12 men and 18 boys working `7(1)/(2)` hours a day can do a work in 60 days. If one man works equal to 2 boys, then the number of boys required to help 21 men to do twice the work in 50 days working 9 hours a day will be:

Options

A. 42
B. 44
C. 46
D. None of these

Answer

To solve the problem step by step, we will use the information provided to find the number of boys required to help 21 men do twice the work in 50 days while working 9 hours a day.

### Step 1: Determine the total work done by 12 men and 18 boys.
Given:
- 12 men and 18 boys can complete a work in 60 days working 7.5 hours a day.

First, we convert 7.5 hours into a fraction:
\[ 7.5 \text{ hours} = \frac{15}{2} \text{ hours} \]

Now, we calculate the total work done (in man-hours) by 12 men and 18 boys:
\[ \text{Total work} = \text{Number of workers} \times \text{Days} \times \text{Hours per day} \]
Where:
- Number of workers = 12 men + 18 boys
- Days = 60
- Hours per day = 7.5 = \(\frac{15}{2}\)

Since 1 man is equivalent to 2 boys, we can convert the boys to men:
\[ 12 \text{ men} + 18 \text{ boys} = 12 + \frac{18}{2} = 12 + 9 = 21 \text{ men equivalent} \]

Now, calculate the total work:
\[ \text{Total work} = 21 \text{ men} \times 60 \text{ days} \times \frac{15}{2} \text{ hours} \]
\[ = 21 \times 60 \times \frac{15}{2} = 21 \times 30 \times 15 = 9450 \text{ man-hours} \]

### Step 2: Calculate the work required for 21 men to do twice the work.
Since we need to do twice the work:
\[ \text{Total work required} = 2 \times 9450 = 18900 \text{ man-hours} \]

### Step 3: Determine the number of boys needed to help 21 men complete the work in 50 days working 9 hours a day.
Given:
- 21 men working for 50 days and 9 hours a day.

Calculate the total work done by 21 men:
\[ \text{Work done by 21 men} = 21 \text{ men} \times 50 \text{ days} \times 9 \text{ hours} \]
\[ = 21 \times 50 \times 9 = 9450 \text{ man-hours} \]

### Step 4: Set up the equation to find the number of boys needed.
Let \( n \) be the number of boys required. The work done by \( n \) boys in the same time is:
\[ \text{Work done by } n \text{ boys} = n \text{ boys} \times 50 \text{ days} \times 9 \text{ hours} \]
\[ = n \times 50 \times 9 = 450n \text{ boy-hours} \]

### Step 5: Combine the work done by men and boys and set equal to total work required.
The total work done by 21 men and \( n \) boys must equal the total work required:
\[ 9450 + 450n = 18900 \]

### Step 6: Solve for \( n \).
\[ 450n = 18900 - 9450 \]
\[ 450n = 9450 \]
\[ n = \frac{9450}{450} = 21 \]

### Conclusion
The number of boys required to help 21 men to do twice the work in 50 days working 9 hours a day is **21 boys**.

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