4 men and 3 women finish a job in 6 days, and 5 men and 7 women can do the same job in 4 days. How long will I man and 1 woman take to do the work?

Options

A. `22 2/7` days
B. `25 1/2` days
C. `5 1/7` days
D. `12 7/22` days

Answer

To solve the problem step by step, we will first determine the work done by men and women, then find out how long it will take for one man and one woman to complete the job.

### Step 1: Define Variables
Let the efficiency of one man be \( M \) and the efficiency of one woman be \( W \).

### Step 2: Set Up the Equations
From the problem, we have two scenarios:
1. **4 Men and 3 Women finish the job in 6 days**:
\[ 6 \times (4M + 3W) = \text{Total Work} \]
This implies:
\[ 4M + 3W = \frac{\text{Total Work}}{6} \quad \text{(Equation 1)} \]

2. **5 Men and 7 Women finish the job in 4 days**:
\[ 4 \times (5M + 7W) = \text{Total Work} \]
This implies:
\[ 5M + 7W = \frac{\text{Total Work}}{4} \quad \text{(Equation 2)} \]

### Step 3: Equate the Total Work
Since both equations represent the same total work, we can set them equal to each other:
\[ 6(4M + 3W) = 4(5M + 7W) \]

### Step 4: Expand and Simplify
Expanding both sides gives:
\[ 24M + 18W = 20M + 28W \]
Rearranging the equation:
\[ 24M - 20M = 28W - 18W \]
This simplifies to:
\[ 4M = 10W \]
Thus, we can express the relationship between men and women:
\[ \frac{M}{W} = \frac{10}{4} = \frac{5}{2} \]

### Step 5: Substitute Back to Find Total Work
Now, we can substitute \( M = \frac{5}{2}W \) back into either Equation 1 or Equation 2. We will use Equation 1:
\[ 4\left(\frac{5}{2}W\right) + 3W = \frac{\text{Total Work}}{6} \]
This simplifies to:
\[ 10W + 3W = \frac{\text{Total Work}}{6} \]
\[ 13W = \frac{\text{Total Work}}{6} \]
Thus, the total work can be expressed as:
\[ \text{Total Work} = 78W \]

### Step 6: Find the Efficiency of 1 Man and 1 Woman
The combined efficiency of 1 man and 1 woman is:
\[ M + W = \frac{5}{2}W + W = \frac{7}{2}W \]

### Step 7: Calculate the Time Taken by 1 Man and 1 Woman
To find out how long it will take for 1 man and 1 woman to complete the job:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency}} = \frac{78W}{\frac{7}{2}W} \]
The \( W \) cancels out:
\[ \text{Time} = \frac{78}{\frac{7}{2}} = 78 \times \frac{2}{7} = \frac{156}{7} \text{ days} \]

### Step 8: Convert to Mixed Number
\[ \frac{156}{7} = 22 \frac{2}{7} \text{ days} \]

### Final Answer
So, the time taken by 1 man and 1 woman to complete the work is \( 22 \frac{2}{7} \) days.