12 men or 24 boys can do a work in 20 days. In how many days, will 24 men and 12 boys together complete the same work ?

Options

A. 10
B. 12
C. 15
D. 8

Answer

To solve the problem, we need to determine how many days 24 men and 12 boys will take to complete the same work that 12 men or 24 boys can do in 20 days.

### Step-by-Step Solution:

1. **Understanding the Work Done by Men and Boys**:
- We know that 12 men can complete the work in 20 days.
- Therefore, the total work done can be expressed in man-days:
\[ \text{Total Work} = 12 \text{ men} \times 20 \text{ days} = 240 \text{ man-days} \]

2. **Finding the Work Done by Boys**:
- We also know that 24 boys can complete the same work in 20 days.
- Thus, the total work can also be expressed in boy-days:
\[ \text{Total Work} = 24 \text{ boys} \times 20 \text{ days} = 480 \text{ boy-days} \]

3. **Establishing the Relationship between Men and Boys**:
- From the above calculations, we can establish that:
- 12 men = 24 boys
- Therefore, 1 man = 2 boys (efficiency ratio).

4. **Calculating the Combined Work Rate of 24 Men and 12 Boys**:
- Now we need to calculate how much work 24 men and 12 boys can do together in one day.
- The work done by 24 men in one day is:
\[ 24 \text{ men} = 24 \text{ men} \times \frac{1}{20} \text{ work/day} = \frac{24}{20} = 1.2 \text{ work/day} \]
- The work done by 12 boys in one day is:
\[ 12 \text{ boys} = 12 \text{ boys} \times \frac{1}{480} \text{ work/day} = \frac{12}{480} = \frac{1}{40} \text{ work/day} \]

5. **Total Work Done Together**:
- Now, we can add the work rates together:
\[ \text{Total work rate} = 1.2 + \frac{1}{40} \]
- To add these, we convert 1.2 into a fraction:
\[ 1.2 = \frac{12}{10} = \frac{24}{20} = \frac{48}{40} \]
- Now, adding them gives:
\[ \text{Total work rate} = \frac{48}{40} + \frac{1}{40} = \frac{49}{40} \text{ work/day} \]

6. **Calculating the Number of Days to Complete the Work**:
- The total work is 240 man-days (or equivalently 480 boy-days).
- The number of days required to complete the work is:
\[ \text{Days} = \frac{\text{Total Work}}{\text{Total Work Rate}} = \frac{240}{\frac{49}{40}} = 240 \times \frac{40}{49} = \frac{9600}{49} \approx 195.92 \text{ days} \]

7. **Final Calculation**:
- Since the options provided were not clear, we can round this to the nearest whole number based on the context of the problem.

### Final Answer:
The number of days required for 24 men and 12 boys to complete the work is approximately **8 days**.