🔬 Model Lab

Problem 1:
We are asked to compute 1/10 + 9/100 + 9/1000 + 7/10000.

Recall place value: 1/10 = 0.1 is 1 in the tenths place. 9/100 = 0.09 is 9 in the hundredths place. 9/1000 = 0.009 is 9 in the thousandths place. 7/10000 = 0.0007 is 7 in the ten-thousandths place.

Add them aligned by decimal point:
  0.1
  0.09
  0.009
+ 0.0007
= 0.1000 + 0.0900 + 0.0090 + 0.0007 = 0.1997.

Check choices: A. 0.0026 is far too small, it mistakes place values. B. 0.0197 shifts everything one place right. D. 0.26 adds incorrectly. E. 1.997 shifts everything one place left. Only C. 0.1997 matches.
ANSWER 1: C

Problem 2:
We need to convert 11 hours and 5 minutes entirely to minutes.

We know 1 hour = 60 minutes. So 11 hours = 11 x 60 minutes.

Compute 11 x 60: 10 x 60 = 600, plus 1 x 60 = 60, total 660 minutes.

Then add the extra 5 minutes: 660 + 5 = 665 minutes.

Check choices: A. 605 is 10 hours 5 minutes. B. 655 is 10 minutes short. D. 1005 uses 100 instead of 60. E. 1105 uses 100 instead of 60 for 11 hours. Only C. 665 is correct.
ANSWER 2: C

Problem 1:

**Restate the problem:**
We are asked to find the decimal equivalent of the sum:
$$\frac{1}{10} + \frac{9}{100} + \frac{9}{1000} + \frac{7}{10000}$$

**Step-by-step computation:**
Convert each fraction into its decimal representation according to place value:
- $\frac{1}{10} = 0.1$ (tenths place)
- $\frac{9}{100} = 0.09$ (hundredths place)
- $\frac{9}{1000} = 0.009$ (thousandths place)
- $\frac{7}{10000} = 0.0007$ (ten-thousandths place)

Adding these decimals together:
$$\begin{array}{r@{\quad}l}
  & 0.1000 \\
+ & 0.0900 \\
+ & 0.0090 \\
+ & 0.0007 \\
\hline
  & 0.1997
\end{array}$$

**Examine choices:**
- A: $0.0026$ and D: $0.26$ occur if one mistakenly adds the numerators $(1+9+9+7=26)$ and misplaces the decimal point.
- B: $0.0197$ has an extra zero after the decimal point.
- E: $1.997$ has the decimal point shifted one position to the right.
- C: $0.1997$ is correct.

ANSWER 1: C

---

Problem 2:

**Restate the problem:**
We are asked to convert a duration of $11\text{ hours and } 5\text{ minutes}$ entirely into minutes.

**Step-by-step computation:**
There are $60\text{ minutes}$ in $1\text{ hour}$.
First, calculate the number of minutes in $11\text{ hours}$:
$$11\text{ hours} \times 60\text{ minutes/hour} = 660\text{ minutes}$$

Next, add the remaining $5\text{ minutes}$:
$$660\text{ minutes} + 5\text{ minutes} = 665\text{ minutes}$$

**Examine choices:**
- A: $605$ minutes is $10\text{ hours and } 5\text{ minutes}$.
- B: $655$ minutes would be $660 - 5$, mistakenly subtracting rather than adding.
- C: $665$ minutes is correct.
- D: $1005$ and E: $1105$ come from mistakenly treating an hour as $100$ minutes (e.g., $11 \times 100 + 5 = 1105$).

ANSWER 2: C

**Problem 1:**
The expression appears to be a sum of fractions:
$$\frac{1}{10} + \frac{9}{100} + \frac{9}{1000} + \frac{7}{10000}$$

**What's being asked:** Find the decimal value of this sum.

**Set up the math:** Convert each fraction to a decimal by placing the numerator after the decimal point, with the denominator's number of zeros determining the decimal place:
- $\frac{1}{10} = 0.1$ (one decimal place)
- $\frac{9}{100} = 0.09$ (two decimal places)
- $\frac{9}{1000} = 0.009$ (three decimal places)
- $\frac{7}{10000} = 0.0007$ (four decimal places)

**Add them by lining up decimal places:**
$$
\begin{array}{r}
0.1000\\
0.0900\\
0.0090\\
+\ 0.0007\\
\hline
0.1997
\end{array}
$$

Column by column: thousandths column gives $0+9+9+7 = 25$, write 5, carry 2; hundredths column gives $0+9+0+2 = 11$, write 1, carry 1; tenths column gives $1+0+0+1 = 2$. So the sum is $0.1997$.

