16 children and 24 men complete a certain workin 18days. If each child takes twice the time taken by a man to finish the work, in how many days will 24 men finish the same work?

Options

A. 12 days
B. 24 days
C. 36 days
D. 48 days

Answer

To solve the problem step-by-step, we need to analyze the given information and apply the concepts of work and time.

### Step 1: Understand the Work Done by Children and Men
We know that 16 children and 24 men complete a certain work in 18 days.

### Step 2: Establish the Relationship Between the Time Taken by Children and Men
It is given that each child takes twice the time taken by a man to finish the work. Therefore, if we denote the time taken by a man to complete the work as \( T \), then the time taken by a child will be \( 2T \).

### Step 3: Determine the Efficiency Ratio
Since work is inversely proportional to time, the efficiency ratio of children to men can be established as follows:
- If one man's efficiency is \( M \), then one child's efficiency will be \( \frac{M}{2} \) because a child takes twice as long.

Thus, the efficiency ratio of children to men is:
\[ \text{Efficiency of 1 child} = \frac{M}{2}, \quad \text{Efficiency of 1 man} = M \]
This gives us the ratio:
\[ \frac{\text{Efficiency of 1 child}}{\text{Efficiency of 1 man}} = \frac{1}{2} \]

### Step 4: Calculate Total Efficiency of Children and Men
Now, we can calculate the total efficiency of 16 children and 24 men:
- Total efficiency of 16 children = \( 16 \times \frac{M}{2} = 8M \)
- Total efficiency of 24 men = \( 24 \times M = 24M \)

Adding these together:
\[ \text{Total efficiency} = 8M + 24M = 32M \]

### Step 5: Calculate Total Work Done
The total work done can be calculated using the formula:
\[ \text{Total Work} = \text{Total Efficiency} \times \text{Time} \]
Substituting the values we have:
\[ \text{Total Work} = 32M \times 18 = 576M \]

### Step 6: Calculate the Time Taken by 24 Men to Complete the Same Work
Now, we need to find out how many days \( X \) it will take for 24 men to complete the same work. The efficiency of 24 men is:
\[ \text{Efficiency of 24 men} = 24M \]
Using the total work calculated:
\[ \text{Total Work} = \text{Efficiency of 24 men} \times \text{Time taken by 24 men} \]
Thus:
\[ 576M = 24M \times X \]

### Step 7: Solve for \( X \)
Now, we can solve for \( X \):
\[ X = \frac{576M}{24M} = \frac{576}{24} = 24 \]

### Conclusion
Thus, 24 men will take **24 days** to complete the same work.