Female listeners = 58 → Male listeners = 136 − 58 = 78.  

Percent of males who listen = (78/104)·100 % = (39/52)·100 % = 75 %.

**Answer 12: E**

---

**Final Answers**

ANSWER 1: C  
ANSWER 2: D  
ANSWER 3: D  
ANSWER 4: B  
ANSWER 5: E  
ANSWER 6: D  
ANSWER 7: D  
ANSWER 8: A  
ANSWER 9: B  
ANSWER 10: A  
ANSWER 11: E  
ANSWER 12: E

Problem 1:  
A 30-liter paint mixture is 25% red tint, 30% yellow tint, and 45% water. Five liters of yellow tint are added.  
Original yellow tint: $0.30 \times 30 = 9$ liters.  
After adding 5 liters: $9 + 5 = 14$ liters yellow.  
New total volume: $30 + 5 = 35$ liters.  
Percent yellow: $\frac{14}{35} = 0.40 = 40\%$.  
Tempting wrong choices: 35% (original yellow percent) or 45% (water percent).  
ANSWER 1: D  

Problem 2:  
Georgie enters through one of six windows and exits through a different one.  
Entry choices: 6. Exit choices: 5 (excluding entry window).  
Total ways: $6 \times 5 = 30$.  
Tempting wrong choices: 15 (if order didn’t matter) or 36 (if same window allowed).  
ANSWER 2: D  

Problem 3:  
$\frac{1}{10} + \frac{2}{20} + \frac{3}{30} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3$.  
Tempting wrong choices: 0.1 (one term) or 0.2 (two terms).  
ANSWER 3: D  

Problem 4:  
Susan starts with \$50.  
Food: \$12.  
Rides: $2 \times 12 = \$24$.  
Total spent: $12 + 24 = \$36$.  
Left: $50 - 36 = \$14$.  
Tempting wrong choices: \$26 (if spent \$24 total) or \$38 (if spent \$12 total).  
ANSWER 4: B  

Problem 5:  
$200,000 \times 200,000 = 40,000,000,000 = 40$ billion.  
Tempting wrong choices: 400,000 (added zeros) or 4 million (multiplied digits).  
ANSWER 5: E  

Problem 6:  
Total paid: \$1A2, an integer divisible by 11.  
$1A2 \mod 11 = 0$.  
$1 + 2 - A \equiv 0 \mod 11 \Rightarrow 3 - A \equiv 0 \mod 11 \Rightarrow A = 3$.  
Tempting wrong choices: 0, 1, 2, or 4 (incorrect divisibility).  
ANSWER 6: E (Wait, let's recheck: 1A2 must be divisible by 11. 132 ÷ 11 = 12, so A=3. But 132 is 11×12, so correct. However, the answer choices are A=0,B=1,C=2,D=3,E=4. So D is 3. So correct answer is D, not E. I made a mistake in the initial answer. Correcting:)  
**Correction**: $1A2 = 132$ when $A=3$, and $132 \div 11 = 12$, so $A=3$.  
ANSWER 6: D  

Problem 7:  
Wire length: 24 cm.  
- Regular hexagon with side 5 cm: Perimeter $6 \times 5 = 30$ cm (not 24).  
- Square with area 36 cm²: Side $6$ cm, perimeter $24$ cm.  
- Right triangle with legs 6 and 8 cm: Hypotenuse 10 cm, perimeter $6+8+10=24$ cm.  
Can make square and triangle.  
Tempting wrong choices: Hexagon (incorrect perimeter).  
ANSWER 7: D  

Problem 8:  
Largest reciprocal means smallest original number.  
$\frac{1}{3} \approx 0.333$, $\frac{2}{5} = 0.4$, $1$, $5$, $1986$.  
Smallest: $\frac{1}{3}$. Reciprocal: 3.  
Others: 2.5, 1, 0.2, 0.0005.  
Tempting wrong choices: $\frac{2}{5}$ (larger than $\frac{1}{3}$).  
ANSWER 8: A  

Problem 9:  
Maximum heart rate: $220 - 26 = 194$.  
Target: $0.80 \times 194 = 155.2 \approx 155$.  
Tempting wrong choices: 134 (70%), 176 (90%), or 243 (incorrect max).  
ANSWER 9: B  

Problem 10:  
Harry: $8 - (2 + 5) = 1$.  
Terry: $8 - 2 + 5 = 11$.  
Difference: $1 - 11 = -10$.  
Tempting wrong choices: 10 (absolute difference) or 0 (if parentheses ignored by both).  
ANSWER 10: A  

Problem 11:  
Sum: $81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99$.  
Average: 90, 10 terms: $90 \times 10 = 900$.  
$2 \times 900 = 1800$.  
Tempting wrong choices: 1600 (min+max=180, times 5=900, times 2=1800, but if miscount terms).  
ANSWER 11: E  

Problem 12:  
Total surveyed: 200.  
Females: 96, so males: $200 - 96 = 104$.  
Total listeners: 136.  
Female listeners: 58, so male listeners: $136 - 58 = 78$.  
Percent male listeners: $\frac{78}{104} = 0.75 = 75\%$.  
Tempting wrong choices: 39 (male non-listeners percent) or 52 (incorrect calculation).  
ANSWER 12: E

### Problem 1:
What is the percentage of yellow tint in the new 35-liter paint mixture, after adding 5 liters of yellow tint to an original 30-liter mixture that was 30% yellow?
First, calculate the original volume of yellow tint: 30% of 30 liters = \(0.3 \times 30 = 9\) liters. Add the 5 new liters of yellow tint to get \(9 + 5 = 14\) liters of yellow in the new mixture. The total new volume of the mixture is \(30 + 5 = 35\) liters. The percentage of yellow is \(\frac{14}{35} = 0.4 = 40\%\). Tempting wrong answers come from forgetting to update the total mixture volume when calculating the new percentage, or miscalculating the original yellow volume.
ANSWER 1: C

---

### Problem 2:
How many ways can Georgie enter through one window and leave through a different window, out of 6 total windows?
Georgie has 6 choices for an entry window. Since he must leave through a different window, he has 5 remaining choices for an exit window. The total number of ways is \(6 \times 5 = 30\). A common wrong answer is 15, which incorrectly treats entry/exit as unordered pairs, when order matters (entering window A and leaving B is different from entering B and leaving A).
ANSWER 2: D

---

### Problem 3:
Simplify \(\frac{1}{10} + \frac{2}{20} + \frac{3}{30}\):
Simplify each fraction first: \(\frac{2}{20} = \frac{1}{10}\) and \(\frac{3}{30} = \frac{1}{10}\). Add them together: \(\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3\). A common wrong answer is 0.1, which incorrectly adds numerators and denominators directly instead of simplifying first.
ANSWER 3: D

---

### Problem 4:
How much money does Susan have left after spending $12 on food and twice that amount on rides, starting with $50?
First calculate the amount spent on rides: twice $12 is \(2 \times 12 = 24\) dollars. Total spent is \(12 + 24 = 36\) dollars. The amount left is \(50 - 36 = 14\) dollars. A common wrong answer is 26, which incorrectly uses $12 instead of $24 for ride costs.
ANSWER 4: B

---

### Problem 5:
Calculate \(200,000 \times 200,000\):
Rewrite 200,000 as \(2 \times 10^5\). Multiplying two copies gives \((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\), which equals 40,000,000,000, or forty billion. Wrong answers come from miscounting the number of zeros in the product.
ANSWER 5: E

---

### Problem 6:
The 3-digit total cost \(1A2\) dollars is divisible by 11 (11 members paid the same integer amount). What is the missing digit A?
The divisibility rule for 11 states that alternating sum of digits must be divisible by 11. For 1A2, the alternating sum is \(1 - A + 2 = 3 - A\). The only value of A (0-9) that makes this sum divisible by 11 is A=3, since \(3-3=0\), which is divisible by 11. Verify: \(132 \div 11 = 12\), a valid integer.
ANSWER 6: D

---

### Problem 7:
Haruki has a 24cm wire. Which of the three shapes can he form? Calculate the perimeter of each shape:
1. Regular hexagon with side 5cm: perimeter = \(6 \times 5 = 30\)cm, which is longer than 24cm, so impossible.
2. Square with area 36cm²: side length = \(\sqrt{36}=6\)cm, perimeter = \(4 \times 6 = 24\)cm, which fits.
3. Right triangle with legs 6cm and 8cm: hypotenuse = 10cm (6-8-10 Pythagorean triple), perimeter = \(6+8+10=24\)cm, which fits.
Only the square and triangle can be formed.
ANSWER 7: D

---

### Problem 8:
Which number has the largest reciprocal? The reciprocal of \(x\) is \(\frac{1}{x}\), so smaller values of \(x\) produce larger reciprocals. List the values of all options:
A. \(\frac{1}{3} \approx 0.333\), B. \(\frac{2}{5}=0.4\), C.1, D.5, E.1986. The smallest value is \(\frac{1}{3}\), whose reciprocal is 3, the largest of all options.
ANSWER 8: A

---

### Problem 9:
Calculate the target heart rate for a 26-year-old athlete: target is 80% of max heart rate, where max = 220 - age.
First calculate max heart rate: \(220 - 26 = 194\) bpm. 80% of 194 is \(0.8 \times 194 = 155.2\), which rounds to 155 bpm. A common wrong answer is 176, which uses 220 instead of the correct max heart rate of 194.
ANSWER 9: B

---

### Problem 10:
Calculate \(H-T\), where H is Harry's correct answer to \(8-(2+5)\), and T is Terry's incorrect answer of \(8-2+5\).
Harry's answer: \(8 - 7 = 1 = H\). Terry's answer: \(6 + 5 = 11 = T\). The difference \(H-T = 1 - 11 = -10\). A common wrong answer is 10, which calculates \(T-H\) instead of \(H-T\).
ANSWER 10: A

---

### Problem 11:
Simplify \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\):
Pair the terms inside the parentheses to simplify addition: \((81+99)+(83+97)+(85+95)+(87+93)+(89+91) = 5 \times 180 = 900\). Multiply by 2: \(2 \times 900 = 1800\).
ANSWER 11: E

---

### Problem 12:
What percentage of males surveyed listen to the radio station? First calculate total number of males: total surveyed = 200, females = 96, so males = \(200 - 96 = 104\). Males who do not listen = 26, so males who listen = \(104 - 26 = 78\). The percentage of males who listen is \(\frac{78}{104} = 0.75 = 75\%\). A common wrong answer is 39%, which divides male listeners by the total surveyed population instead of the total male population.
ANSWER 12: E

### Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.
1.  Calculate original yellow tint volume: 25%? No wait, original is 30% yellow. So 30% of 30 L = $0.3 \times 30 = 9$ liters of yellow.
2.  Add 5 liters of yellow: total yellow = $9 + 5 = 14$ liters.
3.  Total new mixture volume: $30 + 5 = 35$ liters.
4.  Percent yellow: $\frac{14}{35} \times 100 = 40\%$.
ANSWER 1: C
---
### Problem 2:
We count the number of ways to choose an entry window and a different exit window from 6 windows.
1.  Entry window: 6 possible choices.
2.  Exit window: cannot be the entry window, so 5 remaining choices.
3.  Total ways: $6 \times 5 = 30$ (order matters because entry and exit are distinct actions).
ANSWER 2: D
---
### Problem 3:
Simplify and add the fractions:
1.  $\frac{1}{10} = 0.1$, $\frac{2}{20} = \frac{1}{10} = 0.1$, $\frac{3}{30} = \frac{1}{10} = 0.1$.
2.  Sum: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D
---
### Problem 4:
Calculate remaining money after expenses:
1.  Total initial money: $50.
2.  Money spent on rides: twice the food cost = $2 \times 12 = 24$ dollars.
3.  Total spent: $12 + 24 = 36$ dollars.
4.  Remaining: $50 - 36 = 14$ dollars.
ANSWER 4: B
---
### Problem 5:
Compute $200,000 \times 200,000$:
1.  Rewrite as $(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
2.  $10^{10} = 10,000,000,000$, so $4 \times 10^{10} = 40,000,000,000 = 40$ billion.
ANSWER 5: E
---
### Problem 6:
The total payment $1A2$ (a 3-digit number) must be divisible by 11 (since 11 people paid equal integer amounts).
1.  Use the divisibility rule for 11: the alternating sum of digits is divisible by 11. For $1A2$, this is $1 - A + 2 = 3 - A$.
2.  $A$ is a digit (0-9), so $3 - A$ must equal 0 (the only multiple of 11 in range: $3 - A = 11$ gives $A=-8$, $3-A=-11$ gives $A=14$, both invalid).
3.  $3 - A = 0 \implies A=3$. Verify: $132 \div 11 = 12$, which is an integer.
ANSWER 6: D
---
### Problem 7:
Check if the 24 cm wire can form each shape (perimeter of the shape must equal 24 cm):
1.  Regular hexagon (side 5 cm): Perimeter = $6 \times 5 = 30$ cm > 24 cm → cannot make.
2.  Square (area 36 cm²): Side length = $\sqrt{36} = 6$ cm. Perimeter = $4 \times 6 = 24$ cm → can make.
3.  Right triangle (legs 6, 8 cm): Hypotenuse = $\sqrt{6^2 + 8^2} = 10$ cm. Perimeter = $6 + 8 + 10 = 24$ cm → can make.
Only square and triangle can be made.
ANSWER 7: D
---
### Problem 8:
The reciprocal of a positive number $x$ is $\frac{1}{x}$; smaller $x$ gives larger reciprocal.
1.  List values: $\frac{1}{3} \approx 0.33$, $\frac{2}{5}=0.4$, $1$, $5$, $1986$.
2.  Smallest value is $\frac{1}{3}$, whose reciprocal is $3$ (largest among all reciprocals: $\frac{5}{2}=2.5$, $1$, $0.2$, $\frac{1}{1986} \approx 0.0005$).
ANSWER 8: A
---
### Problem 9:
Calculate target heart rate for a 26-year-old:
1.  Maximum heart rate: $220 - 26 = 194$ bpm.
2.  Target heart rate: 80% of 194 = $0.8 \times 194 = 155.2$, which rounds to 155.
ANSWER 9: B
---
### Problem 10:
Compute $H$ (Harry's correct answer) and $T$ (Terry's incorrect answer):
1.  Harry: $8 - (2 + 5) = 8 - 7 = 1$, so $H=1$.
2.  Terry (ignores parentheses, left-to-right): $8 - 2 + 5 = 6 + 5 = 11$, so $T=11$.
3.  Difference: $H - T = 1 - 11 = -10$.
ANSWER 10: A
---
### Problem 11:
Calculate $2 \times (81 + 83 + ... + 99)$:
1.  The sum inside is 10 consecutive odd numbers. Average of first and last term: $\frac{81 + 99}{2} = 90$.
2.  Sum of the 10 terms: $90 \times 10 = 900$.
3.  Multiply by 2: $2 \times 900 = 1800$.
ANSWER 11: E
---
### Problem 12:
Find the percentage of males who listen to the station:
1.  Total males surveyed: $200 - 96 = 104$.
2.  Males who do not listen: 26, so males who listen: $104 - 26 = 78$.
3.  Verify with total listeners: Total listeners = 136, female listeners = 58, so male listeners = $136 - 58 = 78$ (matches).
4.  Percentage: $\frac{78}{104} \times 100 = 75\%$.
ANSWER 12: E

Problem 1:
Original yellow = 30% of 30 = 9 L. Add 5 → 14 L yellow. New total = 35 L. Percent = 14/35 = 40%.

ANSWER 1: C

Problem 2:
Enter: 6 choices, leave: 5 different choices. 6×5 = 30.

ANSWER 2: D

Problem 3:
Each fraction = 0.1. Sum = 0.3.

ANSWER 3: D

Problem 4:
Food 12, rides twice = 24. Spent 36. Left = 50 − 36 = 14.

ANSWER 4: B

Problem 5:
200,000 × 200,000 = 40,000,000,000 = forty billion.

ANSWER 5: E

Problem 6:
1A2 divisible by 11. Alternating sum: 1 − A + 2 = 3 − A must be divisible by 11 → A = 3 (gives 0). Check 132/11 = 12. Yes.

ANSWER 6: D

Problem 7:
Hexagon: 6×5 = 30 > 24, no. Square area 36 → side 6, perimeter 24, yes. Triangle legs 6,8 → hypotenuse 10, perimeter 24, yes. Square and triangle only.

ANSWER 7: D

Problem 8:
Largest reciprocal = smallest number. Smallest is 1/3.

ANSWER 8: A

Problem 9:
Max = 220 − 26 = 194. Target = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

Problem 10:
H = 8 − 7 = 1. T = 8 − 2 + 5 = 11. H − T = 1 − 11 = −10.

ANSWER 10: A

Problem 11:
Sum of 81..99 odd numbers (10 terms): average (81+99)/2 = 90, ×10 = 900. ×2 = 1800.

ANSWER 11: E

Problem 12:
Males total = 200 − 96 = 104. Males who don't listen = 26 → males who listen = 104 − 26 = 78. Percent = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem systematically.

---

**Problem 1:**
Original mixture: 30 liters, 30% yellow = 9 liters yellow. Add 5 liters yellow → new yellow = 14 liters, new total = 35 liters.
Percent yellow = 14/35 = 40%.

ANSWER 1: C

---

**Problem 2:**
Georgie can enter through any of 6 windows, then leave through any of the remaining 5 windows.
6 × 5 = 30.

ANSWER 2: D

---

**Problem 3:**
1/10 = 0.1, 2/20 = 0.1, 3/30 = 0.1. Sum = 0.3.

ANSWER 3: D

---

**Problem 4:**
Food: $12. Rides: 2 × $12 = $24. Total spent: $12 + $24 = $36. Left: $50 − $36 = $14.

ANSWER 4: B

---

**Problem 5:**
200,000 × 200,000 = (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion.

ANSWER 5: E

---

**Problem 6:**
The number 1A2 must be divisible by 11. 1A2 ÷ 11 must be an integer.
11 × 10 = 110, 11 × 11 = 121 (no), 11 × 12 = 132 ✓ → A = 3.
Check: 132/11 = 12 ✓, each paid $12.

ANSWER 6: D

---

**Problem 7:**
Wire = 24 cm.
- Regular hexagon, side 5: perimeter = 6 × 5 = 30 cm > 24. ✗
- Square, area 36: side = 6, perimeter = 24 cm ✓
- Right triangle, legs 6 and 8: hypotenuse = 10, perimeter = 6+8+10 = 24 cm ✓

ANSWER 7: D

---

**Problem 8:**
The reciprocal of x is 1/x. The largest reciprocal comes from the smallest positive number.
Among 1/3, 2/5, 1, 5, 1986 — the smallest is 1/3, and its reciprocal is 3, which is the largest.

ANSWER 8: A

---

**Problem 9:**
Maximum heart rate = 220 − 26 = 194. Target = 80% × 194 = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

---

**Problem 10:**
Harry: 8 − (2 + 5) = 8 − 7 = 1, so H = 1.
Terry: 8 − 2 + 5 = 11, so T = 11.
H − T = 1 − 11 = −10.

ANSWER 10: A

---

**Problem 11:**
The ten odd numbers from 81 to 99: sum = (81+99)/2 × 10 = 90 × 10 = 900.
2 × 900 = 1800.

ANSWER 11: E

---

**Problem 12:**
Total surveyed = 200. Females = 96, so Males = 104.
Total non-listeners = 64. Males who don't listen = 26, so females who don't listen = 64 − 26 = 38.
Males who listen = 104 − 26 = 78.
Percentage = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem step-by-step.

## Problem 1:
**What is asked:** Find the percentage of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.

**Original mixture:**
- Yellow tint: 30% of 30 = 0.30 × 30 = 9 liters
- Total volume: 30 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total volume: 30 + 5 = 35 liters

**Percentage of yellow tint:**
$$\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%$$

ANSWER 1: C

## Problem 2:
**What is asked:** In how many ways can the ghost enter by one window and leave by a different window?

**Solution:**
- Choices for entering: 6 windows
- Choices for leaving: 5 windows (must be different from entry)
- Total ways: 6 × 5 = 30

ANSWER 2: D

## Problem 3:
**What is asked:** Calculate 1/10 + 2/20 + 3/30

**Solution:**
Simplify each fraction:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

ANSWER 3: D

## Problem 4:
**What is asked:** How much money does Susan have left?

**Solution:**
- Started with: $50
- Spent on food: $12
- Spent on rides: 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Left: $50 - $36 = $14

ANSWER 4: B

## Problem 5:
**What is asked:** Calculate 200,000 × 200,000

**Solution:**
$$200,000 × 200,000 = (2 × 10^5) × (2 × 10^5) = 4 × 10^{10} = 40,000,000,000 = 40 \text{ billion}$$

ANSWER 5: E

## Problem 6:
**What is asked:** Find the missing digit A in the 3-digit number 1A2, where 11 club members each paid an equal integer amount that totals 1A2.

**Solution:**
The number 1A2 must be divisible by 11.

Testing divisibility by 11 (alternating sum of digits):
- 1 - A + 2 = 0 (mod 11)
- 3 - A ≡ 0 (mod 11)
- A = 3

Check: 132 ÷ 11 = 12 ✓

ANSWER 6: D

## Problem 7:
**What is asked:** Which shapes can Haruki make with 24 cm of wire?

**Solution:**
Test each shape:

**Regular hexagon with side 5 cm:**
- Perimeter: 6 × 5 = 30 cm
- Need 30 cm, have 24 cm ✗

**Square with area 36 cm²:**
- Side length: √36 = 6 cm
- Perimeter: 4 × 6 = 24 cm ✓

**Right triangle with legs 6 and 8 cm:**
- Hypotenuse: √(6² + 8²) = √(36 + 64) = √100 = 10 cm
- Perimeter: 6 + 8 + 10 = 24 cm ✓

Only square and triangle can be made.

ANSWER 7: D

## Problem 8:
**What is asked:** Which number has the largest reciprocal?

**Solution:**
Find reciprocals:
- 1/(1/3) = 3
- 1/(2/5) = 5/2 = 2.5
- 1/1 = 1
- 1/5 = 0.2
- 1/1986 ≈ 0.0005

The largest reciprocal is 5 (from the number 1/5).

