If one man or two women or three boys can finish a work in 88 days, then how many days will one man, one woman and one boy together take to finish the same work?

Options

A. 46 days
B. 54 days
C. 48 days
D. 44 days

Answer

To solve the problem, we need to determine how many days one man, one woman, and one boy together will take to finish the work, given that one man, two women, or three boys can complete the work in 88 days.

### Step-by-Step Solution:

1. **Understand the Work Rate**:
- Let the work done by one man in one day be \( M \).
- Let the work done by one woman in one day be \( W \).
- Let the work done by one boy in one day be \( B \).

2. **Establish Relationships**:
- From the problem, we know:
- 1 man = 2 women
- 1 man = 3 boys
- Therefore, we can express the relationships as:
- \( M = 2W \)
- \( M = 3B \)

3. **Express Everything in Terms of Boys**:
- From \( M = 3B \), we can express:
- \( W = \frac{M}{2} = \frac{3B}{2} \)
- Now we can express the work done by one man, one woman, and one boy:
- Work done by 1 man = \( M \)
- Work done by 1 woman = \( \frac{3B}{2} \)
- Work done by 1 boy = \( B \)

4. **Combine the Work Rates**:
- The total work done by one man, one woman, and one boy in one day is:
\[ M + W + B = M + \frac{3B}{2} + B \]
- Substitute \( M = 3B \):
\[ = 3B + \frac{3B}{2} + B \]
- Convert everything to a common denominator (which is 2):
\[ = \frac{6B}{2} + \frac{3B}{2} + \frac{2B}{2} = \frac{11B}{2} \]

5. **Determine Total Work**:
- If 1 man, 2 women, or 3 boys can finish the work in 88 days, then the total work \( W_{total} \) can be expressed as:
\[ W_{total} = \text{Work rate} \times \text{Time} = M \times 88 = 3B \times 88 = 264B \]

6. **Calculate Days Taken by One Man, One Woman, and One Boy**:
- Let \( D \) be the number of days taken by one man, one woman, and one boy to complete the work:
\[ \frac{11B}{2} \times D = 264B \]
- Cancel \( B \) from both sides (assuming \( B \neq 0 \)):
\[ \frac{11}{2} D = 264 \]
- Multiply both sides by 2:
\[ 11D = 528 \]
- Divide by 11:
\[ D = \frac{528}{11} = 48 \]

### Conclusion:
One man, one woman, and one boy together will take **48 days** to finish the work.