Let \(M\) be the amount of work one man completes in one day, and \(W\) be the amount of work one woman completes in one day. The total work done is the rate multiplied by the time. Since the work is finished in one day in both given scenarios, the total work equals the combined rate of men and women.
From the first statement:
\(8M + 24W = 1 \quad (Equation \, 1)\)
From the second statement:
\(12M + 18W = 1 \quad (Equation \, 2)\)
We can simplify the equations before solving. Divide Equation 1 by 8:
\(M + 3W = \frac{1}{8} \quad (Equation \, 3)\)
Divide Equation 2 by 6:
\(2M + 3W = \frac{1}{6} \quad (Equation \, 4)\)
Now, subtract Equation 3 from Equation 4 to find the value of \(M\):
\((2M + 3W) - (M + 3W) = \frac{1}{6} - \frac{1}{8}\)
\(M = \frac{4}{24} - \frac{3}{24}\)
\(M = \frac{1}{24}\)
Substitute the value of \(M\) back into Equation 3 to find the value of \(W\):
\(\frac{1}{24} + 3W = \frac{1}{8}\)
\(3W = \frac{1}{8} - \frac{1}{24}\)
\(3W = \frac{3}{24} - \frac{1}{24}\)
\(3W = \frac{2}{24} = \frac{1}{12}\)
\(W = \frac{1}{36}\)
So, one man's work rate is \(\frac{1}{24}\) of the work per day, and one woman's work rate is \(\frac{1}{36}\) of the work per day.
We need to find the time taken by 24 men and 72 women. Calculate their combined work rate:
Total Rate = \(24M + 72W\)
\(\text{Total Rate} = 24 \times \left(\frac{1}{24}\right) + 72 \times \left(\frac{1}{36}\right)\)
\(\text{Total Rate} = 1 + 2\)
\(\text{Total Rate} = 3\)
This means 24 men and 72 women together complete 3 units of work per day.
To find the time taken to finish 1 unit of work:
\(\text{Time} = \frac{\text{Total Work}}{\text{Total Rate}}\)
\(\text{Time} = \frac{1}{3} \, \text{day}\)
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