ANSWER 8: D

## Problem 9:
**What is asked:** Find the target heart rate for a 26-year-old athlete.

**Solution:**
- Maximum heart rate: 220 - 26 = 194 beats per minute
- Target heart rate: 80% of 194 = 0.80 × 194 = 155.2 ≈ 155

ANSWER 9: B

## Problem 10:
**What is asked:** Find H - T where H is Harry's answer and T is Terry's answer.

**Solution:**
**Harry's calculation (correct):**
$$H = 8 - (2 + 5) = 8 - 7 = 1$$

**Terry's calculation (ignoring parentheses):**
$$T = 8 - 2 + 5 = 6 + 5 = 11$$

**Difference:**
$$H - T = 1 - 11 = -10$$

ANSWER 10: A

## Problem 11:
**What is asked:** Calculate 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

**Solution:**
This is an arithmetic sequence with 10 terms. Sum = n(first + last)/2
$$81 + 83 + ... + 99 = \frac{10(81 + 99)}{2} = \frac{10 × 180}{2} = 900$$

$$2 × 900 = 1800$$

ANSWER 11: E

## Problem 12:
**What is asked:** What percentage of males surveyed listen to the station?

**Solution:**
- Total surveyed: 200
- Females: 96
- Males: 200 - 96 = 104

- Total non-listeners: 64
- Males who don't listen: 26
- Males who listen: 104 - 26 = 78

**Percentage of males who listen:**
$$\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%$$

ANSWER 12: E

Problem 1:

The original mixture is 30 liters total, with 30% yellow tint.

So the amount of yellow tint at first is:
- \(30\%\) of \(30 = 0.30 \times 30 = 9\) liters

Then 5 liters of yellow tint are added, so the new amount of yellow tint is:
- \(9 + 5 = 14\) liters

The new total mixture is:
- \(30 + 5 = 35\) liters

So the percent of yellow tint is:
- \(\frac{14}{35} = \frac{2}{5} = 0.4 = 40\%\)

Answer choice C matches 40%.

ANSWER 1: C

Problem 2:

Georgie can:
- choose 1 of the 6 windows to enter
- then choose a different window to leave

For the entry, there are 6 choices.

For the exit, there are 5 remaining choices, since it must be a different window.

So the total number of ways is:
- \(6 \times 5 = 30\)

ANSWER 2: D

Problem 3:

Compute each fraction:
- \(\frac{1}{10} = 0.1\)
- \(\frac{2}{20} = \frac{1}{10} = 0.1\)
- \(\frac{3}{30} = \frac{1}{10} = 0.1\)

Add them:
- \(0.1 + 0.1 + 0.1 = 0.3\)

So the value is \(0.3\).

ANSWER 3: D

Problem 4:

Susan had \$50.

She spent:
- \$12 on food
- twice as much on rides, so \(2 \times 12 = 24\)

Total spent:
- \(12 + 24 = 36\)

Money left:
- \(50 - 36 = 14\)

ANSWER 4: B

Problem 5:

Two hundred thousand is:
- \(200{,}000 = 2 \times 10^5\)

Multiply:
- \(200{,}000 \times 200{,}000 = (2 \times 10^5)(2 \times 10^5) = 4 \times 10^{10}\)

And \(4 \times 10^{10} = 40{,}000{,}000{,}000\), which is forty billion.

ANSWER 5: E

Problem 6:

The total is written as \(1A2\), and 11 members each paid the same integer amount.

So \(1A2\) must be divisible by 11.

Test the answer choices:
- 102 ÷ 11 is not an integer
- 112 ÷ 11 is not an integer
- 122 ÷ 11 = 11 remainder 1
- 132 ÷ 11 = 12 exactly
- 142 ÷ 11 is not an integer

So the number is 132, which means \(A = 3\).

ANSWER 6: D

Problem 7:

We check each shape.

### Hexagon
A regular hexagon has 6 equal sides.
If each side is 5 cm, perimeter is:
- \(6 \times 5 = 30\) cm

The wire is only 24 cm, so he cannot make the hexagon.

### Square
A square with area 36 has side length:
- \(\sqrt{36} = 6\) cm

Perimeter:
- \(4 \times 6 = 24\) cm

So he can make the square exactly.

### Right triangle
The legs are 6 and 8, so the hypotenuse is:
- \(\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)

Perimeter:
- \(6 + 8 + 10 = 24\) cm

So he can make the triangle exactly.

Therefore he can make the square and the triangle only.

ANSWER 7: D

Problem 8:

The reciprocal is largest when the original number is smallest, assuming positive numbers.

Compare:
- \(\frac13 \approx 0.333\)
- \(\frac25 = 0.4\)
- \(1\)
- \(5\)
- \(1986\)

The smallest number is \(\frac13\), so it has the largest reciprocal.

Its reciprocal is 3, which is larger than the reciprocals of all the others.

ANSWER 8: A

Problem 9:

Maximum heart rate:
- \(220 - 26 = 194\)

Target heart rate is 80% of that:
- \(0.8 \times 194 = 155.2\)

To the nearest whole number, that is 155.

ANSWER 9: B

Problem 10:

First, Harry uses the parentheses:
- \(8 - (2 + 5) = 8 - 7 = 1\)

So \(H = 1\).

Terry ignores parentheses:
- \(8 - 2 + 5 = 6 + 5 = 11\)

So \(T = 11\).

Difference:
- \(H - T = 1 - 11 = -10\)

ANSWER 10: A

Problem 11:

We need to evaluate:
- \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

These are 10 odd numbers from 81 to 99.

Instead of adding one by one, use the average:
- average of 81 and 99 is \( \frac{81+99}{2} = 90\)

Since there are 10 numbers, their sum is:
- \(10 \times 90 = 900\)

Now multiply by 2:
- \(2 \times 900 = 1800\)

ANSWER 11: E

Problem 12:

We want the percentage of males surveyed who listen.

From the table:
- Total surveyed = 200
- Females = 96
- So males = \(200 - 96 = 104\)

Males who don’t listen = 26, so males who do listen:
- \(104 - 26 = 78\)

Now find the percent of males who listen:
- \(\frac{78}{104} = \frac{3}{4} = 75\%\)

ANSWER 12: E

Problem 1:  
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?  

- Original yellow tint amount = 30 × 0.30 = 9 liters.  
- Add 5 liters of yellow tint → new yellow amount = 9 + 5 = 14 liters.  
- New total volume = 30 + 5 = 35 liters.  
- New percent yellow = (14 / 35) × 100% = (2/5) × 100% = 40%.  

So the correct choice is 40.  

ANSWER 1: C  

---

Problem 2:  
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?  

- Choose an entry window: 6 choices.  
- Choose a different exit window: 5 choices (cannot be the same as entry).  
- Total ways = 6 × 5 = 30.  

ANSWER 2: D  

---

Problem 3:  
Compute: \( \frac{1}{10} + \frac{2}{20} + \frac{3}{30} \).  

Convert each to a common denominator, 60:  
- \( \frac{1}{10} = \frac{6}{60} \)  
- \( \frac{2}{20} = \frac{2}{20} = \frac{6}{60} \) (since multiply top and bottom by 3)  
- \( \frac{3}{30} = \frac{3}{30} = \frac{6}{60} \) (since multiply top and bottom by 2)  

Add:  
\( \frac{6}{60} + \frac{6}{60} + \frac{6}{60} = \frac{18}{60} = \frac{3}{10} = 0.3 \)

So the value is .3.  

ANSWER 3: D  

---

Problem 4:  
Susan had $50 to spend. She spent $12 on food and twice as much on rides. How much left?  

- Food = 12  
- Rides = twice 12 = 24  
- Total spent = 12 + 24 = 36  
- Money left = 50 − 36 = 14  

ANSWER 4: B  

---

Problem 5:  
Two hundred thousand times two hundred thousand equals ?  

\(200{,}000 \times 200{,}000 = (2\times 10^5)(2\times 10^5) = 4 \times 10^{10}\).  
- \(10^{10} = 10{,}000{,}000{,}000\)  
- \(4 \times 10^{10} = 40{,}000{,}000{,}000 = 40\) billion  

ANSWER 5: E  

---

Problem 6:  
Eleven members each paid the same integer amount for a guest speaker. In all, they paid $1A2 (a 3-digit number). Find missing digit A.  

- Total = \(1A2\) means:  
  \(1A2 = 100 + 10A + 2 = 102 + 10A\).  
- This total is divided equally among 11 people:  
  So \(102 + 10A\) must be divisible by 11.  

Check \(102 + 10A \mod 11\):  
- \(102 \div 11 = 9\) remainder 3, so \(102 \equiv 3 \pmod{11}\).  
- \(10A \equiv -A \pmod{11}\) (since 10 ≡ −1 mod 11).  

So:  
\(102 + 10A \equiv 3 - A \equiv 0 \pmod{11}\)  
Thus \(A \equiv 3 \pmod{11}\).  
Since \(A\) is a digit 0–9, \(A=3\).  

ANSWER 6: D  

---

Problem 7:  
Wire length = 24 cm. Which shapes can be made (using all 24 cm of wire)?  

We need the perimeter (total boundary length) of each shape:

**(1) Regular hexagon with side length 5 cm**  
Perimeter = 6 × 5 = 30 cm → too long. Cannot.

**(2) Square with area 36 cm²**  
Side length = √36 = 6 cm  
Perimeter = 4 × 6 = 24 cm → fits. Can.

**(3) Right triangle with legs 6 and 8 cm**  
Right triangle sides are 6 and 8 legs, and hypotenuse:  
\( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)  
Perimeter = 6 + 8 + 10 = 24 cm → fits. Can.

So the shapes possible: **square and triangle only**.  

ANSWER 7: D  

---

Problem 8:  
Which number has the largest reciprocal? Recall: the reciprocal of \(x\) is \(1/x\).  
We compare \(1/3, 2/5, 1, 1/5, 1/1986\).  

Compute roughly:  
- \(1/3 \approx 0.333\)  
- \(2/5 = 0.4\)  
- \(1 = 1\)  
- \(1/5 = 0.2\)  
- \(1/1986\) is tiny  

Largest is 1 (reciprocal of 1).  

ANSWER 8: C  

---

Problem 9:  
Target heart rate = 80% of (maximum heart rate).  
Maximum heart rate = 220 − age.  
Age = 26.

- Maximum = 220 − 26 = 194  
- Target = 0.80 × 194 = 155.2  
Nearest whole number = 155  

ANSWER 9: B  

---

Problem 10:  
Expression: \(8 - (2 + 5)\)  
- Harry computes correctly:  
  \(2 + 5 = 7\), so \(8 - 7 = 1\).  
- Terry ignores parentheses:  
  \(8 - 2 + 5 = 6 + 5 = 11\).  

Difference \(H - T = 1 - 11 = -10\).  

ANSWER 10: A  

---

Problem 11:  
Compute:  
\(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\)

Notice these are all odd numbers from 81 to 99 stepping by 2.  
There are 10 terms.

Pair from ends:  
- 81 + 99 = 180  
- 83 + 97 = 180  
- 85 + 95 = 180  
- 87 + 93 = 180  
- 89 + 91 = 180  

There are 5 pairs, each sum 180:  
Sum = 5 × 180 = 900  

Now multiply by 2:  
2 × 900 = 1800  

ANSWER 11: E  

---

Problem 12:  
Total surveyed = 200  
Females = 96  
Females who listen = 58  
Males who don’t listen = 26  
Total listeners = 136  

We want: % of males who listen.

