If 2 men or 3 women or 4 boys can do a piece of work in 52 days, then the same piece of work will be done by 1 man, 1 woman and 1 boy in :

Options

A. 48 days
B. 36 days
C. 45 days
D. none of these

Answer

To solve the problem, we need to determine how long it will take for 1 man, 1 woman, and 1 boy to complete the same piece of work that can be done by 2 men, 3 women, or 4 boys in 52 days.

### Step-by-Step Solution:

1. **Understanding the Work Capacity**:
- We know that:
- 2 men can complete the work in 52 days.
- 3 women can complete the work in 52 days.
- 4 boys can complete the work in 52 days.
- This means that the total work can be represented in terms of man-days, woman-days, and boy-days.

2. **Calculating Work Done**:
- Let's denote the total work as \( W \).
- The work done by 2 men in 52 days:
\[ W = 2 \text{ men} \times 52 \text{ days} = 104 \text{ man-days} \]
- Similarly, for women:
\[ W = 3 \text{ women} \times 52 \text{ days} = 156 \text{ woman-days} \]
- And for boys:
\[ W = 4 \text{ boys} \times 52 \text{ days} = 208 \text{ boy-days} \]

3. **Finding the Work Rate**:
- From the equations above, we can derive the work rates:
- Work rate of 1 man:
\[ 1 \text{ man} = \frac{104}{2} = 52 \text{ days} \quad \Rightarrow \quad \text{Work rate of 1 man} = \frac{1}{52} \]
- Work rate of 1 woman:
\[ 1 \text{ woman} = \frac{156}{3} = 52 \text{ days} \quad \Rightarrow \quad \text{Work rate of 1 woman} = \frac{1}{52} \]
- Work rate of 1 boy:
\[ 1 \text{ boy} = \frac{208}{4} = 52 \text{ days} \quad \Rightarrow \quad \text{Work rate of 1 boy} = \frac{1}{52} \]

4. **Combining Work Rates**:
- Now, we need to find the combined work rate of 1 man, 1 woman, and 1 boy:
\[ \text{Combined work rate} = \frac{1}{52} + \frac{1}{52} + \frac{1}{52} = \frac{3}{52} \]

5. **Calculating Time Taken**:
- If the combined work rate of 1 man, 1 woman, and 1 boy is \( \frac{3}{52} \), then the time taken \( D \) to complete the work \( W \) is given by:
\[ D = \frac{W}{\text{Combined work rate}} = \frac{1}{\frac{3}{52}} = \frac{52}{3} \text{ days} \]

6. **Final Answer**:
- Therefore, the time taken by 1 man, 1 woman, and 1 boy to complete the work is:
\[ D = \frac{52}{3} \approx 17.33 \text{ days} \]