If 6 men and 8 boys can do a piece of work in 10 days and 26 men and 48 boys can complete the work in 2 days then in how many days the work will be completed by 15 men and 20 boys?

Options

A. 10 days
B. 4 days
C. 8 days
D. 5 days

Answer

To solve the problem, we need to determine how many days it will take for 15 men and 20 boys to complete a piece of work, given the work rates of different groups.

### Step-by-Step Solution:

1. **Determine the Work Done by 6 Men and 8 Boys in 10 Days:**
- Let the work done by 1 man in 1 day be \( M \) and the work done by 1 boy in 1 day be \( B \).
- The total work done by 6 men and 8 boys in 10 days can be expressed as:
\[ \text{Total Work} = (6M + 8B) \times 10 \]

2. **Determine the Work Done by 26 Men and 48 Boys in 2 Days:**
- Similarly, the total work done by 26 men and 48 boys in 2 days is:
\[ \text{Total Work} = (26M + 48B) \times 2 \]

3. **Set the Two Expressions for Total Work Equal:**
- Since both expressions represent the same total work, we can set them equal to each other:
\[ (6M + 8B) \times 10 = (26M + 48B) \times 2 \]

4. **Simplify the Equation:**
- Expanding both sides gives:
\[ 60M + 80B = 52M + 96B \]
- Rearranging the equation yields:
\[ 60M - 52M = 96B - 80B \]
\[ 8M = 16B \]
- Dividing both sides by 8 gives:
\[ M = 2B \]

5. **Calculate the Total Work in Units:**
- Now, substituting \( M = 2B \) back into the work equation:
\[ \text{Total Work} = (6M + 8B) \times 10 = (6(2B) + 8B) \times 10 \]
\[ = (12B + 8B) \times 10 = 20B \times 10 = 200B \]
- Thus, the total work is 200 units.

6. **Calculate the Efficiency of 15 Men and 20 Boys:**
- The efficiency of 15 men and 20 boys can be calculated as:
\[ \text{Efficiency} = 15M + 20B \]
- Substituting \( M = 2B \):
\[ = 15(2B) + 20B = 30B + 20B = 50B \]

7. **Determine the Number of Days Required to Complete the Work:**
- To find the number of days required to complete 200 units of work with an efficiency of 50B:
\[ \text{Days} = \frac{\text{Total Work}}{\text{Efficiency}} = \frac{200B}{50B} = 4 \text{ days} \]

### Final Answer:
The work will be completed by 15 men and 20 boys in **4 days**.

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