3 men 4 women and 6 boys togethher 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. 7
B. 8
C. 14
D. 5

Answer

To solve the problem step by step, we will first determine the work done by each individual (men, women, and boys) and then calculate how many women are needed to complete the work alone in 7 days.

### Step 1: Define the efficiencies
Let the efficiency of one man be \( M \). According to the problem:
- A woman does double the work of a man, so the efficiency of one woman \( W \) is \( 2M \).
- A boy does half the work of a man, so the efficiency of one boy \( B \) is \( \frac{1}{2}M \).

### Step 2: Calculate the total efficiency of the group
We have:
- 3 men, so their total efficiency is \( 3M \).
- 4 women, so their total efficiency is \( 4 \times 2M = 8M \).
- 6 boys, so their total efficiency is \( 6 \times \frac{1}{2}M = 3M \).

Now, we can sum these efficiencies:
\[ \text{Total Efficiency} = 3M + 8M + 3M = 14M \]

### Step 3: Calculate the total work done
The group can complete the work in 5 days. Therefore, the total work \( W \) can be calculated as:
\[ W = \text{Total Efficiency} \times \text{Number of Days} = 14M \times 5 = 70M \]

### Step 4: Determine how many women are needed to complete the work in 7 days
Let \( x \) be the number of women required to complete the work in 7 days. The efficiency of \( x \) women is:
\[ \text{Efficiency of } x \text{ women} = x \times 2M = 2xM \]

The total work done by \( x \) women in 7 days is:
\[ \text{Total Work} = \text{Efficiency} \times \text{Number of Days} = 2xM \times 7 = 14xM \]

### Step 5: Set up the equation
We know the total work \( W \) is equal to \( 70M \). Therefore, we can set up the equation:
\[ 14xM = 70M \]

### Step 6: Solve for \( x \)
Dividing both sides by \( 14M \):
\[ x = \frac{70M}{14M} = 5 \]

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