An engineer undertakes a project to build a road15 km long in 300 days and employs 45 men for the purpose. After 100 days, he finds 2.5 km of the road has been completed. Find the (approx.) numbe of extra men he must employ to finish the work in time.

Options

A. 43
B. 45
C. 55
D. 68

Answer

To solve the problem step by step, we need to determine how many extra men are required to complete the road project on time. Here’s how we can do it:

### Step 1: Determine the total work required
The total length of the road to be built is 15 km.

### Step 2: Calculate the work completed in 100 days
After 100 days, the engineer has completed 2.5 km of the road.

### Step 3: Calculate the remaining work
Remaining work = Total work - Work completed
Remaining work = 15 km - 2.5 km = 12.5 km

### Step 4: Calculate the remaining time
Total time for the project = 300 days
Time already spent = 100 days
Remaining time = 300 days - 100 days = 200 days

### Step 5: Determine the work rate of the current workforce
The current workforce is 45 men. We need to find out how much work these men can do in the remaining time.

### Step 6: Use the formula for work
The formula for work is:
\[ M_1 \times D_1 \times W_2 = M_2 \times D_2 \times W_1 \]

Where:
- \( M_1 \) = initial number of men = 45
- \( D_1 \) = initial time spent = 100 days
- \( W_2 \) = remaining work = 12.5 km
- \( M_2 \) = required number of men (unknown)
- \( D_2 \) = remaining time = 200 days
- \( W_1 \) = work completed = 2.5 km

### Step 7: Rearranging the formula
We can rearrange the formula to find \( M_2 \):
\[ M_2 = \frac{M_1 \times D_1 \times W_2}{D_2 \times W_1} \]

### Step 8: Substitute the values into the equation
Substituting the known values:
\[ M_2 = \frac{45 \times 100 \times 12.5}{200 \times 2.5} \]

### Step 9: Calculate the right side of the equation
Calculating the numerator:
\[ 45 \times 100 \times 12.5 = 56250 \]

Calculating the denominator:
\[ 200 \times 2.5 = 500 \]

Now, substituting these values:
\[ M_2 = \frac{56250}{500} = 112.5 \]

### Step 10: Round up the number of men
Since we cannot have a fraction of a man, we round up to the nearest whole number, which is 113 men.

### Step 11: Calculate the extra men required
Extra men required = \( M_2 - M_1 \)
Extra men required = \( 113 - 45 = 68 \)

### Final Answer
The engineer must employ approximately **68 extra men** to finish the work on time.

---