A contractor undertook to finish a certain work in 124 days and employed 120 men. After 64 days, he found that he had already done `(2)/(3)` of the work. How many men can be discharge now so that the work may finish in time?

Options

A. 48
B. 56
C. 40
D. 50

Answer

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

### Step 1: Understand the total work and the initial setup
The contractor has a total of 124 days to complete the work with 120 men.

### Step 2: Calculate the work done in the first 64 days
In the first 64 days, the contractor completed \( \frac{2}{3} \) of the work.

### Step 3: Determine the remaining work
The remaining work after 64 days is:
\[ \text{Remaining work} = 1 - \frac{2}{3} = \frac{1}{3} \]

### Step 4: Calculate the remaining days
The total time is 124 days, and 64 days have already been used. Therefore, the remaining days are:
\[ \text{Remaining days} = 124 - 64 = 60 \text{ days} \]

### Step 5: Set up the equation for the remaining work
Let \( x \) be the number of men to be discharged. The number of men remaining after discharging \( x \) men will be:
\[ 120 - x \]
The work done by \( 120 - x \) men in the remaining 60 days must equal the remaining \( \frac{1}{3} \) of the work.

### Step 6: Calculate the work done by the original setup
The work done by 120 men in 64 days can be expressed as:
\[ \text{Work done} = 120 \text{ men} \times 64 \text{ days} \]

### Step 7: Set up the equation for the remaining work
The work done by \( 120 - x \) men in 60 days is:
\[ \text{Work done} = (120 - x) \text{ men} \times 60 \text{ days} \]
Setting the work done equal to the remaining work:
\[ (120 - x) \times 60 = \frac{1}{3} \times (120 \times 124) \]

### Step 8: Calculate the total work
The total work can be calculated as:
\[ \text{Total work} = 120 \text{ men} \times 124 \text{ days} = 14880 \text{ man-days} \]
Thus, the remaining work is:
\[ \frac{1}{3} \times 14880 = 4960 \text{ man-days} \]

### Step 9: Set up the final equation
Now we can equate:
\[ (120 - x) \times 60 = 4960 \]

### Step 10: Solve for \( x \)
Expanding the equation gives:
\[ 7200 - 60x = 4960 \]
Rearranging gives:
\[ 60x = 7200 - 4960 \]
\[ 60x = 2240 \]
\[ x = \frac{2240}{60} = 37.33 \]
Since \( x \) must be a whole number, we round it to \( 37 \).

### Conclusion
The contractor can discharge **37 men** to finish the work on time.