**Rule out tempting wrong choices:**
- (A) $0.0026$ is far too small — the term $\frac{1}{10}$ alone is $0.1$.
- (B) $0.0197$ results from misreading $\frac{1}{10}$ as $\frac{1}{100}$ — a common alignment error.
- (D) $0.26$ is too large; the sum of $0.1 + 0.09 + 0.009$ is only about $0.199$.
- (E) $1.997$ is ten times too large — all four fractions are less than 1, and the largest is only $0.1$.

ANSWER 1: C

**Problem 2:**
**What's being asked:** Convert 11 hours and 5 minutes into total minutes.

**Set up the math:** There are 60 minutes in 1 hour, so 11 hours contains:
$$11 \times 60 = 660 \text{ minutes}$$

**Add the extra minutes:**
$$660 + 5 = 665 \text{ minutes}$$

**Rule out tempting wrong choices:**
- (A) $605$ incorrectly uses "11 hours ≈ 600 minutes" rounding down and misses the $+5$... it's a miscalculation ($10 \times 60 + 5 = 605$ — forgetting one hour).
- (B) $655$ would come from $11 \times 60 = 660$ computed as $650$, or subtracting 5 instead of adding.
- (D) $1005$ and (E) $1105$ mistakenly treat hours as roughly 100 minutes each (e.g., $11 \times 100 + 5 = 1105$) — but an hour is 60 minutes, not 100.

ANSWER 2: C

Problem 1:

We are evaluating

\[
\frac{1}{10}+\frac{9}{100}+\frac{9}{1000}+\frac{7}{10000}.
\]

To add these fractions, rewrite them all with denominator \(10000\):

\[
\frac{1}{10}=\frac{1000}{10000},
\]

\[
\frac{9}{100}=\frac{900}{10000},
\]

\[
\frac{9}{1000}=\frac{90}{10000},
\]

\[
\frac{7}{10000}=\frac{7}{10000}.
\]

Now add the numerators:

\[
\frac{1000+900+90+7}{10000}=\frac{1997}{10000}=0.1997.
\]

So the correct choice is C.

ANSWER 1: C

Problem 2:

We need to convert 11 hours and 5 minutes into total minutes.

Since 1 hour equals 60 minutes,

\[
11\text{ hours}=11\cdot 60=660\text{ minutes}.
\]

Now add the extra 5 minutes:

\[
660+5=665.
\]

So the total is 665 minutes.

ANSWER 2: C

## Problem 1:

We are asked to add the fractions $\frac{1}{10} + \frac{9}{100} + \frac{9}{1000} + \frac{7}{10000}$ and express the result as a decimal.

**Convert each fraction to a decimal:**
- $\frac{1}{10} = 0.1$ (1 in the tenths place)
- $\frac{9}{100} = 0.09$ (9 in the hundredths place)
- $\frac{9}{1000} = 0.009$ (9 in the thousandths place)
- $\frac{7}{10000} = 0.0007$ (7 in the ten-thousandths place)

**Add by stacking place values:**

$$0.1000 + 0.0900 + 0.0090 + 0.0007 = 0.1997$$

Since each fraction contributes exactly one digit to its own place value, the digits simply line up as 1, 9, 9, 7 after the decimal point.

**Ruling out the distractors:**
- **0.0197** would come from mistakenly putting the 1 in the hundredths place (shifting everything one place right).
- **1.997** would come from shifting everything one place left (treating $\frac{1}{10}$ as 1).
- **0.26** and **0.0026** come from just adding the numerators ($1+9+9+7=26$) and ignoring the different denominators.

The correct sum is $0.1997$.