Step 1: Find males total.  
Males = 200 − 96 = 104

Step 2: Find males who don’t listen is given = 26.  
So males who listen = 104 − 26 = 78

Step 3: Convert to percent of males who listen:  
Percent = (78 / 104) × 100%  
Simplify 78/104 = 3/4 (divide by 26):  
So percent = (3/4) × 100% = 75%

ANSWER 12: E

Problem 1:
The original mixture is 30 liters. The amount of yellow tint is 30% of 30 liters: $0.30 \times 30 = 9$ liters.
Five liters of yellow tint are added, so the new amount of yellow tint is $9 + 5 = 14$ liters.
The new total volume of the mixture is $30 + 5 = 35$ liters.
The percentage of yellow tint is $\frac{14}{35} \times 100\%$.
Simplifying the fraction: $\frac{14}{35} = \frac{2}{5} = 0.4$.
$0.4 \times 100\% = 40\%$.
ANSWER 1: C

Problem 2:
Georgie has 6 choices for the window he enters. Since he must leave by a different window, he has $6 - 1 = 5$ choices for the window he leaves by.
The total number of ways is $6 \times 5 = 30$.
ANSWER 2: D

Problem 3:
Simplify each fraction:
$1/10 = 0.1$
$2/20 = 1/10 = 0.1$
$3/30 = 1/10 = 0.1$
Summing them: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides, which is $2 \times 12 = 24$.
Total spent: $12 + 24 = 36$.
Amount left: $50 - 36 = 14$.
ANSWER 4: B

Problem 5:
Two hundred thousand is $200,000$ (or $2 \times 10^5$).
$(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
$4 \times 10^{10} = 40,000,000,000$, which is forty billion.
ANSWER 5: E

Problem 6:
The total amount $1A2$ must be divisible by 11.
A number is divisible by 11 if the alternating sum of its digits is divisible by 11.
For $1A2$, the alternating sum is $1 - A + 2 = 3 - A$.
For this to be 0 (the only multiple of 11 possible for a single digit $A$), $3 - A = 0$, so $A = 3$.
Check: $132 / 11 = 12$. This works.
ANSWER 6: D

Problem 7:
Hexagon: Perimeter of a regular hexagon with side 5 is $6 \times 5 = 30$ cm. (Cannot make)
Square: Area is 36, so side is $\sqrt{36} = 6$. Perimeter is $4 \times 6 = 24$ cm. (Can make)
Triangle: Right triangle with legs 6 and 8. Hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. Perimeter is $6 + 8 + 10 = 24$ cm. (Can make)
ANSWER 7: D

Problem 8:
Reciprocals:
A: $1/(1/3) = 3$
B: $1/(2/5) = 2.5$
C: $1/1 = 1$
D: $1/5 = 0.2$
E: $1/1986 \approx 0.0005$
The largest is 3.
ANSWER 8: A

Problem 9:
Max heart rate = $220 - 26 = 194$.
Target heart rate = $80\%$ of $194 = 0.8 \times 194 = 155.2$.
To the nearest whole number, this is 155.
ANSWER 9: B

Problem 10:
$H = 8 - (2 + 5) = 8 - 7 = 1$.
$T = 8 - 2 + 5 = 6 + 5 = 11$.
$H - T = 1 - 11 = -10$.
ANSWER 10: A

Problem 11:
Sum inside parentheses: $81+83+85+87+89+91+93+95+97+99$.
This is an arithmetic progression with 10 terms.
Sum = $\frac{n}{2}(\text{first} + \text{last}) = \frac{10}{2}(81 + 99) = 5 \times 180 = 900$.
The expression is $2 \times 900 = 1800$.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96.
Males = $200 - 96 = 104$.
Males who don't listen = 26.
Males who listen = $104 - 26 = 78$.
Percentage of males who listen = $(78 / 104) \times 100\%$.
$\frac{78}{104} = \frac{39}{52} = \frac{3}{4} = 0.75 = 75\%$.
ANSWER 12: E

Problem 1: The original mixture has volume 30 L, of which 30% is yellow tint, so there are \(0.3 \times 30 = 9\) L of yellow tint. Adding 5 L of yellow tint produces a new total of 14 L of yellow tint and a new mixture volume of 35 L. The new percentage of yellow tint is therefore \((14/35) \times 100\% = 40\%\).

ANSWER 1: C

Problem 2: There are 6 choices for the entry window. For each of those choices there are then exactly 5 remaining choices for a different exit window, giving a total of \(6 \times 5 = 30\) ordered pairs of distinct windows.

ANSWER 2: D

Problem 3: Rewrite each term with a common denominator of 10:  
\(1/10 = 0.1\),  
\(2/20 = 1/10 = 0.1\),  
\(3/30 = 1/10 = 0.1\).  
Their sum is \(0.1 + 0.1 + 0.1 = 0.3\).

ANSWER 3: D

Problem 4: Twice the food cost is \(2 \times 12 = 24\) dollars spent on rides. The total spent is therefore \(12 + 24 = 36\) dollars, leaving \(50 - 36 = 14\) dollars.

ANSWER 4: B

Problem 5: Write the factors in scientific notation:  
\(200000 = 2 \times 10^5\), so  
\((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\),  
which is forty billion.

ANSWER 5: E

Problem 6: The three-digit total \(1A2\) must be divisible by 11. The alternating-sum divisibility rule requires that \(1 - A + 2 = 3 - A\) be a multiple of 11. The only admissible digit \(A\) (0-9) that satisfies the condition is \(A = 3\), giving the total 132, which is exactly \(11 \times 12\).

ANSWER 6: D

Problem 7: A regular hexagon with side 5 cm has perimeter 30 cm, which exceeds the 24 cm length of wire, so it is impossible. A square of area 36 cm² has side length 6 cm and perimeter 24 cm, which matches the wire length. A right triangle with legs 6 cm and 8 cm has hypotenuse 10 cm, so its perimeter is \(6 + 8 + 10 = 24\) cm, which also matches. Consequently only the square and the triangle are possible.

ANSWER 7: D

Problem 8: For positive numbers the reciprocal function is strictly decreasing, so the largest reciprocal belongs to the smallest number in the list, which is \(1/3\).

ANSWER 8: A

Problem 9: The theoretical maximum heart rate is \(220 - 26 = 194\) beats per minute. Eighty percent of that value is \(0.8 \times 194 = 155.2\), which rounds to the nearest whole number 155.

ANSWER 9: B

Problem 10: Harry evaluates the parentheses first and obtains \(8 - (2 + 5) = 1\). Terry omits the parentheses and obtains \(8 - 2 + 5 = 11\). Their difference is therefore \(1 - 11 = -10\).

ANSWER 10: A

Problem 11: The sum inside the parentheses consists of the ten consecutive odd integers from 81 to 99. Their average is \((81 + 99)/2 = 90\), so the sum equals \(10 \times 90 = 900\). Multiplying by the outer factor of 2 yields \(2 \times 900 = 1800\).

ANSWER 11: E

Problem 12: There are 200 people surveyed and 96 females, so there are \(200 - 96 = 104\) males. Of the 96 females, 58 listen, leaving \(96 - 58 = 38\) females who do not listen. Given that 26 males do not listen, the number of males who listen is \(104 - 26 = 78\). The required percentage is therefore \((78/104) \times 100\% = 75\%\).

ANSWER 12: E

Problem 1:
The problem asks us to find the percentage of yellow tint in a new mixture after adding 5 liters of yellow tint to the original 30 liters of paint mixture. The original mixture is 25% red tint, 30% yellow tint, and 45% water.

First, let's calculate the amount of yellow tint in the original mixture. 30% of 30 liters is $0.3 \times 30 = 9$ liters.

After adding 5 liters of yellow tint, the total amount of yellow tint becomes $9 + 5 = 14$ liters.

The total volume of the new mixture is $30 + 5 = 35$ liters.

To find the percentage of yellow tint in the new mixture, we divide the amount of yellow tint by the total volume and multiply by 100: $\frac{14}{35} \times 100 = 40\%$.

Let's examine the choices:
- A. 25: Too low because the percentage of yellow tint increased.
- B. 35: Not the correct calculation.
- C. 40: Matches our calculation.
- D. 45: Too high.
- E. 50: Too high.

The correct percentage is 40%.

ANSWER 1: C

Problem 2:
The problem asks us to determine the number of ways Georgie the Ghost can enter a haunted house through one of six windows and leave by a different window.

To enter, Georgie has 6 choices. After entering, there are 5 remaining windows to choose from for leaving.

The total number of ways to enter and leave is $6 \times 5 = 30$.

Let's examine the choices:
- A. 12: Too low.
- B. 15: Too low.
- C. 18: Too low.
- D. 30: Matches our calculation.
- E. 36: Too high because it would imply Georgie can leave by the same window.

The correct number of ways is 30.

ANSWER 2: D

Problem 3:
The problem requires us to simplify the expression $1⁄10 + 2⁄20 + 3⁄30$.

First, simplify each fraction:
- $1⁄10$ stays the same.
- $2⁄20 = 1⁄10$.
- $3⁄30 = 1⁄10$.

Adding them together: $1⁄10 + 1⁄10 + 1⁄10 = 3⁄10 = 0.3$.

Let's examine the choices:
- A. .1: Too low.
- B. .123: Not the correct sum.
- C. .2: Too low.
- D. .3: Matches our calculation.
- E. .6: Too high.

The correct sum is 0.3.

ANSWER 3: D

Problem 4:
The problem asks us to find out how many dollars Susan had left after spending money on food and rides at the carnival. She started with $50, spent $12 on food, and twice as much on rides.

First, calculate the amount spent on rides: $12 \times 2 = $24.

Total amount spent is $12 + $24 = $36.

The amount left is $50 - $36 = $14.

Let's examine the choices:
- A. 12: Too low.
- B. 14: Matches our calculation.
- C. 26: Too high.
- D. 38: Too high.
- E. 50: Incorrect because she spent money.

The correct amount left is $14.

ANSWER 4: B

Problem 5:
The problem asks us to calculate $200,000 \times 200,000$.

$200,000 = 2 \times 10^5$, so $200,000 \times 200,000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10} = 40,000,000,000$.

Let's examine the choices:
- A. four hundred thousand: Too low.
- B. four million: Too low.
- C. forty thousand: Too low.
- D. four hundred million: Too low.
- E. forty billion: Matches our calculation.

The correct result is forty billion.

ANSWER 5: E

Problem 6:
The problem requires us to find the missing digit A in the 3-digit number $1A2 that represents the total amount paid by 11 members of the Math Club.

The total amount paid is divisible by 11 since there are 11 members who paid the same amount.

To check divisibility by 11, we use the rule: the difference between the sum of digits at odd places and the sum of digits at even places should be divisible by 11.

For $1A2, the sum of digits at odd places is $1 + 2 = 3$, and the sum at even places is $A$. The difference is $3 - A$.

For $3 - A$ to be divisible by 11 (or 0, since $A$ is a single digit), $A = 3$ makes $3 - A = 0$, which is divisible by 11.

Let's examine the choices:
- A. 0: Not correct because $3 - 0 = 3$, not divisible by 11.
- B. 1: Not correct because $3 - 1 = 2$, not divisible by 11.
- C. 2: Not correct because $3 - 2 = 1$, not divisible by 11.
- D. 3: Correct because $3 - 3 = 0$, which is divisible by 11.
- E. 4: Not correct because $3 - 4 = -1$, not divisible by 11.

The missing digit A is 3, but we should verify if 132 is indeed divisible by 11. $132 / 11 = 12$, so it is.

ANSWER 6: D

Problem 7:
The problem asks us to determine which shapes Haruki can form using a 24 cm long wire.

1. A regular hexagon with side length 5 cm: The perimeter is $6 \times 5 = 30$ cm, which is more than 24 cm. So, Haruki cannot form this hexagon.

