3 men, 4 women and 6 boys together can complete a work in 5 days. A woman does double the work a man does and a boy does half the work a man does in a day. How many women alone can complete this work in 7 days?

Options

A. 14
B. 7
C. 8
D. 5

Answer

To solve the problem step by step, let's break it down:

### Step 1: Define the Work Done by Each Person
Let the work done by 1 man in 1 day be \( m \).
According to the problem:
- A woman does double the work of a man, so the work done by 1 woman in 1 day is \( 2m \).
- A boy does half the work of a man, so the work done by 1 boy in 1 day is \( \frac{m}{2} \).

### Step 2: Calculate Total Work Done by 3 Men, 4 Women, and 6 Boys
Now, we can express the total work done by 3 men, 4 women, and 6 boys in 1 day:
- Work done by 3 men: \( 3m \)
- Work done by 4 women: \( 4 \times 2m = 8m \)
- Work done by 6 boys: \( 6 \times \frac{m}{2} = 3m \)

Adding these together gives:
\[ 3m + 8m + 3m = 14m \]
So, the total work done by 3 men, 4 women, and 6 boys in 1 day is \( 14m \).

### Step 3: Calculate Total Work for 5 Days
Since they can complete the work in 5 days, the total work \( W \) can be calculated as:
\[ W = \text{Work done in 1 day} \times \text{Number of days} = 14m \times 5 = 70m \]

### Step 4: Determine Work Done by Women Alone in 7 Days
Let \( n \) be the number of women required to complete the work in 7 days. The work done by \( n \) women in 1 day is:
\[ n \times 2m \]
Therefore, the total work done by \( n \) women in 7 days is:
\[ 7 \times n \times 2m = 14nm \]

### Step 5: Set Up the Equation
To complete the total work \( W \), we set up the equation:
\[ 14nm = 70m \]

### Step 6: Solve for \( n \)
Dividing both sides by \( 14m \) (assuming \( m \neq 0 \)):
\[ n = \frac{70m}{14m} = 5 \]

### Conclusion
Thus, the number of women required to complete the work in 7 days is \( 5 \).