2 men and 7 women can do a piece of work in 14 days whereas 3 men and 8 women can do it in 11 days. In how many days 5 men and 4 women can do the same work?

Options

A. 11
B. 12
C. 14
D. 10

Answer

To solve the problem step by step, we first need to establish the work rates of men and women based on the information given.

### Step 1: Define Variables
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
From the problem, we know:
1. **Equation 1**: \( 2m + 7w \) can complete the work in 14 days.
- Therefore, the total work \( W \) can be expressed as:
\[ W = (2m + 7w) \times 14 \]
\[ W = 28m + 98w \]

2. **Equation 2**: \( 3m + 8w \) can complete the work in 11 days.
- Therefore, the total work \( W \) can also be expressed as:
\[ W = (3m + 8w) \times 11 \]
\[ W = 33m + 88w \]

### Step 3: Set the Two Equations Equal
Since both expressions represent the same total work \( W \), we can set them equal to each other:
\[ 28m + 98w = 33m + 88w \]

### Step 4: Rearrange the Equation
Rearranging the equation gives:
\[ 28m - 33m = 88w - 98w \]
\[ -5m = -10w \]
Dividing both sides by -5:
\[ m = 2w \]

### Step 5: Substitute \( m \) in Terms of \( w \)
Now we can express \( m \) in terms of \( w \):
\[ m = 2w \]

### Step 6: Calculate Work Done by 5 Men and 4 Women
Now we need to find out how many days \( 5m + 4w \) can complete the work:
\[ 5m + 4w = 5(2w) + 4w = 10w + 4w = 14w \]

### Step 7: Find Total Work in Terms of \( w \)
Using either of the earlier equations to find \( W \) in terms of \( w \):
Using \( W = 28m + 98w \):
\[ W = 28(2w) + 98w = 56w + 98w = 154w \]

### Step 8: Calculate Days Taken by 5 Men and 4 Women
Now, we can find the number of days \( D \) it takes for \( 5m + 4w \) to complete the work:
\[ D = \frac{W}{5m + 4w} = \frac{154w}{14w} = \frac{154}{14} = 11 \]

### Final Answer
Thus, \( 5 \) men and \( 4 \) women can complete the work in **11 days**.

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