2. A square of area 36 cm$^2$: The side length is $\sqrt{36} = 6$ cm. The perimeter is $4 \times 6 = 24$ cm, which matches the wire length. So, Haruki can form this square.

3. A right triangle with legs 6 and 8 cm: Using the Pythagorean theorem, the hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm. The perimeter is $6 + 8 + 10 = 24$ cm, which matches the wire length. So, Haruki can form this triangle.

Let's examine the choices:
- A. Triangle only: Incorrect because Haruki can also form the square.
- B. Hexagon and square only: Incorrect because Haruki cannot form the hexagon.
- C. Hexagon and triangle only: Incorrect because Haruki cannot form the hexagon.
- D. Square and triangle only: Matches our findings.
- E. Hexagon, triangle, and square: Incorrect because Haruki cannot form the hexagon.

The correct shapes Haruki can form are the square and the triangle.

ANSWER 7: D

Problem 8:
The problem asks us to find which of the given numbers has the largest reciprocal.

The reciprocal of a number $x$ is $1/x$. The larger $x$ is, the smaller its reciprocal is, and vice versa.

Let's examine the choices:
- A. $1⁄3$: The reciprocal is 3.
- B. $2⁄5$: The reciprocal is $5/2 = 2.5$.
- C. 1: The reciprocal is 1.
- D. 5: The reciprocal is $1/5 = 0.2$.
- E. 1986: The reciprocal is $1/1986$, very small.

Comparing the reciprocals: $3 > 2.5 > 1 > 0.2 > 1/1986$. So, $1/3$ has the largest reciprocal.

Let's examine the choices again:
- A. $1⁄3$: Correct.
- B. $2⁄5$: Not correct.
- C. 1: Not correct.
- D. 5: Not correct.
- E. 1986: Not correct.

The number with the largest reciprocal is $1/3$.

ANSWER 8: A

Problem 9:
The proble

Problem 1:
Original mixture: 30 L. Yellow tint = 30% of 30 = 0.30 × 30 = 9 L.
Add 5 L yellow: new yellow = 9 + 5 = 14 L, new total volume = 30 + 5 = 35 L.
Percent yellow = (14/35) × 100% = 40%.
ANSWER 1: C

Problem 2:
6 choices for entry window, 5 remaining choices for a different exit window.
Total ways = 6 × 5 = 30.
ANSWER 2: D

Problem 3:
1/10 + 2/20 + 3/30 = 1/10 + 1/10 + 1/10 = 3/10 = 0.3.
ANSWER 3: D

Problem 4:
Spent on food: $12. Spent on rides: 2 × $12 = $24.
Total spent = $12 + $24 = $36. Left = $50 − $36 = $14.
ANSWER 4: B

Problem 5:
200,000 × 200,000 = (2×10⁵) × (2×10⁵) = 4×10¹⁰ = 40,000,000,000, which is forty billion.
ANSWER 5: E

Problem 6:
Total paid is $1A2, a multiple of 11. Divisibility by 11: (1 − A + 2) = 3 − A must be a multiple of 11.
With A a digit 0–9, 3 − A = 0 ⇒ A = 3. (Check: 132 ÷ 11 = 12.)
ANSWER 6: D

Problem 7:
Wire length = 24 cm.
- Regular hexagon, side 5 cm: perimeter = 6×5 = 30 cm > 24 cm → cannot make.
- Square, area 36 cm²: side = 6 cm, perimeter = 4×6 = 24 cm → can make.
- Right triangle, legs 6 and 8 cm: hypotenuse = 10 cm, perimeter = 6+8+10 = 24 cm → can make.
So square and triangle only.
ANSWER 7: D

Problem 8:
Reciprocals: 1/(1/3)=3, 1/(2/5)=2.5, 1/1=1, 1/5=0.2, 1/1986≈0.0005. Largest is 3, from 1/3.
ANSWER 8: A

Problem 9:
Maximum heart rate = 220 − 26 = 194 bpm.
Target = 80% of 194 = 0.8 × 194 = 155.2 → 155 to nearest whole number.
ANSWER 9: B

Problem 10:
Harry: H = 8 − (2+5) = 8 − 7 = 1.
Terry: T = 8 − 2 + 5 = 6 + 5 = 11.
H − T = 1 − 11 = −10.
ANSWER 10: A

Problem 11:
Sum inside parentheses: 10 terms, average = (81+99)/2 = 90, sum = 10×90 = 900.
2 × 900 = 1800.
ANSWER 11: E

Problem 12:
Total surveyed = 200. Females = 96 ⇒ Males = 200 − 96 = 104.
Total listeners = 136, female listeners = 58 ⇒ male listeners = 136 − 58 = 78.
Percent of males who listen = (78/104) × 100% = 75%.
ANSWER 12: E

Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint.

**Original mixture (30 liters):**
- Red tint: 25% of 30 = 7.5 liters
- Yellow tint: 30% of 30 = 9 liters
- Water: 45% of 30 = 13.5 liters

**After adding 5 liters of yellow tint:**
- Yellow tint: 9 + 5 = 14 liters
- Total mixture: 30 + 5 = 35 liters

**New percentage of yellow tint:**
14 / 35 = 2/5 = 0.40 = 40%

Tempting wrong answers: Choice B (35) might come from just adding 5 to the original 30%, but we must recalculate based on the new total.

ANSWER 1: C

---

Problem 2:
Georgie enters through one window and leaves through a *different* window.

- Number of choices for the entry window: 6
- Number of choices for the exit window (must be different): 5
- Total ways: 6 × 5 = 30

Tempting wrong answers: Choice E (36) would be 6 × 6, which allows entering and leaving through the same window. Choice A (12) would be 6 + 6, which doesn't account for all combinations.

ANSWER 2: D

---

Problem 3:
Simplify each fraction first:
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

Sum: 1/10 + 1/10 + 1/10 = 3/10 = 0.3

Tempting wrong answers: Choice B (.123) might come from concatenating the digits 1, 2, 3 instead of actually computing. Choice C (.2) might come from miscounting the terms.

ANSWER 3: D

---

Problem 4:
Susan starts with $50.
- Spent on food: $12
- Spent on rides: twice the food amount = 2 × $12 = $24
- Total spent: $12 + $24 = $36
- Amount left: $50 − $36 = $14

Tempting wrong answers: Choice C (26) might come from subtracting only the food ($50 − $24 = $26, forgetting food). Choice D (38) comes from subtracting only $12.

ANSWER 4: B

---

Problem 5:
200,000 × 200,000

Write in scientific notation:
(2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰

4 × 10¹⁰ = 40,000,000,000 = forty billion

Tempting wrong answers: Choice A (four hundred thousand) comes from adding instead of multiplying. Choice D (four hundred million) comes from miscounting zeros.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, so the total $1A2 must be divisible by 11.

**Divisibility rule for 11:** The alternating sum of digits must be a multiple of 11.
Alternating sum: 1 − A + 2 = 3 − A

For 3 − A to be divisible by 11 (and A to be a single digit 0–9):
3 − A = 0 → A = 3

**Check:** $132 ÷ 11 = $12 per member. This is a valid integer amount.

Tempting wrong answers: Other values of A don't make 3 − A a multiple of 11 within the digit range.

ANSWER 6: D

---

Problem 7:
Wire length = 24 cm. We check if each shape's perimeter ≤ 24 cm.

**Regular hexagon (side 5 cm):**
Perimeter = 6 × 5 = 30 cm. Since 30 > 24, he **cannot** make this.

**Square (area 36 cm²):**
Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Since 24 = 24, he **can** make this.

**Right triangle (legs 6 and 8 cm):**
Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Perimeter = 6 + 8 + 10 = 24 cm. Since 24 = 24, he **can** make this.

Haruki can make the square and triangle only.

Tempting wrong answers: Choice E includes the hexagon, but its perimeter exceeds the wire length.

ANSWER 7: D

---

Problem 8:
The reciprocal of a number x is 1/x. The *smaller* the positive number, the *larger* its reciprocal.

| Choice | Number | Reciprocal |
|--------|--------|------------|
| A | 1/3 | 3 |
| B | 2/5 | 5/2 = 2.5 |
| C | 1 | 1 |
| D | 5 | 1/5 = 0.2 |
| E | 1986 | 1/1986 ≈ 0.0005 |

The largest reciprocal is 3, which comes from 1/3.

Tempting wrong answers: Choice E (1986) is the largest number, so it has the *smallest* reciprocal.

ANSWER 8: A

---

Problem 9:
Athlete's age = 26 years.

**Step 1:** Maximum heart rate = 220 − 26 = 194 beats per minute.

**Step 2:** Target heart rate = 80% of 194 = 0.80 × 194 = 155.2

**Step 3:** Round to nearest whole number: 155

Tempting wrong answers: Choice C (176) might come from taking 80% of 220 without subtracting the age. Choice D (194) is the maximum heart rate itself.

ANSWER 9: B

---

Problem 10:
**Harry (correct calculation):**
H = 8 − (2 + 5) = 8 − 7 = 1

**Terry (ignores parentheses):**
T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:**
H − T = 1 − 11 = −10

Tempting wrong answers: Choice E (10) is |H − T|, but the question asks for H − T specifically, which is negative since Terry's answer is larger.

ANSWER 10: A

---

Problem 11:
We need: 2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)

The sum inside is an arithmetic sequence:
- First term: 81, Last term: 99
- Number of terms: 10
- Sum = (number of terms) × (first + last) / 2 = 10 × (81 + 99) / 2 = 10 × 180 / 2 = 10 × 90 = 900

Multiply by 2: 2 × 900 = 1800

Tempting wrong answers: Choice C (1700) might come from miscounting terms or miscalculating the average.

ANSWER 11: E

---

Problem 12:
Let me organize the data from the survey:

- Total surveyed: 200
- Females: 96 → Males: 200 − 96 = 104
- Total listeners: 136
- Females who listen: 58 → Males who listen: 136 − 58 = 78
- Males who don't listen: 26 (Check: 78 + 26 = 104 ✓)

**Percentage of males who listen:**
(78 / 104) × 100% = (3/4) × 100% = 75%

Tempting wrong answers: Choice C (52) might come from computing 104/200 (percentage of total who are male). Choice D (55) might come from 78/136 (percentage of listeners who are male).

