12 men and 16 boys can do a piece of work in 5 days. 13 men and 24 boys can do the same work in 4 days. How long will 7 men and 10 boys take to do the same work?

Answer

To solve the problem, we need to determine how long it will take for 7 men and 10 boys to complete the same piece of work given the efficiencies of men and boys based on the information provided.

### Step-by-Step Solution:

1. **Define Variables:**
Let the efficiency of one man be \( m \) and the efficiency of one boy be \( b \).

2. **Set Up Equations from Given Information:**
From the first scenario:
- 12 men and 16 boys can complete the work in 5 days.
- The total work done can be expressed as:
\[ \text{Total Work} = \text{Efficiency} \times \text{Time} = (12m + 16b) \times 5 \]
Thus, we have:
\[ 60m + 80b = W \quad \text{(1)} \]

From the second scenario:
- 13 men and 24 boys can complete the work in 4 days.
- The total work done can be expressed as:
\[ \text{Total Work} = (13m + 24b) \times 4 \]
Thus, we have:
\[ 52m + 96b = W \quad \text{(2)} \]

3. **Equate the Two Expressions for Total Work:**
Since both expressions equal the total work \( W \), we can set them equal to each other:
\[ 60m + 80b = 52m + 96b \]

4. **Simplify the Equation:**
Rearranging gives:
\[ 60m - 52m = 96b - 80b \]
\[ 8m = 16b \]
Dividing both sides by 8 gives:
\[ m = 2b \quad \text{(3)} \]

5. **Substitute \( m \) in One of the Work Equations:**
We can substitute \( m \) from equation (3) into equation (1):
\[ 60(2b) + 80b = W \]
\[ 120b + 80b = W \]
\[ 200b = W \]
Thus, the total work \( W \) is \( 200b \).

6. **Find the Efficiency of 7 Men and 10 Boys:**
Now, we calculate the efficiency of 7 men and 10 boys:
\[ \text{Efficiency} = 7m + 10b \]
Substituting \( m = 2b \):
\[ = 7(2b) + 10b = 14b + 10b = 24b \]

7. **Calculate the Time Taken by 7 Men and 10 Boys:**
The time taken to complete the work is given by:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency}} = \frac{200b}{24b} \]
The \( b \) cancels out:
\[ = \frac{200}{24} = \frac{25}{3} \text{ days} \]
Converting this to a mixed number:
\[ = 8 \frac{1}{3} \text{ days} \]

### Final Answer:
Therefore, the time required for 7 men and 10 boys to complete the work is \( 8 \frac{1}{3} \) days.