15 men can complete a work in 8 days while 10 women can complete the same work in 20 days. 7 men starts working and after 12 days they are replaced by 10 women. Find time taken by 10 women to complete the remaining work.

Options

A. A)5 days
B. B)8 days
C. C)7 days
D. D)6 days

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Calculate the total work done by men and women.
1. **Men's Work Rate**:
- 15 men can complete the work in 8 days.
- Therefore, the total work can be calculated as:
\[ \text{Total Work} = \text{Number of Men} \times \text{Days} = 15 \times 8 = 120 \text{ man-days} \]

2. **Women's Work Rate**:
- 10 women can complete the same work in 20 days.
- Therefore, the total work can also be calculated as:
\[ \text{Total Work} = \text{Number of Women} \times \text{Days} = 10 \times 20 = 200 \text{ woman-days} \]

### Step 2: Find the work done by 7 men in 12 days.
1. **Efficiency of 1 Man**:
- Since 15 men can complete the work in 120 man-days, the work done by 1 man in 1 day is:
\[ \text{Work done by 1 man in 1 day} = \frac{1}{120} \text{ of the work} \]

2. **Efficiency of 7 Men**:
- Therefore, the work done by 7 men in 1 day is:
\[ \text{Work done by 7 men in 1 day} = 7 \times \frac{1}{120} = \frac{7}{120} \]

3. **Work done by 7 Men in 12 Days**:
- The total work done by 7 men in 12 days is:
\[ \text{Total work done} = 12 \times \frac{7}{120} = \frac{84}{120} = \frac{7}{10} \text{ of the work} \]

### Step 3: Calculate the remaining work.
1. **Remaining Work**:
- If 7 men completed \(\frac{7}{10}\) of the work, the remaining work is:
\[ \text{Remaining Work} = 1 - \frac{7}{10} = \frac{3}{10} \text{ of the work} \]

### Step 4: Calculate the time taken by 10 women to complete the remaining work.
1. **Efficiency of 10 Women**:
- From the previous calculations, we know that 10 women can complete the work in 200 woman-days.
- Therefore, the work done by 1 woman in 1 day is:
\[ \text{Work done by 1 woman in 1 day} = \frac{1}{200} \]
- Thus, the work done by 10 women in 1 day is:
\[ \text{Work done by 10 women in 1 day} = 10 \times \frac{1}{200} = \frac{10}{200} = \frac{1}{20} \]

2. **Time Taken by 10 Women to Complete Remaining Work**:
- To find the time taken \(T\) by 10 women to complete \(\frac{3}{10}\) of the work:
\[ \frac{1}{20} \times T = \frac{3}{10} \]
- Solving for \(T\):
\[ T = \frac{3}{10} \times 20 = 6 \text{ days} \]

### Final Answer:
The time taken by 10 women to complete the remaining work is **6 days**.

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