ANSWER 12: E

 Problem 1:
We start with 30 liters of paint. The original amount of yellow tint is 30% of 30 liters:
\[
0.30 \times 30 = 9 \text{ liters.}
\]
Five liters of yellow tint are added, so the new amount of yellow tint is:
\[
9 + 5 = 14 \text{ liters.}
\]
The new total volume of the mixture is:
\[
30 + 5 = 35 \text{ liters.}
\]
The percent of yellow tint in the new mixture is:
\[
\frac{14}{35} = \frac{2}{5} = 0.40 = 40\%.
\]
(Choice A is the original red percentage, D is the original water percentage, and E would require 17.5 liters of yellow, which we do not have.)
ANSWER 1: C

Problem 2:
Georgie chooses one of 6 windows to enter. Since he must leave through a different window, there are 5 remaining choices for his exit. By the multiplication principle:
\[
6 \times 5 = 30 \text{ ways.}
\]
(Choice E, 36, would count entering and leaving through the same window; the other choices are too small.)
ANSWER 2: D

Problem 3:
Simplify each fraction:
\[
\frac{2}{20} = \frac{1}{10}, \qquad \frac{3}{30} = \frac{1}{10}.
\]
So the sum is:
\[
\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3.
\]
(Choice A is just one term; C is 0.2; E is 0.6.)
ANSWER 3: D

Problem 4:
Susan spent $12 on food. She spent twice as much on rides:
\[
2 \times 12 = 24 \text{ dollars.}
\]
Total spent:
\[
12 + 24 = 36 \text{ dollars.}
\]
Amount left:
\[
50 - 36 = 14 \text{ dollars.}
\]
(Choice C is the total spent; D is the amount left if she had spent only $12.)
ANSWER 4: B

Problem 5:
\[
200{,}000 \times 200{,}000 = (2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}.
\]
\[
4 \times 10^{10} = 40{,}000{,}000{,}000,
\]
which is read as forty billion.
(Choice D is $4 \times 10^8$; B is $4 \times 10^6$.)
ANSWER 5: E

Problem 6:
The total amount paid is the three-digit number $1A2$, which equals:
\[
100 + 10A + 2 = 102 + 10A.
\]
Since 11 members each paid the same integer amount, $102 + 10A$ must be divisible by 11. Testing the choices:
- $A=0$: $102 \div 11 \approx 9.27$ (not an integer)
- $A=1$: $112 \div 11 \approx 10.18$ (not an integer)
- $A=2$: $122 \div 11 \approx 11.09$ (not an integer)
- $A=3$: $132 \div 11 = 12$ (integer!)
- $A=4$: $142 \div 11 \approx 12.9$ (not an integer)
Thus the missing digit is 3.
ANSWER 6: D

Problem 7:
The wire is 24 cm long, so a shape is possible only if its perimeter is at most 24 cm.
- **Regular hexagon, side 5 cm:** Perimeter $= 6 \times 5 = 30$ cm. $30 > 24$, so impossible.
- **Square of area 36 cm²:** Side $= \sqrt{36} = 6$ cm. Perimeter $= 4 \times 6 = 24$ cm. Exactly fits, so possible.
- **Right triangle with legs 6 and 8 cm:** Hypotenuse $= \sqrt{6^2 + 8^2} = \sqrt{100} = 10$ cm. Perimeter $= 6 + 8 + 10 = 24$ cm. Exactly fits, so possible.
Only the square and the triangle can be formed.
ANSWER 7: D

Problem 8:
For positive numbers, the reciprocal function $f(x) = \frac{1}{x}$ is decreasing, so the smallest number has the largest reciprocal. Compare the two smallest choices:
\[
\frac{1}{3} \approx 0.333\ldots, \qquad \frac{2}{5} = 0.4.
\]
Since $\frac{1}{3} < \frac{2}{5} < 1 < 5 < 1986$, the smallest number is $\frac{1}{3}$. Its reciprocal is $3$, which is larger than the reciprocals of all the other choices.
ANSWER 8: A

Problem 9:
First find the theoretical maximum heart rate:
\[
220 - 26 = 194 \text{ beats per minute.}
\]
The target rate is 80% of this:
\[
0.80 \times 194 = 155.2.
\]
Rounded to the nearest whole number, this is 155.
(Choice C is the maximum heart rate itself.)
ANSWER 9: B

Problem 10:
Harry computes correctly:
\[
H = 8 - (2 + 5) = 8 - 7 = 1.
\]
Terry ignores the parentheses:
\[
T = 8 - 2 + 5 = 6 + 5 = 11.
\]
The difference is:
\[
H - T = 1 - 11 = -10.
\]
ANSWER 10: A

Problem 11:
The ten numbers inside the parentheses form an arithmetic sequence: 81, 83, …, 99. The sum of an arithmetic sequence is:
\[
\frac{\text{number of terms}}{2} \times (\text{first} + \text{last}) = \frac{10}{2} \times (81 + 99) = 5 \times 180 = 900.
\]
Now multiply by the factor of 2 outside:
\[
2 \times 900 = 1800.
\]
(One can also pair them: $(81+99)=180$, $(83+97)=180$, etc., giving five pairs that each sum to 180.)
ANSWER 11: E

Problem 12:
Total surveyed = 200, and females surveyed = 96, so:
\[
\text{Males surveyed} = 200 - 96 = 104.
\]
Total listeners = 136, and females who listen = 58, so:
\[
\text{Males who listen} = 136 - 58 = 78.
\]
(We can check: total non-listeners = 64, males who don't listen = 26, so females who don't listen = 38; $58+38=96$ and $78+26=104$, which checks out.)

The percentage of males who listen is:
\[
\frac{78}{104} = \frac{3}{4} = 0.75 = 75\%.
\]
ANSWER 12: E

Problem 1:
A mixture of 30 liters is 25% red, 30% yellow, and 45% water. Five liters of yellow tint are added. What is the percent of yellow tint in the new mixture?

**Setup:** Find the amount of yellow tint originally, add 5 liters, then find the new percentage.

**Original yellow tint:** 30% of 30 = 0.30 × 30 = 9 liters

**After adding 5 liters of yellow tint:**
- New yellow tint: 9 + 5 = 14 liters
- New total volume: 30 + 5 = 35 liters
- New percentage: 14/35 = 2/5 = 40%

**Checking wrong choices:** 25% would be if we just added the original percentages; 35% would be if we just added 5% to 30%; 45% and 50% are too high.

ANSWER 1: C

---

Problem 2:
Six windows. Georgie enters by one and leaves by a different one. How many ways?

**Setup:** Choose an entry window (6 choices), then choose a different exit window (5 choices).

**Calculation:** 6 × 5 = 30 ways

**Checking wrong choices:** 15 would be if we divided by 2 (treating order as irrelevant); 12 would be 6×2; 18 would be 6×3; 36 would be 6×6 (allowing same window).

ANSWER 2: D

---

Problem 3:
1/10 + 2/20 + 3/30 = ?

**Simplify each fraction:**
- 1/10 = 1/10
- 2/20 = 1/10
- 3/30 = 1/10

**Sum:** 1/10 + 1/10 + 1/10 = 3/10 = 0.3

**Checking wrong choices:** 0.1 would be just one term; 0.2 would be two terms; 0.6 would be adding numerators and denominators incorrectly.

ANSWER 3: D

---

Problem 4:
Susan had $50. She spent $12 on food and twice as much on rides. How much left?

**Setup:** Food = $12, Rides = 2 × $12 = $24

**Total spent:** $12 + $24 = $36

**Amount left:** $50 - $36 = $14

**Checking wrong choices:** 12 is just the food amount; 26 is 50-24 (forgetting food); 38 is 50-12 (forgetting rides).

ANSWER 4: B

---

Problem 5:
200,000 × 200,000 = ?

**Setup:** Express in scientific notation: 200,000 = 2 × 10⁵

**Calculation:** (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion

**Checking wrong choices:** Four hundred thousand is 4×10⁵; four million is 4×10⁶; forty thousand is 4×10⁴; four hundred million is 4×10⁸.

ANSWER 5: E

---

Problem 6:
11 members each paid the same integer amount, totaling $1A2. Find the missing digit A.

**Setup:** The total $1A2 must be divisible by 11.

**Divisibility test for 11:** A number is divisible by 11 if the alternating sum of its digits is divisible by 11.

For 1A2: (1 + 2) - A = 3 - A must be divisible by 11.

Since A is a single digit (0-9), 3 - A can only equal 0 (the only multiple of 11 in range -6 to 3).

So A = 3, giving $132 = 11 × $12 ✓

**Checking:** With A=0: $102 ÷ 11 ≈ 9.27 (not integer); A=1: $112 ÷ 11 ≈ 10.18 (not integer); A=3: $132 ÷ 11 = 12 ✓

ANSWER 6: D

---

Problem 7:
Wire is 24 cm. Which shapes can be made?

**Regular hexagon with side 5 cm:** Perimeter = 6 × 5 = 30 cm. Need 30 cm, only have 24 cm. ❌

**Square with area 36 cm²:** Side = √36 = 6 cm. Perimeter = 4 × 6 = 24 cm. Need 24 cm, have 24 cm. ✓

**Right triangle with legs 6 and 8 cm:** Hypotenuse = √(36 + 64) = √100 = 10 cm. Perimeter = 6 + 8 + 10 = 24 cm. Need 24 cm, have 24 cm. ✓

**Conclusion:** Can make the square and triangle, but not the hexagon.

ANSWER 7: D

---

Problem 8:
Which number has the largest reciprocal?

**Find each reciprocal:**
- A. 1/3 → reciprocal = 3
- B. 2/5 → reciprocal = 5/2 = 2.5
- C. 1 → reciprocal = 1
- D. 5 → reciprocal = 1/5 = 0.2
- E. 1986 → reciprocal ≈ 0.0005

**Key insight:** Smaller positive numbers have larger reciprocals. The smallest number listed is 1/3, so it has the largest reciprocal (3).

ANSWER 8: A

---

Problem 9:
Target heart rate = 80% of (220 - age). Find target for 26-year-old.

**Maximum heart rate:** 220 - 26 = 194 bpm

**Target heart rate:** 80% of 194 = 0.80 × 194 = 155.2 ≈ 155 bpm

**Checking wrong choices:** 134 would be about 69%; 176 would be about 91%; 194 is the max (100%); 243 is impossible.

ANSWER 9: B

---

Problem 10:
Harry calculates 8 − (2 + 5) correctly. Terry calculates 8 − 2 + 5. Find H − T.

**Harry's answer:** H = 8 − (2 + 5) = 8 − 7 = 1

**Terry's answer:** T = 8 − 2 + 5 = 6 + 5 = 11

**Difference:** H − T = 1 − 11 = −10

**Checking wrong choices:** −6 would be if Terry got 7; 0 would be if they got the same answer; 6 or 10 would be T − H.

ANSWER 10: A

---

Problem 11:
2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99) = ?

**Setup:** This is 10 consecutive odd numbers. The average of the first and last is (81 + 99)/2 = 90.

**Sum of the 10 numbers:** 10 × 90 = 900

**Multiply by 2:** 2 × 900 = 1800

**Alternative verification:** Pair the numbers: (81+99) + (83+97) + (85+95) + (87+93) + (89+91) = 180 + 180 + 180 + 180 + 180 = 900. Then 2 × 900 = 1800.

ANSWER 11: E

---

Problem 12:
Find the percentage of males who listen to the station.

**Given information:**
- Total surveyed: 200
- Females: 96
- Females who listen: 58
- Males who don't listen: 26
- Total listeners: 136
- Total non-listeners: 64

**Find males:** Males = 200 - 96 = 104

**Find males who listen:** Males who listen = Total listeners - Females who listen = 136 - 58 = 78

**Verification:** Males who don't listen = 104 - 78 = 26 ✓ (matches given)

**Percentage of males who listen:** 78/104 × 100 = 75%

**Checking wrong choices:** 39% would be 78/200 (using total instead of males); 48% would be about 50/104; 52% would be about 54/104; 55% would be about 57/104.

ANSWER 12: E

**Problem 1**

The original mixture is 30 L with 30 % yellow tint → yellow = 0.30·30 = 9 L.  
Add 5 L of yellow → total yellow = 9 + 5 = 14 L.  
Total volume after addition = 30 + 5 = 35 L.  

Percent yellow = (14/35)·100 % = 40 %.

**Answer 1: C**

---

**Problem 2**

Choose an entry window (6 ways) and a different exit window (5 ways).  
Number of ways = 6 × 5 = 30.

**Answer 2: D**

---

**Problem 3**

\[
\frac{1}{10}=0.1,\qquad\frac{2}{20}=0.1,\qquad\frac{3}{30}=0.1
\]
Sum = 0.1 + 0.1 + 0.1 = 0.3.

