If 6 men or 8 women can reap a field in 86 days, how long will 14 men and 10 women take to reap it?

Answer

To solve the problem, we need to determine how long it will take for 14 men and 10 women to reap the field, given that 6 men or 8 women can complete the task in 86 days.

### Step-by-step Solution:

1. **Determine the total work done in man-days:**
- If 6 men can reap the field in 86 days, the total work can be calculated as:
\[ \text{Total Work} = \text{Number of Men} \times \text{Number of Days} = 6 \times 86 = 516 \text{ man-days} \]

2. **Calculate the efficiency of men and women:**
- We know that 6 men are equivalent to 8 women. Therefore, we can find the ratio of their efficiencies:
\[ \text{Efficiency of Men} : \text{Efficiency of Women} = 8 : 6 = 4 : 3 \]
- Let the efficiency of one man be 4 units and that of one woman be 3 units.

3. **Calculate the daily work done by 14 men and 10 women:**
- The total daily work done by 14 men is:
\[ \text{Work by Men} = 14 \times 4 = 56 \text{ units} \]
- The total daily work done by 10 women is:
\[ \text{Work by Women} = 10 \times 3 = 30 \text{ units} \]
- Therefore, the total daily work done by 14 men and 10 women together is:
\[ \text{Total Daily Work} = 56 + 30 = 86 \text{ units} \]

4. **Calculate the time taken to complete the work:**
- We already calculated the total work as 516 man-days. To find out how many days it will take for 14 men and 10 women to complete the work, we divide the total work by the daily work:
\[ \text{Time Taken} = \frac{\text{Total Work}}{\text{Total Daily Work}} = \frac{516}{86} = 6 \text{ days} \]

### Final Answer:
Thus, 14 men and 10 women will take **6 days** to reap the field.