A contractor undertakes to do a piece of work in 96 days. He engages 100 men at the begin ning. But in the 1/6 of the scheduled time, 1/7 of work is completed. How many additional men should be employed so that the work will be com pleted in time?

Options

A. 30
B. 20
C. 15
D. 10

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Determine the total work and the time taken
The contractor has undertaken to complete the work in 96 days. We denote the total work as \( W \).

### Step 2: Calculate the time elapsed and work completed
In \( \frac{1}{6} \) of the scheduled time:
\[ \text{Time elapsed} = \frac{96}{6} = 16 \text{ days} \]
In this time, \( \frac{1}{7} \) of the work is completed:
\[ \text{Work completed} = \frac{W}{7} \]

### Step 3: Calculate the remaining work
The remaining work after 16 days is:
\[ \text{Remaining work} = W - \frac{W}{7} = \frac{7W}{7} - \frac{W}{7} = \frac{6W}{7} \]

### Step 4: Determine the remaining time
The total time for the project is 96 days, and 16 days have already been used. Thus, the remaining time is:
\[ \text{Remaining time} = 96 - 16 = 80 \text{ days} \]

### Step 5: Set up the equation for the remaining work
Let \( x \) be the additional number of men needed. The total number of men working now is \( 100 + x \). The work rate for \( 100 + x \) men over the remaining 80 days must equal the remaining work:
\[ (100 + x) \times \text{Rate} \times 80 = \frac{6W}{7} \]

### Step 6: Calculate the rate of work
The rate of work done by 100 men in 16 days is:
\[ 100 \times \text{Rate} \times 16 = \frac{W}{7} \]
From this, we can express the rate:
\[ \text{Rate} = \frac{W}{7 \times 100 \times 16} = \frac{W}{11200} \]

### Step 7: Substitute the rate into the remaining work equation
Substituting the rate back into the equation:
\[ (100 + x) \times \frac{W}{11200} \times 80 = \frac{6W}{7} \]

### Step 8: Simplify the equation
Cancelling \( W \) from both sides:
\[ (100 + x) \times \frac{80}{11200} = \frac{6}{7} \]
\[ (100 + x) \times \frac{1}{140} = \frac{6}{7} \]
Multiplying both sides by 140:
\[ 100 + x = \frac{6 \times 140}{7} \]
\[ 100 + x = 120 \]

### Step 9: Solve for \( x \)
\[ x = 120 - 100 = 20 \]

### Final Answer
The contractor needs to employ **20 additional men** to complete the work on time.

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