If 12 men and 16 boys can complete a piece of work in 5 days and 13 men and 24 boys can complete the same work in 4 days, then in how many days 7 men and 10 boys will complete this work?

Options

A. 8 days
B. 9 days
C. `8(1)/(3)` days
D. `10(1)/(3)` days

Answer

To solve the problem step by step, we need to determine the amount of work done by men and boys, and then find out how long it will take for 7 men and 10 boys to complete the same work.

### Step 1: Define the work done by men and boys
Let the work done by 1 man in 1 day be \( M \) and the work done by 1 boy in 1 day be \( B \).

### Step 2: Set up the equations based on the given information
From the first scenario:
- 12 men and 16 boys can complete the work in 5 days.
- Therefore, the total work can be expressed as:
\[ 5 \times (12M + 16B) = W \quad \text{(1)} \]

From the second scenario:
- 13 men and 24 boys can complete the work in 4 days.
- Therefore, the total work can be expressed as:
\[ 4 \times (13M + 24B) = W \quad \text{(2)} \]

### Step 3: Equate the two expressions for work
Since both expressions equal the total work \( W \), we can set them equal to each other:
\[ 5(12M + 16B) = 4(13M + 24B) \]

### Step 4: Expand both sides
Expanding both sides gives:
\[ 60M + 80B = 52M + 96B \]

### Step 5: Rearrange the equation
Rearranging the equation to isolate terms gives:
\[ 60M - 52M = 96B - 80B \]
\[ 8M = 16B \]

### Step 6: Simplify the relationship between men and boys
Dividing both sides by 8 gives:
\[ M = 2B \]
This means that 1 man is equivalent to the work done by 2 boys.

### Step 7: Substitute \( M \) in terms of \( B \)
Now, we can substitute \( M \) in terms of \( B \) into either equation (1) or (2). Let's use equation (1):
\[ W = 5 \times (12(2B) + 16B) = 5 \times (24B + 16B) = 5 \times 40B = 200B \]
So, the total work \( W \) is \( 200B \).

### Step 8: Calculate the work done by 7 men and 10 boys
Now, we need to find out how many days \( x \) it will take for 7 men and 10 boys to complete the work:
\[ W = x \times (7M + 10B) \]
Substituting \( M = 2B \):
\[ W = x \times (7(2B) + 10B) = x \times (14B + 10B) = x \times 24B \]

### Step 9: Set the equation for total work
Now, we know \( W = 200B \):
\[ 200B = x \times 24B \]

### Step 10: Solve for \( x \)
Dividing both sides by \( B \) (assuming \( B \neq 0 \)):
\[ 200 = 24x \]
\[ x = \frac{200}{24} = \frac{25}{3} \approx 8.33 \]

### Step 11: Convert to mixed number
Converting \( \frac{25}{3} \) to a mixed number gives:
\[ 8 \frac{1}{3} \]

### Final Answer
Thus, 7 men and 10 boys will complete the work in **8 \( \frac{1}{3} \) days**.