This problem involves calculating the number of men required for a task based on given work rates (men, days, hours) and a change in the amount of work.
Men and Work Calculation
The total amount of work done is directly proportional to the number of men, the number of days they work, and the number of hours they work per day. This relationship can be expressed using the formula:
$ \frac{W_1}{M_1 \times D_1 \times H_1} = \frac{W_2}{M_2 \times D_2 \times H_2} $Where:
- $ W $ = Amount of Work
- $ M $ = Number of Men
- $ D $ = Number of Days
- $ H $ = Hours worked per day
- Subscript 1 refers to the initial scenario, and subscript 2 refers to the target scenario.
Initial Work Scenario
We are given the details for the first piece of work:
- Number of men ($ M_1 $): 16
- Number of days ($ D_1 $): 8
- Hours per day ($ H_1 $): 12
- Amount of work ($ W_1 $): Let's denote this as $ W_1 $.
Target Work Scenario
For the second, larger piece of work, we have:
- Amount of work ($ W_2 $): Three times the first work, so $ W_2 = 3 \times W_1 $.
- Number of days ($ D_2 $): 24
- Hours per day ($ H_2 $): 8
- Number of men ($ M_2 $): This is what we need to find.
Men Calculation for Target Work
Now, we substitute the known values into the formula:
$ \frac{W_1}{16 \times 8 \times 12} = \frac{3 \times W_1}{M_2 \times 24 \times 8} $We can cancel out $ W_1 $ from both sides of the equation since it's non-zero:
$ \frac{1}{16 \times 8 \times 12} = \frac{3}{M_2 \times 24 \times 8} $To solve for $ M_2 $, we rearrange the equation:
$ M_2 = \frac{3 \times (16 \times 8 \times 12)}{(24 \times 8)} $Now, simplify the calculation:
- Cancel out the 8 from the numerator and denominator: $ M_2 = \frac{3 \times 16 \times 12}{24} $
- Simplify $ \frac{12}{24} $ to $ \frac{1}{2} $: $ M_2 = 3 \times 16 \times \frac{1}{2} $
- Calculate the result: $ M_2 = 3 \times 8 $ $ M_2 = 24 $
Therefore, 24 men are needed to complete the second work.
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