12 men and 15 women can do a work in 6 days, and 6 men and 12 women can do it in 10 days, in how many days can 8 men and 10 women do the same work?

Answer

To solve the problem, we need to find out how many days 8 men and 10 women can complete the same work given the information about the work done by 12 men and 15 women in 6 days, and 6 men and 12 women in 10 days.

### Step-by-Step Solution:

1. **Calculate the total work done in terms of man-days and woman-days.**
- Let the work done by 1 man in 1 day be \( M \) and the work done by 1 woman in 1 day be \( W \).
- From the first scenario:
\[ \text{Total work} = \text{(Number of workers)} \times \text{(Days)} = (12M + 15W) \times 6 \]
- From the second scenario:
\[ \text{Total work} = (6M + 12W) \times 10 \]

2. **Set the two expressions for total work equal to each other:**
\[ (12M + 15W) \times 6 = (6M + 12W) \times 10 \]

3. **Expand both sides:**
\[ 72M + 90W = 60M + 120W \]

4. **Rearrange the equation:**
\[ 72M - 60M = 120W - 90W \]
\[ 12M = 30W \]

5. **Simplify to find the ratio of men to women:**
\[ \frac{M}{W} = \frac{30}{12} = \frac{5}{2} \]
This means 1 man does the work equivalent to 2.5 women.

6. **Substitute \( M \) in terms of \( W \):**
\[ M = \frac{5}{2}W \]

7. **Calculate the work done by 8 men and 10 women in one day:**
\[ \text{Work done by 8 men} = 8M = 8 \times \frac{5}{2}W = 20W \]
\[ \text{Total work done by 8 men and 10 women in one day} = 20W + 10W = 30W \]

8. **Calculate the total work in terms of \( W \):**
- From the first scenario, total work can be calculated:
\[ \text{Total work} = (12M + 15W) \times 6 = (12 \times \frac{5}{2}W + 15W) \times 6 \]
\[ = (30W + 15W) \times 6 = 45W \times 6 = 270W \]

9. **Find the number of days required for 8 men and 10 women to complete the work:**
\[ \text{Days} = \frac{\text{Total Work}}{\text{Work done in one day}} = \frac{270W}{30W} = 9 \text{ days} \]

### Final Answer:
8 men and 10 women can complete the work in **9 days**.