If 6 men and 8 boys can do a piece of work in 10 days and 26 men and 48 boys can do the same work in 2 days, the time taken by 15 men and 20 boys to do the same type of work will be
A)6 days
B)4 days
C)8 days
D)7 days

Options

A. 6 days
B. 4 days
C. 8 days
D. 7 days

Answer

To solve the problem, we need to determine the time taken by 15 men and 20 boys to complete a piece of work, given the work rates of 6 men and 8 boys over 10 days, and 26 men and 48 boys over 2 days.

### Step-by-Step Solution:

1. **Calculate the Total Work Done**:
- From the first scenario, 6 men and 8 boys can complete the work in 10 days.
- Total work (in man-days) = (Number of men + Number of boys) × Days
- Total work = (6 + 8) × 10 = 14 × 10 = 140 man-days.

2. **Calculate the Total Work Done in the Second Scenario**:
- From the second scenario, 26 men and 48 boys can complete the work in 2 days.
- Total work = (26 + 48) × 2 = 74 × 2 = 148 man-days.

3. **Equate the Total Work**:
- Since both scenarios represent the same piece of work, we can equate the total work calculated from both scenarios.
- 140 man-days = 148 man-days (This indicates a discrepancy, so we need to find the work rate of men and boys).

4. **Find the Work Rate of 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 \( B \).
- From the first scenario:
\[ 6M + 8B = \frac{140}{10} = 14 \quad \text{(1)} \]
- From the second scenario:
\[ 26M + 48B = \frac{148}{2} = 74 \quad \text{(2)} \]

5. **Solve the Equations**:
- From equation (1):
\[ 6M + 8B = 14 \quad \text{(1)} \]
- From equation (2):
\[ 26M + 48B = 74 \quad \text{(2)} \]
- We can simplify equation (1) by dividing everything by 2:
\[ 3M + 4B = 7 \quad \text{(3)} \]
- Now, we can solve equations (3) and (2) simultaneously.

6. **Express B in terms of M**:
- From equation (3):
\[ 4B = 7 - 3M \implies B = \frac{7 - 3M}{4} \quad \text{(4)} \]

7. **Substitute B in equation (2)**:
- Substitute equation (4) into equation (2):
\[ 26M + 48\left(\frac{7 - 3M}{4}\right) = 74 \]
- Simplifying this gives:
\[ 26M + 12(7 - 3M) = 74 \]
\[ 26M + 84 - 36M = 74 \]
\[ -10M + 84 = 74 \implies -10M = -10 \implies M = 1 \]

8. **Find B**:
- Substitute \( M = 1 \) back into equation (4):
\[ B = \frac{7 - 3(1)}{4} = \frac{4}{4} = 1 \]

9. **Calculate the Efficiency of 15 Men and 20 Boys**:
- Work done by 15 men and 20 boys in one day:
\[ 15M + 20B = 15(1) + 20(1) = 15 + 20 = 35 \text{ units/day} \]

10. **Calculate the Time Taken**:
- Total work = 140 units (from the first scenario).
- Time taken = Total work / Daily work rate = \( \frac{140}{35} = 4 \) days.

### Final Answer:
The time taken by 15 men and 20 boys to do the same work is **4 days**.