**Answer 3: D**

---

**Problem 4**

Food cost = \$12.  
Rides cost = twice food = 2·12 = \$24.  
Total spent = 12 + 24 = \$36.  
Money left = 50 − 36 = \$14.

**Answer 4: B**

---

**Problem 5**

\(200{,}000 \times 200{,}000 = (2\times10^{5})^{2}=4\times10^{10}=40{,}000{,}000{,}000\).

That is 40 billion.

**Answer 5: E**

---

**Problem 6**

Total paid = \$1A2 = 100 + 10A + 2 = 102 + 10A.  
Since the 11 members each paid an integer amount, 102 + 10A must be divisible by 11.

\(102\equiv3\pmod{11}\).  
We need \(10A\equiv-3\equiv8\pmod{11}\).  
Because \(10\equiv-1\pmod{11}\), this is \(-A\equiv8\pmod{11}\) → \(A\equiv3\pmod{11}\).  
A digit, so \(A=3\).

**Answer 6: D**

---

**Problem 7**

- Hexagon: perimeter = 6·5 = 30 cm > 24 cm → impossible.  
- Square: side = √36 = 6 cm, perimeter = 4·6 = 24 cm → possible.  
- Right triangle: legs 6 cm, 8 cm, hypotenuse = √(6²+8²)=10 cm, perimeter = 6+8+10 = 24 cm → possible.

Only the square and triangle can be made.

**Answer 7: D**

---

**Problem 8**

Reciprocals:
- \(1/3 → 3\)  
- \(2/5 → 5/2 = 2.5\)  
- \(1 → 1\)  
- \(5 → 1/5 = 0.2\)  
- \(1986 → 1/1986 ≈ 0.0005\)

Largest reciprocal is 3, from \(1/3\).

**Answer 8: A**

---

**Problem 9**

Maximum heart rate = 220 − 26 = 194 bpm.  
Target = 80 % of 194 = 0.8·194 = 155.2 → rounded to 155 bpm.

**Answer 9: B**

---

**Problem 10**

Harry: \(8-(2+5)=8-7=1\).  
Terry (ignoring parentheses): \(8-2+5 = 6+5 = 11\).  
Difference \(H-T = 1-11 = -10\).

**Answer 10: A**

---

**Problem 11**

Sum of the ten terms: average = (81+99)/2 = 90, so sum = 90·10 = 900.  
Multiply by 2: \(2·900 = 1800\).

**Answer 11: E**

---

**Problem 12**

Total surveyed = 200.  
Females = 96 → Males = 200 − 96 = 104.  

Female listeners = 58 → Male listeners = 136 − 58 = 78.  

Percent of males who listen = (78/104)·100 % = (39/52)·100 % = 75 %.

**Answer 12: E**

---

**Final Answers**

ANSWER 1: C  
ANSWER 2: D  
ANSWER 3: D  
ANSWER 4: B  
ANSWER 5: E  
ANSWER 6: D  
ANSWER 7: D  
ANSWER 8: A  
ANSWER 9: B  
ANSWER 10: A  
ANSWER 11: E  
ANSWER 12: E

Problem 1:  
A 30-liter paint mixture is 25% red tint, 30% yellow tint, and 45% water. Five liters of yellow tint are added.  
Original yellow tint: $0.30 \times 30 = 9$ liters.  
After adding 5 liters: $9 + 5 = 14$ liters yellow.  
New total volume: $30 + 5 = 35$ liters.  
Percent yellow: $\frac{14}{35} = 0.40 = 40\%$.  
Tempting wrong choices: 35% (original yellow percent) or 45% (water percent).  
ANSWER 1: D  

Problem 2:  
Georgie enters through one of six windows and exits through a different one.  
Entry choices: 6. Exit choices: 5 (excluding entry window).  
Total ways: $6 \times 5 = 30$.  
Tempting wrong choices: 15 (if order didn’t matter) or 36 (if same window allowed).  
ANSWER 2: D  

Problem 3:  
$\frac{1}{10} + \frac{2}{20} + \frac{3}{30} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3$.  
Tempting wrong choices: 0.1 (one term) or 0.2 (two terms).  
ANSWER 3: D  

Problem 4:  
Susan starts with \$50.  
Food: \$12.  
Rides: $2 \times 12 = \$24$.  
Total spent: $12 + 24 = \$36$.  
Left: $50 - 36 = \$14$.  
Tempting wrong choices: \$26 (if spent \$24 total) or \$38 (if spent \$12 total).  
ANSWER 4: B  

Problem 5:  
$200,000 \times 200,000 = 40,000,000,000 = 40$ billion.  
Tempting wrong choices: 400,000 (added zeros) or 4 million (multiplied digits).  
ANSWER 5: E  

Problem 6:  
Total paid: \$1A2, an integer divisible by 11.  
$1A2 \mod 11 = 0$.  
$1 + 2 - A \equiv 0 \mod 11 \Rightarrow 3 - A \equiv 0 \mod 11 \Rightarrow A = 3$.  
Tempting wrong choices: 0, 1, 2, or 4 (incorrect divisibility).  
ANSWER 6: E (Wait, let's recheck: 1A2 must be divisible by 11. 132 ÷ 11 = 12, so A=3. But 132 is 11×12, so correct. However, the answer choices are A=0,B=1,C=2,D=3,E=4. So D is 3. So correct answer is D, not E. I made a mistake in the initial answer. Correcting:)  
**Correction**: $1A2 = 132$ when $A=3$, and $132 \div 11 = 12$, so $A=3$.  
ANSWER 6: D  

Problem 7:  
Wire length: 24 cm.  
- Regular hexagon with side 5 cm: Perimeter $6 \times 5 = 30$ cm (not 24).  
- Square with area 36 cm²: Side $6$ cm, perimeter $24$ cm.  
- Right triangle with legs 6 and 8 cm: Hypotenuse 10 cm, perimeter $6+8+10=24$ cm.  
Can make square and triangle.  
Tempting wrong choices: Hexagon (incorrect perimeter).  
ANSWER 7: D  

Problem 8:  
Largest reciprocal means smallest original number.  
$\frac{1}{3} \approx 0.333$, $\frac{2}{5} = 0.4$, $1$, $5$, $1986$.  
Smallest: $\frac{1}{3}$. Reciprocal: 3.  
Others: 2.5, 1, 0.2, 0.0005.  
Tempting wrong choices: $\frac{2}{5}$ (larger than $\frac{1}{3}$).  
ANSWER 8: A  

Problem 9:  
Maximum heart rate: $220 - 26 = 194$.  
Target: $0.80 \times 194 = 155.2 \approx 155$.  
Tempting wrong choices: 134 (70%), 176 (90%), or 243 (incorrect max).  
ANSWER 9: B  

Problem 10:  
Harry: $8 - (2 + 5) = 1$.  
Terry: $8 - 2 + 5 = 11$.  
Difference: $1 - 11 = -10$.  
Tempting wrong choices: 10 (absolute difference) or 0 (if parentheses ignored by both).  
ANSWER 10: A  

Problem 11:  
Sum: $81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99$.  
Average: 90, 10 terms: $90 \times 10 = 900$.  
$2 \times 900 = 1800$.  
Tempting wrong choices: 1600 (min+max=180, times 5=900, times 2=1800, but if miscount terms).  
ANSWER 11: E  

Problem 12:  
Total surveyed: 200.  
Females: 96, so males: $200 - 96 = 104$.  
Total listeners: 136.  
Female listeners: 58, so male listeners: $136 - 58 = 78$.  
Percent male listeners: $\frac{78}{104} = 0.75 = 75\%$.  
Tempting wrong choices: 39 (male non-listeners percent) or 52 (incorrect calculation).  
ANSWER 12: E

### Problem 1:
What is the percentage of yellow tint in the new 35-liter paint mixture, after adding 5 liters of yellow tint to an original 30-liter mixture that was 30% yellow?
First, calculate the original volume of yellow tint: 30% of 30 liters = \(0.3 \times 30 = 9\) liters. Add the 5 new liters of yellow tint to get \(9 + 5 = 14\) liters of yellow in the new mixture. The total new volume of the mixture is \(30 + 5 = 35\) liters. The percentage of yellow is \(\frac{14}{35} = 0.4 = 40\%\). Tempting wrong answers come from forgetting to update the total mixture volume when calculating the new percentage, or miscalculating the original yellow volume.
ANSWER 1: C

---

### Problem 2:
How many ways can Georgie enter through one window and leave through a different window, out of 6 total windows?
Georgie has 6 choices for an entry window. Since he must leave through a different window, he has 5 remaining choices for an exit window. The total number of ways is \(6 \times 5 = 30\). A common wrong answer is 15, which incorrectly treats entry/exit as unordered pairs, when order matters (entering window A and leaving B is different from entering B and leaving A).
ANSWER 2: D

---

### Problem 3:
Simplify \(\frac{1}{10} + \frac{2}{20} + \frac{3}{30}\):
Simplify each fraction first: \(\frac{2}{20} = \frac{1}{10}\) and \(\frac{3}{30} = \frac{1}{10}\). Add them together: \(\frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} = 0.3\). A common wrong answer is 0.1, which incorrectly adds numerators and denominators directly instead of simplifying first.
ANSWER 3: D

---

### Problem 4:
How much money does Susan have left after spending $12 on food and twice that amount on rides, starting with $50?
First calculate the amount spent on rides: twice $12 is \(2 \times 12 = 24\) dollars. Total spent is \(12 + 24 = 36\) dollars. The amount left is \(50 - 36 = 14\) dollars. A common wrong answer is 26, which incorrectly uses $12 instead of $24 for ride costs.
ANSWER 4: B

---

### Problem 5:
Calculate \(200,000 \times 200,000\):
Rewrite 200,000 as \(2 \times 10^5\). Multiplying two copies gives \((2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}\), which equals 40,000,000,000, or forty billion. Wrong answers come from miscounting the number of zeros in the product.
ANSWER 5: E

---

### Problem 6:
The 3-digit total cost \(1A2\) dollars is divisible by 11 (11 members paid the same integer amount). What is the missing digit A?
The divisibility rule for 11 states that alternating sum of digits must be divisible by 11. For 1A2, the alternating sum is \(1 - A + 2 = 3 - A\). The only value of A (0-9) that makes this sum divisible by 11 is A=3, since \(3-3=0\), which is divisible by 11. Verify: \(132 \div 11 = 12\), a valid integer.
ANSWER 6: D

---

### Problem 7:
Haruki has a 24cm wire. Which of the three shapes can he form? Calculate the perimeter of each shape:
1. Regular hexagon with side 5cm: perimeter = \(6 \times 5 = 30\)cm, which is longer than 24cm, so impossible.
2. Square with area 36cm²: side length = \(\sqrt{36}=6\)cm, perimeter = \(4 \times 6 = 24\)cm, which fits.
3. Right triangle with legs 6cm and 8cm: hypotenuse = 10cm (6-8-10 Pythagorean triple), perimeter = \(6+8+10=24\)cm, which fits.
Only the square and triangle can be formed.
ANSWER 7: D

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### Problem 8:
Which number has the largest reciprocal? The reciprocal of \(x\) is \(\frac{1}{x}\), so smaller values of \(x\) produce larger reciprocals. List the values of all options:
A. \(\frac{1}{3} \approx 0.333\), B. \(\frac{2}{5}=0.4\), C.1, D.5, E.1986. The smallest value is \(\frac{1}{3}\), whose reciprocal is 3, the largest of all options.
ANSWER 8: A

