1 man or 2 boys or 3 girls can do a piece of work in 88 days. In how many days one man , one boy and one girl can do the same work ?

Answer

To solve the problem, we need to determine how many days one man, one boy, and one girl can complete a piece of work together, given that one man, two boys, or three girls can complete the work in 88 days.

### Step-by-Step Solution:

1. **Understanding Work Rates**:
- One man (M), two boys (2B), and three girls (3G) can complete the work in 88 days. This means they all have the same work capacity.
- Let’s denote the work done by one man in one day as \( M \), one boy as \( B \), and one girl as \( G \).

2. **Establishing Relationships**:
- From the information given:
- \( 1M = 2B \) (1 man is equivalent to 2 boys)
- \( 1M = 3G \) (1 man is equivalent to 3 girls)
- From \( 1M = 2B \), we can express \( B \) in terms of \( M \):
\[ B = \frac{1}{2}M \]
- From \( 1M = 3G \), we can express \( G \) in terms of \( M \):
\[ G = \frac{1}{3}M \]

3. **Finding Combined Work Rate**:
- Now, we want to find the total work rate of one man, one boy, and one girl together:
\[ \text{Total work rate} = M + B + G \]
- Substitute \( B \) and \( G \):
\[ = M + \frac{1}{2}M + \frac{1}{3}M \]
- To combine these, we need a common denominator. The least common multiple of 1, 2, and 3 is 6:
\[ = \frac{6}{6}M + \frac{3}{6}M + \frac{2}{6}M = \frac{11}{6}M \]

4. **Calculating Total Work**:
- The total work done by 3 girls in 88 days is:
\[ \text{Total Work} = 3G \times 88 = 3 \times \frac{1}{3}M \times 88 = M \times 88 \]
- Thus, the total work is equivalent to \( 88M \).

5. **Finding Days Taken by One Man, One Boy, and One Girl**:
- If the combined work rate of one man, one boy, and one girl is \( \frac{11}{6}M \), we can find the number of days \( D \) required to complete the total work:
\[ \text{Total Work} = \text{Work Rate} \times \text{Days} \]
\[ 88M = \frac{11}{6}M \times D \]
- Dividing both sides by \( M \):
\[ 88 = \frac{11}{6}D \]
- Rearranging gives:
\[ D = 88 \times \frac{6}{11} = \frac{528}{11} = 48 \]

### Final Answer:
One man, one boy, and one girl can complete the work in **48 days**.