If 12 men and 16 boys can do a piece of work in 5 days and, 13 men and 24 boys can do it in 4 days, how long will 7 men and 10 boys take to do it ?

Answer

To solve the problem step by step, we will first establish the relationships between men and boys in terms of work done, and then we will calculate how long it will take for 7 men and 10 boys to complete the same work.

### Step 1: Establish the equations based on the given information.

1. **Equation from the first scenario**:
- 12 men and 16 boys can complete the work in 5 days.
- Work done = (Number of workers) × (Time)
- Work done = (12M + 16B) × 5

2. **Equation from the second scenario**:
- 13 men and 24 boys can complete the work in 4 days.
- Work done = (13M + 24B) × 4

Since both scenarios complete the same work, we can set the two equations equal to each other:
\[ (12M + 16B) \times 5 = (13M + 24B) \times 4 \]

### Step 2: Expand and simplify the equations.

1. Expanding both sides:
- Left side: \(60M + 80B\)
- Right side: \(52M + 96B\)

So, we have:
\[ 60M + 80B = 52M + 96B \]

### Step 3: Rearranging the equation.

1. Move all terms involving M to one side and B to the other:
\[ 60M - 52M = 96B - 80B \]
\[ 8M = 16B \]

### Step 4: Find the relationship between M and B.

1. Dividing both sides by 8:
\[ M = 2B \]

### Step 5: Substitute M in terms of B into the first equation.

1. Substitute \(M = 2B\) into the first work equation:
\[ 12(2B) + 16B = 24B + 16B = 40B \]
2. So, the first equation becomes:
\[ 40B \times 5 = 200B \]

### Step 6: Calculate the total work done.

1. The total work done by 12 men and 16 boys in 5 days is:
\[ 200B \]

### Step 7: Calculate the work done by 7 men and 10 boys.

1. First, express 7 men and 10 boys in terms of boys:
- 7 men = \(7 \times 2B = 14B\)
- 10 boys = \(10B\)
- Total = \(14B + 10B = 24B\)

### Step 8: Set up the equation for time taken by 7 men and 10 boys.

1. Let \(X\) be the number of days it takes for 7 men and 10 boys to complete the work:
\[ 24B \times X = 200B \]

### Step 9: Solve for X.

1. Dividing both sides by \(24B\):
\[ X = \frac{200B}{24B} = \frac{200}{24} = \frac{25}{3} \approx 8.33 \text{ days} \]

### Final Answer:
The time taken by 7 men and 10 boys to complete the work is approximately **8.33 days**.