Four men and three women can do a job in six days. When five men and six women work on the same job, the work gets completed in four days. How long will two women and, three men take to do the job?

Options

A. 18 days
B. 13 4/8 days
C. 8 4/13 days
D. 7 days

Answer

To solve the problem step by step, we will first determine the work done by men and women, and then calculate how long it will take for 2 women and 3 men to complete the job.

### Step 1: Define Work Done by Men and Women
Let the work done by one man in one day be \( M \) and the work done by one woman in one day be \( W \).

### Step 2: Set Up Equations Based on Given Information
From the problem, we know:
1. **4 men and 3 women can complete the job in 6 days.**
\[ \text{Total work} = \text{(Work done per day)} \times \text{(Number of days)} \]
\[ \text{Total work} = (4M + 3W) \times 6 \]
\[ \text{Total work} = 24M + 18W \quad \text{(Equation 1)} \]

2. **5 men and 6 women can complete the job in 4 days.**
\[ \text{Total work} = (5M + 6W) \times 4 \]
\[ \text{Total work} = 20M + 24W \quad \text{(Equation 2)} \]

### Step 3: Set the Two Equations Equal
Since both equations represent the total work, we can set them equal to each other:
\[ 24M + 18W = 20M + 24W \]

### Step 4: Rearrange the Equation
Rearranging gives:
\[ 24M - 20M = 24W - 18W \]
\[ 4M = 6W \]
\[ \frac{M}{W} = \frac{6}{4} = \frac{3}{2} \]
This means that the work done by one man is equivalent to the work done by 1.5 women.

### Step 5: Substitute \( M \) in Terms of \( W \)
Let \( M = \frac{3}{2}W \).

### Step 6: Substitute \( M \) Back into One of the Equations
Substituting \( M \) into Equation 1:
\[ 24\left(\frac{3}{2}W\right) + 18W = \text{Total Work} \]
\[ 36W + 18W = \text{Total Work} \]
\[ \text{Total Work} = 54W \]

### Step 7: Calculate Work Done by 2 Women and 3 Men
Now we need to find out how much work 2 women and 3 men can do in one day:
\[ \text{Work done by 2 women} = 2W \]
\[ \text{Work done by 3 men} = 3M = 3\left(\frac{3}{2}W\right) = \frac{9}{2}W \]
\[ \text{Total work done in one day} = 2W + \frac{9}{2}W = \frac{4}{2}W + \frac{9}{2}W = \frac{13}{2}W \]

### Step 8: Calculate the Number of Days to Complete the Job
To find the number of days \( D \) required to complete the total work of \( 54W \):
\[ D = \frac{\text{Total Work}}{\text{Work done in one day}} = \frac{54W}{\frac{13}{2}W} \]
\[ D = 54W \times \frac{2}{13W} = \frac{108}{13} \approx 8.31 \text{ days} \]

### Conclusion
Thus, it will take approximately \( \frac{108}{13} \) days or about 8.31 days for 2 women and 3 men to complete the job.