If 20 men or 24 women or 40 boys can do a job in 12 days working for 8 hours a day, how many men working with 6 women and 2 boys take to do a job four times as big working for 5 hours a day for 12 days?

Options

A. 8 men
B. 12 men
C. 2 men
D. 24 men

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the total work done in terms of man-hours
Given that:
- 20 men can complete the job in 12 days working 8 hours a day.

The total work done (in man-hours) can be calculated as:
\[ \text{Total Work} = \text{Number of Men} \times \text{Number of Days} \times \text{Hours per Day} \]
\[ \text{Total Work} = 20 \text{ men} \times 12 \text{ days} \times 8 \text{ hours/day} = 1920 \text{ man-hours} \]

### Step 2: Calculate the total work for the new job
The new job is four times as big, so:
\[ \text{Total Work for New Job} = 4 \times 1920 \text{ man-hours} = 7680 \text{ man-hours} \]

### Step 3: Determine the work done by the group of workers
We need to find out how much work can be done by the group of men, women, and boys in the new scenario.

Let’s first convert the workers into equivalent man-hours:
- 6 women can be converted to men using the ratio we found earlier: 1 man = 1.2 women.
\[ 6 \text{ women} = \frac{6}{1.2} \text{ men} = 5 \text{ men} \]

- 2 boys can also be converted to men using the ratio: 1 man = 2 boys.
\[ 2 \text{ boys} = \frac{2}{2} \text{ men} = 1 \text{ man} \]

So, the total number of men equivalent in the group is:
\[ \text{Total equivalent men} = \text{Men} + \text{Equivalent Women} + \text{Equivalent Boys} = x + 5 + 1 = x + 6 \]
where \( x \) is the number of men we need to find.

### Step 4: Calculate the total work done by the group in the given time
The group works for 12 days at 5 hours a day, so the total working hours is:
\[ \text{Total Working Hours} = 12 \text{ days} \times 5 \text{ hours/day} = 60 \text{ hours} \]

The total work done by the group in man-hours is:
\[ \text{Total Work Done} = (x + 6) \text{ men} \times 60 \text{ hours} \]

### Step 5: Set up the equation
We know that the total work done must equal the total work required for the new job:
\[ (x + 6) \times 60 = 7680 \]

### Step 6: Solve for \( x \)
Dividing both sides by 60:
\[ x + 6 = \frac{7680}{60} = 128 \]
\[ x = 128 - 6 = 122 \]

### Conclusion
The number of men required is 122.

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