Greg is 1.7 times faster at cleaning windows than Owen. It takes 39 minutes for them to clean 100 windows together. Using a rational equation, calculate how long it would take Owen to clean 100 window by himself. Round the answer to the nearest tenth. (1 point) minutes

Let the rate at which Owen cleans windows be \( r \) windows per minute. Therefore, Greg, who is 1.7 times faster than Owen, cleans at a rate of \( 1.7r \) windows per minute.

When they work together, their combined rate is the sum of their individual rates:

\[
r + 1.7r = 2.7r \text{ windows per minute.}
\]

They clean 100 windows together in 39 minutes, so we can set up the equation:

\[
2.7r \times 39 = 100.
\]

Now, we can solve for \( r \):

\[
2.7r \times 39 = 100 \\
2.7r = \frac{100}{39} \\
r = \frac{100}{39 \times 2.7}.
\]

Calculating the denominator:

\[
39 \times 2.7 = 105.3.
\]

Now substituting this back in, we have:

\[
r = \frac{100}{105.3} \approx 0.948.
\]

This means Owen can clean approximately 0.948 windows per minute. Next, we need to find out how long it would take Owen to clean 100 windows by himself:

Let \( t \) be the time it takes Owen to clean 100 windows:

\[
r \times t = 100 \\
0.948 \times t = 100 \\
t = \frac{100}{0.948} \approx 105.5 \text{ minutes.}
\]

Therefore, it would take Owen approximately \( 105.5 \) minutes to clean 100 windows by himself.

Rounding to the nearest tenth, the final answer is:

\[
\boxed{105.5} \text{ minutes.}
\]

Are you sure

Let's re-evaluate the calculations step by step to ensure accuracy.

1. **Define the variables**:
- Let \( r \) be Owen's rate in windows per minute.
- Greg's rate is \( 1.7r \).

2. **Combined Rate**:
- Their combined rate when working together is:
\[
r + 1.7r = 2.7r
\]

3. **Time Together**:
- They clean 100 windows in 39 minutes, so we can write:
\[
(2.7r) \times 39 = 100
\]

4. **Solve for \( r \)**:
- This simplifies to:
\[
2.7r = \frac{100}{39}
\]
\[
r = \frac{100}{39 \times 2.7}
\]

5. **Calculating \( 39 \times 2.7 \)**:
- This gives:
\[
39 \times 2.7 = 105.3
\]

6. **Substituting to find \( r \)**:
- Now substitute:
\[
r = \frac{100}{105.3} \approx 0.948 \text{ windows per minute.}
\]

7. **Time for Owen to clean 100 windows alone**:
- Let \( t \) be the time in minutes for Owen to clean 100 windows:
\[
r \times t = 100 \implies 0.948 \times t = 100
\]
- Rearranging gives:
\[
t = \frac{100}{0.948}
\]

8. **Calculating \( t \)**:
- Performing the division:
\[
t \approx 105.5 \text{ minutes.}
\]

9. **Rounding the Answer**:
- When rounded to the nearest tenth, it is still:
\[
\boxed{105.5} \text{ minutes.}
\]

Thus, it looks like the calculations were indeed accurate, and the final conclusion is that it would take Owen approximately \( 105.5 \) minutes to clean 100 windows by himself. If there's anything specific you'd like to clarify or check, please let me know!