If 3 women and 6 men can complete a work in 4 days and 6 women and 4 men can complete same work in 3 days, so 1 woman can alone complete the work in how many days

Options

A. 46 days
B. 24 days
C. 12 days
D. 6 days

Explanation

To solve the problem, we need to find out how many days one woman would take to complete the work alone, given the information about the work done by groups of women and men.

### Step-by-Step Solution:

1. **Understanding the Work Done:**
- We know that 3 women and 6 men can complete the work in 4 days.
- We also know that 6 women and 4 men can complete the same work in 3 days.

2. **Setting Up the Equations:**
- Let the work done by one woman in one day be W (work units).
- Let the work done by one man in one day be M (work units).
- The total work can be expressed in terms of women and men:
- For the first group (3 women and 6 men working for 4 days):
\[ \text{Total Work} = (3W + 6M) \times 4 \]
- For the second group (6 women and 4 men working for 3 days):
\[ \text{Total Work} = (6W + 4M) \times 3 \]

3. **Equating the Total Work:**
- Since both expressions represent the same total work, we can set them equal to each other:
\[ (3W + 6M) \times 4 = (6W + 4M) \times 3 \]

4. **Expanding the Equations:**
- Expanding both sides gives:
\[ 12W + 24M = 18W + 12M \]

5. **Rearranging the Equation:**
- Rearranging the equation to isolate W and M:
\[ 12M - 12W = 18W - 12W \]
\[ 12M = 6W \]
\[ W = 2M \]
- This means that the work done by one woman is equivalent to the work done by two men.

6. **Finding the Total Work in Terms of One Woman:**
- Now, we can express the total work in terms of one woman:
- Using the first group (3 women and 6 men):
\[ \text{Total Work} = (3W + 6M) \times 4 \]
- Substitute \( W = 2M \):
\[ = (3(2M) + 6M) \times 4 \]
\[ = (6M + 6M) \times 4 \]
\[ = 12M \times 4 = 48M \]

7. **Expressing Total Work in Terms of One Woman:**
- Since \( W = 2M \), we can express \( M \) in terms of \( W \):
\[ M = \frac{W}{2} \]
- Substitute this into the total work:
\[ \text{Total Work} = 48M = 48 \times \frac{W}{2} = 24W \]

8. **Finding Days for One Woman to Complete the Work:**
- If one woman does the work alone, we denote the number of days taken by one woman as X:
\[ W \times X = 24W \]
- Dividing both sides by W (assuming W ≠ 0):
\[ X = 24 \text{ days} \]

### Conclusion:
Thus, one woman can complete the work alone in **24 days**.