A contractor undertakes to make a road in 40 days and employs 25 men. After 24 days, he finds that only one-third of the road is made. How many extra men should he employ so that he is able to complete the work 4 days earlier?

Options

A. 70
B. 75
C. 80
D. 85

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Problem**:
The contractor has 40 days to complete the road with 25 men. After 24 days, only one-third of the road is completed. The contractor wants to finish the remaining work 4 days earlier than the original schedule.

2. **Calculate the Total Work**:
Let the total work required to complete the road be represented as \( W \). Since one-third of the road is completed in 24 days, the remaining work is:
\[ W_{\text{remaining}} = W - \frac{1}{3}W = \frac{2}{3}W \]

3. **Determine the Remaining Time**:
Since the contractor originally planned to complete the work in 40 days and has already spent 24 days, he has:
\[ 40 - 24 = 16 \text{ days remaining} \]
However, he wants to finish 4 days earlier, so the new time to complete the remaining work is:
\[ 16 - 4 = 12 \text{ days} \]

4. **Calculate the Work Rate**:
The work done by 25 men in 24 days is:
\[ \text{Work done} = 25 \text{ men} \times 24 \text{ days} = 600 \text{ man-days} \]
Since this represents one-third of the total work:
\[ W = 3 \times 600 = 1800 \text{ man-days} \]

5. **Calculate Remaining Work**:
The remaining work is:
\[ W_{\text{remaining}} = \frac{2}{3}W = \frac{2}{3} \times 1800 = 1200 \text{ man-days} \]

6. **Determine Required Work Rate**:
To complete the remaining work of 1200 man-days in 12 days, the required number of men \( M \) can be calculated as:
\[ M \times 12 = 1200 \implies M = \frac{1200}{12} = 100 \text{ men} \]

7. **Calculate Extra Men Needed**:
The contractor initially had 25 men. Therefore, the number of extra men \( x \) needed is:
\[ x = M - 25 = 100 - 25 = 75 \]

### Final Answer:
The contractor needs to employ **75 extra men** to complete the work 4 days earlier.

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