ANSWER 1: C

## Problem 2:

We must convert 11 hours and 5 minutes entirely into minutes.

**Step 1 — Convert the hours:** There are 60 minutes in one hour, so
$$11 \times 60 = 660 \text{ minutes}$$

**Step 2 — Add the extra minutes:**
$$660 + 5 = 665 \text{ minutes}$$

**Ruling out the distractors:**
- **605** comes from mistakenly computing $600 + 5$ (using 11 × 60 = 600, an error).
- **655** comes from subtracting the 5 instead of adding it ($660 - 5$).
- **1105** and **1005** come from just writing the digits "11" and "05" together rather than converting units.

So the match lasted $665$ minutes.

ANSWER 2: C

Problem 1:

We are asked to evaluate

\[
\frac{1}{10}+\frac{9}{100}+\frac{9}{1000}+\frac{7}{10000}.
\]

Write each fraction as a decimal:

\[
\frac{1}{10}=0.1
\]

\[
\frac{9}{100}=0.09
\]

\[
\frac{9}{1000}=0.009
\]

\[
\frac{7}{10000}=0.0007
\]

Now add carefully:

\[
0.1+0.09=0.19
\]

\[
0.19+0.009=0.199
\]

\[
0.199+0.0007=0.1997
\]

So the correct choice is C.

Choice A, \(0.0026\), is far too small because \(\frac{1}{10}\) alone is already \(0.1\).  
Choice B, \(0.0197\), incorrectly places the \(1\) in the hundredths place rather than the tenths place.  
Choice D, \(0.26\), and choice E, \(1.997\), do not match the place values of the fractions.

ANSWER 1: C

Problem 2:

We are asked to convert 11 hours and 5 minutes into minutes.

Since each hour has 60 minutes,

\[
11 \text{ hours}=11\cdot 60=660 \text{ minutes}.
\]

Then add the extra 5 minutes:

\[
660+5=665.
\]

Therefore, the match lasted 665 minutes.

Choice A, 605, would be \(10\) hours and \(5\) minutes.  
Choice B, 655, is 10 minutes too few.  
Choices D and E are much too large.

ANSWER 2: C

Problem 1:
We are asked to compute 1/10 + 9/100 + 9/1000 + 7/10000.

Recall place value: 1/10 = 0.1 is 1 in the tenths place. 9/100 = 0.09 is 9 in the hundredths place. 9/1000 = 0.009 is 9 in the thousandths place. 7/10000 = 0.0007 is 7 in the ten-thousandths place.

Add them aligned by decimal point:
  0.1
  0.09
  0.009
+ 0.0007
= 0.1000 + 0.0900 + 0.0090 + 0.0007 = 0.1997.

Check choices: A. 0.0026 is far too small, it mistakes place values. B. 0.0197 shifts everything one place right. D. 0.26 adds incorrectly. E. 1.997 shifts everything one place left. Only C. 0.1997 matches.
ANSWER 1: C

Problem 2:
We need to convert 11 hours and 5 minutes entirely to minutes.

We know 1 hour = 60 minutes. So 11 hours = 11 x 60 minutes.

Compute 11 x 60: 10 x 60 = 600, plus 1 x 60 = 60, total 660 minutes.

Then add the extra 5 minutes: 660 + 5 = 665 minutes.

Check choices: A. 605 is 10 hours 5 minutes. B. 655 is 10 minutes short. D. 1005 uses 100 instead of 60. E. 1105 uses 100 instead of 60 for 11 hours. Only C. 665 is correct.
ANSWER 2: C

Problem 1:

**Restate the problem:**
We are asked to find the decimal equivalent of the sum:
$$\frac{1}{10} + \frac{9}{100} + \frac{9}{1000} + \frac{7}{10000}$$

**Step-by-step computation:**
Convert each fraction into its decimal representation according to place value:
- $\frac{1}{10} = 0.1$ (tenths place)
- $\frac{9}{100} = 0.09$ (hundredths place)
- $\frac{9}{1000} = 0.009$ (thousandths place)
- $\frac{7}{10000} = 0.0007$ (ten-thousandths place)

Adding these decimals together:
$$\begin{array}{r@{\quad}l}
  & 0.1000 \\
+ & 0.0900 \\
+ & 0.0090 \\
+ & 0.0007 \\
\hline
  & 0.1997
\end{array}$$

**Examine choices:**
- A: $0.0026$ and D: $0.26$ occur if one mistakenly adds the numerators $(1+9+9+7=26)$ and misplaces the decimal point.
- B: $0.0197$ has an extra zero after the decimal point.
- E: $1.997$ has the decimal point shifted one position to the right.
- C: $0.1997$ is correct.

