1 man or 2 women or 3 boys can do a work in 44 days. Then, in how many days will 1 man, 1 woman and 1 boy do the work?

Options

A. 12 days
B. 24 days
C. 18 days
D. 36 days

Answer

To solve the problem step by step, we need to first establish the work rates of one man, one woman, and one boy based on the information given.

### Step 1: Establish Work Rates
From the problem, we know:
- 1 man can do the work in 44 days.
- 2 women can do the work in 44 days.
- 3 boys can do the work in 44 days.

Let's denote the work done by:
- 1 man in one day = \( M \)
- 1 woman in one day = \( W \)
- 1 boy in one day = \( B \)

Since 1 man can complete the work in 44 days, the work done by 1 man in one day is:
\[ M = \frac{1}{44} \]

Since 2 women can complete the work in 44 days, the work done by 1 woman in one day is:
\[ 2W = \frac{1}{44} \]
\[ W = \frac{1}{88} \]

Since 3 boys can complete the work in 44 days, the work done by 1 boy in one day is:
\[ 3B = \frac{1}{44} \]
\[ B = \frac{1}{132} \]

### Step 2: Combine Work Rates
Now, we need to find out how much work is done by 1 man, 1 woman, and 1 boy in one day:
\[ \text{Total work done in one day} = M + W + B \]
Substituting the values we found:
\[ M + W + B = \frac{1}{44} + \frac{1}{88} + \frac{1}{132} \]

### Step 3: Find a Common Denominator
To add these fractions, we need a common denominator. The least common multiple of 44, 88, and 132 is 264. Now we convert each fraction:
- \( \frac{1}{44} = \frac{6}{264} \)
- \( \frac{1}{88} = \frac{3}{264} \)
- \( \frac{1}{132} = \frac{2}{264} \)

Now we can add them:
\[ M + W + B = \frac{6}{264} + \frac{3}{264} + \frac{2}{264} = \frac{11}{264} \]

### Step 4: Calculate Total Days
Now that we know the total work done in one day by 1 man, 1 woman, and 1 boy is \( \frac{11}{264} \), we can find out how many days it will take them to complete the work:
Let \( D \) be the number of days required to complete the work:
\[ D \cdot \frac{11}{264} = 1 \]
\[ D = \frac{264}{11} = 24 \]

### Final Answer
Thus, 1 man, 1 woman, and 1 boy will complete the work in **24 days**.

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