Three men, four women and six children can complete a work in seven days. A woman does double the work a man does and a child does half the work a man does. How many women alone can complete this work in 7 days?

Options

A. 7
B. 6
C. 12
D. Cannot be determined

Answer

To solve the problem step by step, we need to determine how many women alone can complete the work in 7 days.

### Step 1: Determine the work done by men, women, and children
We know that:
- 3 men, 4 women, and 6 children can complete the work in 7 days.

Let’s denote the work done by one man in one day as \( M \).
According to the problem:
- One woman does double the work of a man, so the work done by one woman in one day is \( 2M \).
- One child does half the work of a man, so the work done by one child in one day is \( \frac{1}{2}M \).

### Step 2: Calculate total work done in one day
Now, we can express the total work done by 3 men, 4 women, and 6 children in one day:
- Work done by 3 men = \( 3M \)
- Work done by 4 women = \( 4 \times 2M = 8M \)
- Work done by 6 children = \( 6 \times \frac{1}{2}M = 3M \)

Adding these together gives:
\[ \text{Total work done in one day} = 3M + 8M + 3M = 14M \]

### Step 3: Calculate total work for 7 days
Since they complete the work in 7 days, the total work \( W \) can be calculated as:
\[ W = \text{Total work done in one day} \times \text{Number of days} = 14M \times 7 = 98M \]

### Step 4: Determine how many women can complete the work alone in 7 days
Let \( x \) be the number of women required to complete the work in 7 days. The work done by \( x \) women in one day is:
\[ \text{Work done by } x \text{ women in one day} = x \times 2M = 2xM \]

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

### Step 5: Set up the equation
We know that the total work \( W \) is equal to the total work done by \( x \) women in 7 days:
\[ 14xM = 98M \]

### Step 6: Solve for \( x \)
Dividing both sides by \( 14M \):
\[ x = \frac{98M}{14M} = 7 \]

### Conclusion
Thus, **7 women** alone can complete the work in 7 days.

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