ANSWER 1: C

---

Problem 2:

**Restate the problem:**
We are asked to convert a duration of $11\text{ hours and } 5\text{ minutes}$ entirely into minutes.

**Step-by-step computation:**
There are $60\text{ minutes}$ in $1\text{ hour}$.
First, calculate the number of minutes in $11\text{ hours}$:
$$11\text{ hours} \times 60\text{ minutes/hour} = 660\text{ minutes}$$

Next, add the remaining $5\text{ minutes}$:
$$660\text{ minutes} + 5\text{ minutes} = 665\text{ minutes}$$

**Examine choices:**
- A: $605$ minutes is $10\text{ hours and } 5\text{ minutes}$.
- B: $655$ minutes would be $660 - 5$, mistakenly subtracting rather than adding.
- C: $665$ minutes is correct.
- D: $1005$ and E: $1105$ come from mistakenly treating an hour as $100$ minutes (e.g., $11 \times 100 + 5 = 1105$).

ANSWER 2: C

**Problem 1:**
The expression appears to be a sum of fractions:
$$\frac{1}{10} + \frac{9}{100} + \frac{9}{1000} + \frac{7}{10000}$$

**What's being asked:** Find the decimal value of this sum.

**Set up the math:** Convert each fraction to a decimal by placing the numerator after the decimal point, with the denominator's number of zeros determining the decimal place:
- $\frac{1}{10} = 0.1$ (one decimal place)
- $\frac{9}{100} = 0.09$ (two decimal places)
- $\frac{9}{1000} = 0.009$ (three decimal places)
- $\frac{7}{10000} = 0.0007$ (four decimal places)

**Add them by lining up decimal places:**
$$
\begin{array}{r}
0.1000\\
0.0900\\
0.0090\\
+\ 0.0007\\
\hline
0.1997
\end{array}
$$

Column by column: thousandths column gives $0+9+9+7 = 25$, write 5, carry 2; hundredths column gives $0+9+0+2 = 11$, write 1, carry 1; tenths column gives $1+0+0+1 = 2$. So the sum is $0.1997$.

**Rule out tempting wrong choices:**
- (A) $0.0026$ is far too small — the term $\frac{1}{10}$ alone is $0.1$.
- (B) $0.0197$ results from misreading $\frac{1}{10}$ as $\frac{1}{100}$ — a common alignment error.
- (D) $0.26$ is too large; the sum of $0.1 + 0.09 + 0.009$ is only about $0.199$.
- (E) $1.997$ is ten times too large — all four fractions are less than 1, and the largest is only $0.1$.

ANSWER 1: C

**Problem 2:**
**What's being asked:** Convert 11 hours and 5 minutes into total minutes.

**Set up the math:** There are 60 minutes in 1 hour, so 11 hours contains:
$$11 \times 60 = 660 \text{ minutes}$$

**Add the extra minutes:**
$$660 + 5 = 665 \text{ minutes}$$

**Rule out tempting wrong choices:**
- (A) $605$ incorrectly uses "11 hours ≈ 600 minutes" rounding down and misses the $+5$... it's a miscalculation ($10 \times 60 + 5 = 605$ — forgetting one hour).
- (B) $655$ would come from $11 \times 60 = 660$ computed as $650$, or subtracting 5 instead of adding.
- (D) $1005$ and (E) $1105$ mistakenly treat hours as roughly 100 minutes each (e.g., $11 \times 100 + 5 = 1105$) — but an hour is 60 minutes, not 100.

