In a computer game a builder can build a wall in 10 hours while a destroyer can demolish such a wall completely in 14 hours . Both the builder and the destroyer were initially set to work together on level ground . But after 7 hours the destroyer was taken out . What was the total time ( in hours) taken to build the wall ?
A)35
B)17
C)24
D)15

Options

A. 35
B. 17
C. 24
D. 15

Answer

To solve the problem, we need to determine how long it takes to build the wall when both the builder and the destroyer are working together for a certain period, and then the builder continues alone after the destroyer stops.

### Step-by-Step Solution:

1. **Determine the work rates of the builder and destroyer:**
- The builder can build a wall in 10 hours, so the work rate of the builder is:
\[ \text{Builder's rate} = \frac{1 \text{ wall}}{10 \text{ hours}} = 0.1 \text{ walls per hour} \]
- The destroyer can demolish a wall in 14 hours, so the work rate of the destroyer is:
\[ \text{Destroyer's rate} = \frac{1 \text{ wall}}{14 \text{ hours}} = \frac{1}{14} \text{ walls per hour} \]

2. **Calculate the combined work rate when both are working together:**
- When both the builder and the destroyer are working together, their combined work rate is:
\[ \text{Combined rate} = \text{Builder's rate} - \text{Destroyer's rate} = 0.1 - \frac{1}{14} \]
- To perform this calculation, convert 0.1 to a fraction:
\[ 0.1 = \frac{1}{10} \]
- Now find a common denominator (which is 70):
\[ \frac{1}{10} = \frac{7}{70}, \quad \frac{1}{14} = \frac{5}{70} \]
- Therefore:
\[ \text{Combined rate} = \frac{7}{70} - \frac{5}{70} = \frac{2}{70} = \frac{1}{35} \text{ walls per hour} \]

3. **Calculate the work done in the first 7 hours:**
- In 7 hours, the amount of work done together is:
\[ \text{Work done} = \text{Combined rate} \times \text{Time} = \frac{1}{35} \times 7 = \frac{7}{35} = \frac{1}{5} \text{ walls} \]

4. **Determine the remaining work:**
- The total work required to build the wall is 1 wall. After 7 hours, the work done is \(\frac{1}{5}\) walls, so the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{5} = \frac{4}{5} \text{ walls} \]

5. **Calculate how long it takes for the builder to finish the remaining work alone:**
- The builder's work rate is \(\frac{1}{10}\) walls per hour. To find the time required to build \(\frac{4}{5}\) walls:
\[ \text{Time} = \frac{\text{Remaining work}}{\text{Builder's rate}} = \frac{\frac{4}{5}}{\frac{1}{10}} = \frac{4}{5} \times 10 = 8 \text{ hours} \]

6. **Calculate the total time taken to build the wall:**
- The total time taken is the time spent working together plus the time spent by the builder alone:
\[ \text{Total time} = 7 \text{ hours} + 8 \text{ hours} = 15 \text{ hours} \]

### Final Answer:
The total time taken to build the wall is **15 hours**.