A piece of work was to be completed in 40 days a number of men employed upon it did only half the work in 24 days, 16 more men were then set on, and the work was completed in the specified time, how many men were employed at first ?

Options

A. 16 men
B. 32 men
C. 24 men
D. 48 men

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Work Done
The total work is to be completed in 40 days. According to the problem, half of the work was completed in 24 days.

**Hint:** Identify the total work in terms of days and the fraction of work done.

### Step 2: Calculate the Total Work
Let the total work be represented as W. Since half of the work is done in 24 days, we can say:
\[ \text{Work done in 24 days} = \frac{W}{2} \]

### Step 3: Determine the Work Rate
Let the number of men employed initially be \( n \). The work done by \( n \) men in 24 days can be expressed as:
\[ n \times 24 = \frac{W}{2} \]
From this, we can derive the work done by one man in one day:
\[ n = \frac{W}{48} \]

### Step 4: Calculate Remaining Work
Since half of the work is completed, the remaining work is also \( \frac{W}{2} \). This remaining work needs to be completed in the remaining days, which is:
\[ 40 - 24 = 16 \text{ days} \]

### Step 5: Set Up the Equation for Remaining Work
After 24 days, 16 more men are added, making the total number of men:
\[ n + 16 \]
The equation for the remaining work can be set up as:
\[ (n + 16) \times 16 = \frac{W}{2} \]

### Step 6: Substitute \( n \) into the Equation
Now we substitute \( n \) from Step 3 into the equation:
\[ \left(\frac{W}{48} + 16\right) \times 16 = \frac{W}{2} \]

### Step 7: Simplify the Equation
Expanding the left side:
\[ \frac{W \times 16}{48} + 16 \times 16 = \frac{W}{2} \]
This simplifies to:
\[ \frac{W}{3} + 256 = \frac{W}{2} \]

### Step 8: Solve for W
To solve for W, we can first eliminate the fractions by multiplying through by 6 (the least common multiple of 3 and 2):
\[ 2W + 1536 = 3W \]
Rearranging gives:
\[ 3W - 2W = 1536 \]
Thus:
\[ W = 1536 \]

### Step 9: Find the Initial Number of Men
Now that we have W, we can find \( n \):
\[ n = \frac{W}{48} = \frac{1536}{48} = 32 \]

### Conclusion
The number of men employed at first was **32**.

---