4 men and 6 women get Rs. 1600 by doing a piece of work in 5 days. 3 men and 7 women get Rs. 1740 by doing the same work in 6 days. In how many days, 7 men and 6 women can complete the same work getting Rs. 3760?

Answer

To solve the problem, we need to determine how many days it will take for 7 men and 6 women to complete the same work for Rs. 3760. We will first find the daily wages of men and women based on the given information.

### Step-by-Step Solution:

1. **Calculate the total work done in terms of man-days and woman-days for the first scenario:**
- 4 men and 6 women complete the work in 5 days for Rs. 1600.
- Total work = (4 men + 6 women) × 5 days = 20 man-days + 30 woman-days = 50 units of work.

2. **Calculate the daily earnings for the first scenario:**
- Daily earnings = Total earnings / Number of days = Rs. 1600 / 5 = Rs. 320 per day.

3. **Calculate the work done in terms of man-days and woman-days for the second scenario:**
- 3 men and 7 women complete the work in 6 days for Rs. 1740.
- Total work = (3 men + 7 women) × 6 days = 18 man-days + 42 woman-days = 60 units of work.

4. **Calculate the daily earnings for the second scenario:**
- Daily earnings = Total earnings / Number of days = Rs. 1740 / 6 = Rs. 290 per day.

5. **Set up equations based on the daily earnings:**
- Let the daily wage of one man be M and that of one woman be W.
- From the first scenario:
\[ 4M + 6W = 320 \quad \text{(1)} \]
- From the second scenario:
\[ 3M + 7W = 290 \quad \text{(2)} \]

6. **Solve the equations simultaneously:**
- Multiply equation (1) by 3:
\[ 12M + 18W = 960 \quad \text{(3)} \]
- Multiply equation (2) by 4:
\[ 12M + 28W = 1160 \quad \text{(4)} \]
- Subtract equation (3) from equation (4):
\[ (12M + 28W) - (12M + 18W) = 1160 - 960 \]
\[ 10W = 200 \implies W = 20 \]

7. **Substitute W back to find M:**
- Substitute W = 20 into equation (1):
\[ 4M + 6(20) = 320 \]
\[ 4M + 120 = 320 \implies 4M = 200 \implies M = 50 \]

8. **Calculate the total daily earnings for 7 men and 6 women:**
- Daily earnings = (7 men × M) + (6 women × W)
- Daily earnings = (7 × 50) + (6 × 20) = 350 + 120 = Rs. 470.

9. **Determine how many days it takes to earn Rs. 3760:**
- Total days = Total earnings / Daily earnings = Rs. 3760 / Rs. 470.
- Calculate:
\[ 3760 ÷ 470 = 8 \]

### Final Answer:
It will take **8 days** for 7 men and 6 women to complete the work for Rs. 3760.