ANSWER 2: C

Problem 1:

We are evaluating

\[
\frac{1}{10}+\frac{9}{100}+\frac{9}{1000}+\frac{7}{10000}.
\]

To add these fractions, rewrite them all with denominator \(10000\):

\[
\frac{1}{10}=\frac{1000}{10000},
\]

\[
\frac{9}{100}=\frac{900}{10000},
\]

\[
\frac{9}{1000}=\frac{90}{10000},
\]

\[
\frac{7}{10000}=\frac{7}{10000}.
\]

Now add the numerators:

\[
\frac{1000+900+90+7}{10000}=\frac{1997}{10000}=0.1997.
\]

So the correct choice is C.

ANSWER 1: C

Problem 2:

We need to convert 11 hours and 5 minutes into total minutes.

Since 1 hour equals 60 minutes,

\[
11\text{ hours}=11\cdot 60=660\text{ minutes}.
\]

Now add the extra 5 minutes:

\[
660+5=665.
\]

So the total is 665 minutes.

ANSWER 2: C

## Problem 1:

We are asked to add the fractions $\frac{1}{10} + \frac{9}{100} + \frac{9}{1000} + \frac{7}{10000}$ and express the result as a decimal.

**Convert each fraction to a decimal:**
- $\frac{1}{10} = 0.1$ (1 in the tenths place)
- $\frac{9}{100} = 0.09$ (9 in the hundredths place)
- $\frac{9}{1000} = 0.009$ (9 in the thousandths place)
- $\frac{7}{10000} = 0.0007$ (7 in the ten-thousandths place)

**Add by stacking place values:**

$$0.1000 + 0.0900 + 0.0090 + 0.0007 = 0.1997$$

Since each fraction contributes exactly one digit to its own place value, the digits simply line up as 1, 9, 9, 7 after the decimal point.

**Ruling out the distractors:**
- **0.0197** would come from mistakenly putting the 1 in the hundredths place (shifting everything one place right).
- **1.997** would come from shifting everything one place left (treating $\frac{1}{10}$ as 1).
- **0.26** and **0.0026** come from just adding the numerators ($1+9+9+7=26$) and ignoring the different denominators.

The correct sum is $0.1997$.

ANSWER 1: C

## Problem 2:

We must convert 11 hours and 5 minutes entirely into minutes.

**Step 1 — Convert the hours:** There are 60 minutes in one hour, so
$$11 \times 60 = 660 \text{ minutes}$$

**Step 2 — Add the extra minutes:**
$$660 + 5 = 665 \text{ minutes}$$

**Ruling out the distractors:**
- **605** comes from mistakenly computing $600 + 5$ (using 11 × 60 = 600, an error).
- **655** comes from subtracting the 5 instead of adding it ($660 - 5$).
- **1105** and **1005** come from just writing the digits "11" and "05" together rather than converting units.

So the match lasted $665$ minutes.

ANSWER 2: C

Problem 1:

We are asked to evaluate

\[
\frac{1}{10}+\frac{9}{100}+\frac{9}{1000}+\frac{7}{10000}.
\]

Write each fraction as a decimal:

\[
\frac{1}{10}=0.1
\]

\[
\frac{9}{100}=0.09
\]

\[
\frac{9}{1000}=0.009
\]

\[
\frac{7}{10000}=0.0007
\]

Now add carefully:

\[
0.1+0.09=0.19
\]

\[
0.19+0.009=0.199
\]

\[
0.199+0.0007=0.1997
\]

So the correct choice is C.

Choice A, \(0.0026\), is far too small because \(\frac{1}{10}\) alone is already \(0.1\).  
Choice B, \(0.0197\), incorrectly places the \(1\) in the hundredths place rather than the tenths place.  
Choice D, \(0.26\), and choice E, \(1.997\), do not match the place values of the fractions.

ANSWER 1: C

Problem 2:

We are asked to convert 11 hours and 5 minutes into minutes.

Since each hour has 60 minutes,

\[
11 \text{ hours}=11\cdot 60=660 \text{ minutes}.
\]

Then add the extra 5 minutes:

\[
660+5=665.
\]

Therefore, the match lasted 665 minutes.

Choice A, 605, would be \(10\) hours and \(5\) minutes.  
Choice B, 655, is 10 minutes too few.  
Choices D and E are much too large.

ANSWER 2: C