Let's break this problem down step-by-step, considering all the given information and constraints.
### Step 1: Calculating Total Work Done in Man-Hours
First, we need to determine the total amount of work required to complete the original job, expressed in man-hours, woman-hours, and boy-hours.
For men:
- 20 men work for 12 days, 8 hours each day.
- Total work by 20 men = [tex]\(20 \times 12 \times 8 = 1920\)[/tex] man-hours.
For women:
- 24 women work for 12 days, 8 hours each day.
- Total work by 24 women = [tex]\(24 \times 12 \times 8 = 2304\)[/tex] woman-hours.
For boys:
- 40 boys work for 12 days, 8 hours each day.
- Total work by 40 boys = [tex]\(40 \times 12 \times 8 = 3840\)[/tex] boy-hours.
### Step 2: Work Done by One Man, One Woman, and One Boy
Now, we calculate the work done by one individual (man, woman, boy) in the given time:
- Work done by one man in 12 days = [tex]\(1920 \, \text{man-hours} / 20 \, \text{men} = 96 \, \text{man-hours}\)[/tex].
- Work done by one woman in 12 days = [tex]\(2304 \, \text{woman-hours} / 24 \, \text{women} = 96 \, \text{woman-hours}\)[/tex].
- Work done by one boy in 12 days = [tex]\(3840 \, \text{boy-hours} / 40 \, \text{boys} = 96 \, \text{boy-hours}\)[/tex].
### Step 3: Total Work for the New (Four Times Bigger) Job
The new job is four times as big as the original job, so:
- Total work for new job = [tex]\(4 \times 1920 = 7680\)[/tex] man-hours.
### Step 4: Work Contribution by Women and Boys in the New Scenario
Given that 6 women and 2 boys are part of the new workforce, we calculate their contributions:
- Work done by 6 women in new scenario = [tex]\(6 \times 96 = 576\)[/tex] man-hours.
- Work done by 2 boys in new scenario = [tex]\(2 \times 96 = 192\)[/tex] man-hours.
Total work done by women and boys = [tex]\(576 + 192 = 768\)[/tex] man-hours.
### Step 5: Remaining Work to be Done by Men
Next, we calculate the remaining work that needs to be done by men:
- Remaining work = Total work for new job - Work done by women and boys
- Remaining work = [tex]\(7680 - 768 = 6912\)[/tex] man-hours.
### Step 6: Number of Men Required
Finally, we need to determine the number of men required to complete the remaining work in the given timeframe:
- They have 12 days, working 5 hours each day.
- Total man-hours available per man = [tex]\(12 \times 5 = 60\)[/tex] hours.
So, the number of men required [tex]\(= \frac{\text{Remaining work}}{\text{Total man-hours per man}}\)[/tex].
- Number of men required = [tex]\(\frac{6912}{60} = 1.2\)[/tex].
### Conclusion
Hence, 1.2 men (which implies 2 men, realistically, rounding to the nearest whole number) working alongside 6 women and 2 boys will be required to complete the job that is four times as big, working for 5 hours per day for 12 days.
