6 men and 8 women as well as 3 men and 13 women can finish the same work in 10 days. In how many days will 6 women finish the same work if they work alone?

Options

A. 32
B. 25
C. 30
D. 35

Answer

To solve the problem, we need to determine how many days it will take for 6 women to finish the same work alone. We start by analyzing the information given about the work done by men and women together.

### Step-by-Step Solution:

1. **Understanding the Work Done by Groups**:
- We know that 6 men and 8 women can complete the work in 10 days.
- We also know that 3 men and 13 women can complete the same work in 10 days.

2. **Calculating Total Work**:
- The total work can be expressed as:
\[ \text{Work} = \text{Efficiency} \times \text{Time} \]
- For the first group (6 men and 8 women):
\[ \text{Work} = (6M + 8W) \times 10 \]
- For the second group (3 men and 13 women):
\[ \text{Work} = (3M + 13W) \times 10 \]

3. **Equating the Work**:
- Since both groups complete the same work, we can set the equations equal to each other:
\[ (6M + 8W) \times 10 = (3M + 13W) \times 10 \]
- Dividing both sides by 10:
\[ 6M + 8W = 3M + 13W \]

4. **Rearranging the Equation**:
- Rearranging gives us:
\[ 6M - 3M = 13W - 8W \]
\[ 3M = 5W \]
- This implies:
\[ M = \frac{5}{3}W \]

5. **Finding Total Work in Terms of Women’s Efficiency**:
- Now, substituting \(M\) back into one of the work equations:
\[ \text{Work} = (6M + 8W) \times 10 \]
- Replacing \(M\) with \(\frac{5}{3}W\):
\[ \text{Work} = \left(6 \times \frac{5}{3}W + 8W\right) \times 10 \]
\[ = \left(10W + 8W\right) \times 10 \]
\[ = 18W \times 10 = 180W \]

6. **Calculating Days for 6 Women to Complete the Work**:
- Now we need to find out how many days it will take for 6 women to complete 180W:
\[ \text{Efficiency of 6 Women} = 6W \]
- Using the formula for work:
\[ \text{Time} = \frac{\text{Work}}{\text{Efficiency}} = \frac{180W}{6W} = 30 \text{ days} \]

### Final Answer:
Thus, it will take 6 women **30 days** to complete the work alone.