A man, a woman and a boy can together complete a piece of work In 3 days. If a man alone can do it in 6 days and a boy alone in 18 days, how long will a woman alone take to complete the work?

Options

A. 9 days
B. 21 days
C. 24 days
D. 27 days

Answer

To solve the problem step by step, we need to determine how long the woman alone will take to complete the work based on the information provided.

### Step 1: Determine the work done by each person in one day.
- **Man's work**: A man can complete the work in 6 days. Therefore, the work done by the man in one day is:
\[ \text{Work done by Man in 1 day} = \frac{1}{6} \]

- **Boy's work**: A boy can complete the work in 18 days. Therefore, the work done by the boy in one day is:
\[ \text{Work done by Boy in 1 day} = \frac{1}{18} \]

- **Combined work of Man, Woman, and Boy**: Together, they can complete the work in 3 days. Therefore, the work done by all three in one day is:
\[ \text{Work done by Man, Woman, and Boy in 1 day} = \frac{1}{3} \]

### Step 2: Set up the equation for the woman's work.
Let the work done by the woman in one day be represented as \( W \). According to the information:
\[ \text{Work done by Man} + \text{Work done by Woman} + \text{Work done by Boy} = \text{Combined Work} \]
Substituting the values we have:
\[ \frac{1}{6} + W + \frac{1}{18} = \frac{1}{3} \]

### Step 3: Solve for \( W \).
First, we need a common denominator to combine the fractions. The least common multiple of 6, 18, and 3 is 18. We can rewrite the equation:
\[ \frac{3}{18} + W + \frac{1}{18} = \frac{6}{18} \]
Combining the fractions on the left side:
\[ \frac{3 + 1}{18} + W = \frac{6}{18} \]
This simplifies to:
\[ \frac{4}{18} + W = \frac{6}{18} \]
Now, isolate \( W \):
\[ W = \frac{6}{18} - \frac{4}{18} = \frac{2}{18} = \frac{1}{9} \]

### Step 4: Determine how long the woman will take to complete the work.
Since \( W = \frac{1}{9} \), this means that the woman can complete \( \frac{1}{9} \) of the work in one day. Therefore, the total time taken by the woman to complete the work alone is the reciprocal of her one-day work:
\[ \text{Time taken by Woman} = \frac{1}{W} = 9 \text{ days} \]

### Final Answer:
The woman alone will take **9 days** to complete the work.

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