If 1 man or 2 women or 3 boys can complete a piece of work in 88 days, then 1 man 1 women and 1 boy together will complete it in

Options

A. 1. 36 days
B. 2. 42 days
C. 3. 48 days
D. 4. 54 days

Answer

To solve the problem step by step, we will first establish the work rates of each individual (man, woman, boy) based on the information given.

### Step 1: Establish the work rates
We know that:
- 1 man can complete the work in 88 days.
- 2 women can complete the work in 88 days.
- 3 boys can complete the work in 88 days.

From this, we can derive the work rates:
- Work done by 1 man in 1 day = 1/88
- Work done by 2 women in 1 day = 1/88 → Work done by 1 woman in 1 day = 1/(2*88) = 1/176
- Work done by 3 boys in 1 day = 1/88 → Work done by 1 boy in 1 day = 1/(3*88) = 1/264

### Step 2: Calculate the combined work rate of 1 man, 1 woman, and 1 boy
Now, we will find the combined work rate of 1 man, 1 woman, and 1 boy:
- Work done by 1 man in 1 day = 1/88
- Work done by 1 woman in 1 day = 1/176
- Work done by 1 boy in 1 day = 1/264

The combined work rate is:
\[ \text{Combined work rate} = \frac{1}{88} + \frac{1}{176} + \frac{1}{264} \]

### Step 3: Find a common denominator
To add these fractions, we need a common denominator. The least common multiple (LCM) of 88, 176, and 264 is 528.

Now, we convert each fraction:
- \( \frac{1}{88} = \frac{6}{528} \)
- \( \frac{1}{176} = \frac{3}{528} \)
- \( \frac{1}{264} = \frac{2}{528} \)

### Step 4: Add the fractions
Now we can add the fractions:
\[ \text{Combined work rate} = \frac{6}{528} + \frac{3}{528} + \frac{2}{528} = \frac{11}{528} \]

### Step 5: Calculate the time taken to complete the work
The combined work rate of 1 man, 1 woman, and 1 boy is \( \frac{11}{528} \) of the work in one day. To find the number of days \( D \) to complete the work, we take the reciprocal of the combined work rate:
\[ D = \frac{1}{\frac{11}{528}} = \frac{528}{11} = 48 \text{ days} \]

Thus, 1 man, 1 woman, and 1 boy together will complete the work in **48 days**.