2 men or 3 women or 4 boys can do a work in 52 days. Then in how many days will 1 man, 1 women and 1 boy do the work?

Answer

To solve the problem, we need to determine how many days it will take for 1 man, 1 woman, and 1 boy to complete the work together, given that 2 men, 3 women, or 4 boys can complete the work in 52 days.

### Step-by-Step Solution:

1. **Understanding the Work Rate**:
- 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.

2. **Calculating Work Done**:
- Let's denote the total work as \( W \).
- If 2 men complete the work in 52 days, then the work done by 2 men in one day is:
\[ \text{Work by 2 men in 1 day} = \frac{W}{52} \]
- Therefore, the work done by 1 man in one day is:
\[ \text{Work by 1 man in 1 day} = \frac{W}{52} \times \frac{1}{2} = \frac{W}{104} \]

3. **Calculating Work Rate for Women**:
- Similarly, for 3 women:
\[ \text{Work by 3 women in 1 day} = \frac{W}{52} \]
- Thus, the work done by 1 woman in one day is:
\[ \text{Work by 1 woman in 1 day} = \frac{W}{52} \times \frac{1}{3} = \frac{W}{156} \]

4. **Calculating Work Rate for Boys**:
- For 4 boys:
\[ \text{Work by 4 boys in 1 day} = \frac{W}{52} \]
- Therefore, the work done by 1 boy in one day is:
\[ \text{Work by 1 boy in 1 day} = \frac{W}{52} \times \frac{1}{4} = \frac{W}{208} \]

5. **Calculating Combined Work Rate**:
- Now, we can find the combined work done by 1 man, 1 woman, and 1 boy in one day:
\[ \text{Total work in 1 day} = \text{Work by 1 man} + \text{Work by 1 woman} + \text{Work by 1 boy} \]
\[ = \frac{W}{104} + \frac{W}{156} + \frac{W}{208} \]

6. **Finding a Common Denominator**:
- The least common multiple (LCM) of 104, 156, and 208 is 1560.
- Converting each fraction:
\[ \frac{W}{104} = \frac{15W}{1560}, \quad \frac{W}{156} = \frac{10W}{1560}, \quad \frac{W}{208} = \frac{7.5W}{1560} \]
- Adding these together:
\[ \text{Total work in 1 day} = \frac{15W + 10W + 7.5W}{1560} = \frac{32.5W}{1560} \]

7. **Calculating Days to Complete the Work**:
- The total work \( W \) can be expressed in terms of the combined work rate:
\[ \text{Days} = \frac{W}{\text{Total work in 1 day}} = \frac{W}{\frac{32.5W}{1560}} = \frac{1560}{32.5} \]
- Simplifying this gives:
\[ \text{Days} = 48 \]

### Final Answer:
1 man, 1 woman, and 1 boy will complete the work in **48 days**.