25 men and 15 women can complete the work in 12 days. They started working together and after 8 days, women left the work. Now 25 men complete the remaining work in 6 days. Find the time required to complete the work by only 15 women?

Options

A. 36 days
B. 88 days
C. 94 days
D. None of these

Answer

To solve the problem step by step, let's break it down:

### Step 1: Calculate the total work done by 25 men and 15 women in 12 days.
- Let the work done by 1 man in 1 day be \( M \) and the work done by 1 woman in 1 day be \( W \).
- The total work done by 25 men and 15 women in 12 days can be expressed as:
\[ (25M + 15W) \times 12 = \text{Total Work} \]

### Step 2: Calculate the total work done in terms of efficiency.
- We can simplify the equation:
\[ 25M + 15W = \frac{\text{Total Work}}{12} \]

### Step 3: Work done in 8 days by 25 men and 15 women.
- In 8 days, the work done is:
\[ (25M + 15W) \times 8 \]

### Step 4: Calculate the remaining work after 8 days.
- The remaining work after 8 days is:
\[ \text{Remaining Work} = \text{Total Work} - (25M + 15W) \times 8 \]

### Step 5: Work done by 25 men in the next 6 days.
- The remaining work is completed by 25 men in 6 days:
\[ 25M \times 6 = \text{Remaining Work} \]

### Step 6: Set up the equation for total work.
- From the above steps, we can equate the remaining work:
\[ (25M + 15W) \times 12 - (25M + 15W) \times 8 = 25M \times 6 \]
- Simplifying gives:
\[ (25M + 15W) \times 4 = 25M \times 6 \]

### Step 7: Solve for the ratio of men's and women's work.
- Rearranging gives:
\[ 25M + 15W = \frac{25M \times 6}{4} \]
- This simplifies to:
\[ 25M + 15W = 37.5M \]
- Therefore:
\[ 15W = 12.5M \implies \frac{M}{W} = \frac{15}{12.5} = \frac{6}{5} \]

### Step 8: Calculate the total work in terms of women’s work.
- Let’s denote the total work as \( T \):
\[ T = 25M \times 12 + 15W \times 12 \]
- Substitute \( M = \frac{5}{6}W \):
\[ T = 25 \left(\frac{5}{6}W\right) \times 12 + 15W \times 12 \]
\[ T = \frac{125}{6}W \times 12 + 15W \times 12 \]
\[ T = 250W + 180W = 430W \]

### Step 9: Calculate the time taken by 15 women to complete the work.
- The work done by 15 women in 1 day is:
\[ 15W \]
- Therefore, the time taken by 15 women to complete the total work \( T \) is:
\[ \text{Time} = \frac{T}{15W} = \frac{430W}{15W} = \frac{430}{15} = 28.67 \text{ days} \]

### Conclusion
- The time required for 15 women to complete the work is approximately 28.67 days.