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### Problem 9:
Calculate the target heart rate for a 26-year-old athlete: target is 80% of max heart rate, where max = 220 - age.
First calculate max heart rate: \(220 - 26 = 194\) bpm. 80% of 194 is \(0.8 \times 194 = 155.2\), which rounds to 155 bpm. A common wrong answer is 176, which uses 220 instead of the correct max heart rate of 194.
ANSWER 9: B

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### Problem 10:
Calculate \(H-T\), where H is Harry's correct answer to \(8-(2+5)\), and T is Terry's incorrect answer of \(8-2+5\).
Harry's answer: \(8 - 7 = 1 = H\). Terry's answer: \(6 + 5 = 11 = T\). The difference \(H-T = 1 - 11 = -10\). A common wrong answer is 10, which calculates \(T-H\) instead of \(H-T\).
ANSWER 10: A

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### Problem 11:
Simplify \(2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)\):
Pair the terms inside the parentheses to simplify addition: \((81+99)+(83+97)+(85+95)+(87+93)+(89+91) = 5 \times 180 = 900\). Multiply by 2: \(2 \times 900 = 1800\).
ANSWER 11: E

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### Problem 12:
What percentage of males surveyed listen to the radio station? First calculate total number of males: total surveyed = 200, females = 96, so males = \(200 - 96 = 104\). Males who do not listen = 26, so males who listen = \(104 - 26 = 78\). The percentage of males who listen is \(\frac{78}{104} = 0.75 = 75\%\). A common wrong answer is 39%, which divides male listeners by the total surveyed population instead of the total male population.
ANSWER 12: E

### Problem 1:
We need to find the percent of yellow tint in the new mixture after adding 5 liters of yellow tint to the original 30-liter mixture.
1.  Calculate original yellow tint volume: 25%? No wait, original is 30% yellow. So 30% of 30 L = $0.3 \times 30 = 9$ liters of yellow.
2.  Add 5 liters of yellow: total yellow = $9 + 5 = 14$ liters.
3.  Total new mixture volume: $30 + 5 = 35$ liters.
4.  Percent yellow: $\frac{14}{35} \times 100 = 40\%$.
ANSWER 1: C
---
### Problem 2:
We count the number of ways to choose an entry window and a different exit window from 6 windows.
1.  Entry window: 6 possible choices.
2.  Exit window: cannot be the entry window, so 5 remaining choices.
3.  Total ways: $6 \times 5 = 30$ (order matters because entry and exit are distinct actions).
ANSWER 2: D
---
### Problem 3:
Simplify and add the fractions:
1.  $\frac{1}{10} = 0.1$, $\frac{2}{20} = \frac{1}{10} = 0.1$, $\frac{3}{30} = \frac{1}{10} = 0.1$.
2.  Sum: $0.1 + 0.1 + 0.1 = 0.3$.
ANSWER 3: D
---
### Problem 4:
Calculate remaining money after expenses:
1.  Total initial money: $50.
2.  Money spent on rides: twice the food cost = $2 \times 12 = 24$ dollars.
3.  Total spent: $12 + 24 = 36$ dollars.
4.  Remaining: $50 - 36 = 14$ dollars.
ANSWER 4: B
---
### Problem 5:
Compute $200,000 \times 200,000$:
1.  Rewrite as $(2 \times 10^5) \times (2 \times 10^5) = 4 \times 10^{10}$.
2.  $10^{10} = 10,000,000,000$, so $4 \times 10^{10} = 40,000,000,000 = 40$ billion.
ANSWER 5: E
---
### Problem 6:
The total payment $1A2$ (a 3-digit number) must be divisible by 11 (since 11 people paid equal integer amounts).
1.  Use the divisibility rule for 11: the alternating sum of digits is divisible by 11. For $1A2$, this is $1 - A + 2 = 3 - A$.
2.  $A$ is a digit (0-9), so $3 - A$ must equal 0 (the only multiple of 11 in range: $3 - A = 11$ gives $A=-8$, $3-A=-11$ gives $A=14$, both invalid).
3.  $3 - A = 0 \implies A=3$. Verify: $132 \div 11 = 12$, which is an integer.
ANSWER 6: D
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### Problem 7:
Check if the 24 cm wire can form each shape (perimeter of the shape must equal 24 cm):
1.  Regular hexagon (side 5 cm): Perimeter = $6 \times 5 = 30$ cm > 24 cm → cannot make.
2.  Square (area 36 cm²): Side length = $\sqrt{36} = 6$ cm. Perimeter = $4 \times 6 = 24$ cm → can make.
3.  Right triangle (legs 6, 8 cm): Hypotenuse = $\sqrt{6^2 + 8^2} = 10$ cm. Perimeter = $6 + 8 + 10 = 24$ cm → can make.
Only square and triangle can be made.
ANSWER 7: D
---
### Problem 8:
The reciprocal of a positive number $x$ is $\frac{1}{x}$; smaller $x$ gives larger reciprocal.
1.  List values: $\frac{1}{3} \approx 0.33$, $\frac{2}{5}=0.4$, $1$, $5$, $1986$.
2.  Smallest value is $\frac{1}{3}$, whose reciprocal is $3$ (largest among all reciprocals: $\frac{5}{2}=2.5$, $1$, $0.2$, $\frac{1}{1986} \approx 0.0005$).
ANSWER 8: A
---
### Problem 9:
Calculate target heart rate for a 26-year-old:
1.  Maximum heart rate: $220 - 26 = 194$ bpm.
2.  Target heart rate: 80% of 194 = $0.8 \times 194 = 155.2$, which rounds to 155.
ANSWER 9: B
---
### Problem 10:
Compute $H$ (Harry's correct answer) and $T$ (Terry's incorrect answer):
1.  Harry: $8 - (2 + 5) = 8 - 7 = 1$, so $H=1$.
2.  Terry (ignores parentheses, left-to-right): $8 - 2 + 5 = 6 + 5 = 11$, so $T=11$.
3.  Difference: $H - T = 1 - 11 = -10$.
ANSWER 10: A
---
### Problem 11:
Calculate $2 \times (81 + 83 + ... + 99)$:
1.  The sum inside is 10 consecutive odd numbers. Average of first and last term: $\frac{81 + 99}{2} = 90$.
2.  Sum of the 10 terms: $90 \times 10 = 900$.
3.  Multiply by 2: $2 \times 900 = 1800$.
ANSWER 11: E
---
### Problem 12:
Find the percentage of males who listen to the station:
1.  Total males surveyed: $200 - 96 = 104$.
2.  Males who do not listen: 26, so males who listen: $104 - 26 = 78$.
3.  Verify with total listeners: Total listeners = 136, female listeners = 58, so male listeners = $136 - 58 = 78$ (matches).
4.  Percentage: $\frac{78}{104} \times 100 = 75\%$.
ANSWER 12: E

Problem 1:
Original yellow = 30% of 30 = 9 L. Add 5 → 14 L yellow. New total = 35 L. Percent = 14/35 = 40%.

ANSWER 1: C

Problem 2:
Enter: 6 choices, leave: 5 different choices. 6×5 = 30.

ANSWER 2: D

Problem 3:
Each fraction = 0.1. Sum = 0.3.

ANSWER 3: D

Problem 4:
Food 12, rides twice = 24. Spent 36. Left = 50 − 36 = 14.

ANSWER 4: B

Problem 5:
200,000 × 200,000 = 40,000,000,000 = forty billion.

ANSWER 5: E

Problem 6:
1A2 divisible by 11. Alternating sum: 1 − A + 2 = 3 − A must be divisible by 11 → A = 3 (gives 0). Check 132/11 = 12. Yes.

ANSWER 6: D

Problem 7:
Hexagon: 6×5 = 30 > 24, no. Square area 36 → side 6, perimeter 24, yes. Triangle legs 6,8 → hypotenuse 10, perimeter 24, yes. Square and triangle only.

ANSWER 7: D

Problem 8:
Largest reciprocal = smallest number. Smallest is 1/3.

ANSWER 8: A

Problem 9:
Max = 220 − 26 = 194. Target = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

Problem 10:
H = 8 − 7 = 1. T = 8 − 2 + 5 = 11. H − T = 1 − 11 = −10.

ANSWER 10: A

Problem 11:
Sum of 81..99 odd numbers (10 terms): average (81+99)/2 = 90, ×10 = 900. ×2 = 1800.

ANSWER 11: E

Problem 12:
Males total = 200 − 96 = 104. Males who don't listen = 26 → males who listen = 104 − 26 = 78. Percent = 78/104 = 75%.

ANSWER 12: E

I'll solve each problem systematically.

---

**Problem 1:**
Original mixture: 30 liters, 30% yellow = 9 liters yellow. Add 5 liters yellow → new yellow = 14 liters, new total = 35 liters.
Percent yellow = 14/35 = 40%.

ANSWER 1: C

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**Problem 2:**
Georgie can enter through any of 6 windows, then leave through any of the remaining 5 windows.
6 × 5 = 30.

ANSWER 2: D

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**Problem 3:**
1/10 = 0.1, 2/20 = 0.1, 3/30 = 0.1. Sum = 0.3.

ANSWER 3: D

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**Problem 4:**
Food: $12. Rides: 2 × $12 = $24. Total spent: $12 + $24 = $36. Left: $50 − $36 = $14.

ANSWER 4: B

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**Problem 5:**
200,000 × 200,000 = (2 × 10⁵) × (2 × 10⁵) = 4 × 10¹⁰ = 40,000,000,000 = forty billion.

ANSWER 5: E

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**Problem 6:**
The number 1A2 must be divisible by 11. 1A2 ÷ 11 must be an integer.
11 × 10 = 110, 11 × 11 = 121 (no), 11 × 12 = 132 ✓ → A = 3.
Check: 132/11 = 12 ✓, each paid $12.

ANSWER 6: D

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**Problem 7:**
Wire = 24 cm.
- Regular hexagon, side 5: perimeter = 6 × 5 = 30 cm > 24. ✗
- Square, area 36: side = 6, perimeter = 24 cm ✓
- Right triangle, legs 6 and 8: hypotenuse = 10, perimeter = 6+8+10 = 24 cm ✓

ANSWER 7: D

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**Problem 8:**
The reciprocal of x is 1/x. The largest reciprocal comes from the smallest positive number.
Among 1/3, 2/5, 1, 5, 1986 — the smallest is 1/3, and its reciprocal is 3, which is the largest.

ANSWER 8: A

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**Problem 9:**
Maximum heart rate = 220 − 26 = 194. Target = 80% × 194 = 0.8 × 194 = 155.2 ≈ 155.

ANSWER 9: B

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**Problem 10:**
Harry: 8 − (2 + 5) = 8 − 7 = 1, so H = 1.
Terry: 8 − 2 + 5 = 11, so T = 11.
H − T = 1 − 11 = −10.

ANSWER 10: A

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**Problem 11:**
The ten odd numbers from 81 to 99: sum = (81+99)/2 × 10 = 90 × 10 = 900.
2 × 900 = 1800.

ANSWER 11: E

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**Problem 12:**
Total surveyed = 200. Females = 96, so Males = 104.
Total non-listeners = 64. Males who don't listen = 26, so females who don't listen = 64 − 26 = 38.
Males who listen = 104 − 26 = 78.
Percentage = 78/104 = 75%.

ANSWER 12: E