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The work done by a woman in 8 hours is equal to the work done by a man in 6 hours and the work done by a boy in 12 hours. If working 6 hours per day 9 men can complete a work in 6 days, then in how many days can 12 men, 12 women and 12 boys together finish the same work working 8 hours per day??
(A) 1.5
(B) 3
(C) 3 2/3
(D) 4.5
(E) 5
(A) 1.5
(B) 3
(C) 3 2/3
(D) 4.5
(E) 5
\(\text{Work}_{done} = 8 \times W=6 \times M=12 \times B\), where \(W, M, B \implies \) Women, Men and Boy.
or, \(4\cdot W=3 \cdot M=6\cdot B \longrightarrow (1)\) (Taking GCD from the above equation)
Let's rewrite the question as i.e., \((S_1)\),
$$ \text{In how many days can } 12 \text{ men, } 12 \text{ women and } 12 \text{ boys together finish the same work working } 8 \text{ hours per day} $$
Firstly, consider the first half of the question:
\(12\cdot M+12\cdot W+12\cdot B = 12 \cdot M + 12 \cdot \dfrac{3}{4}\cdot M + 12 \cdot \dfrac{3}{6} \cdot M = 27 \cdot M \)
whereas from equation (1), we have, \(4 \cdot W = 3 \cdot M \implies W = \dfrac{3}{4} \cdot M \)
similarly, \(B = \dfrac{3}{6} \cdot M\)
Let \(x\) be the number of days to complete the work working \(8 \) hours a day, and it was given that i.e., \((S_2)\),
$$ \text{if working } 6 \text{ hours per day } 9 \text{ men can complete a work in }6\text{ days} $$, then we can rewite this and the above statement as;
\(27\times 8\times x=9\times 6\times 6\) (same work is completed by different set of groups \(S_1 \in W_{done}, \ S_2 \in W_{done}\))
\(x=\frac{3}{2}\)
Ans A
Note: I have updated the explanation with better comments to understand the solution.
or, \(4\cdot W=3 \cdot M=6\cdot B \longrightarrow (1)\) (Taking GCD from the above equation)
Let's rewrite the question as i.e., \((S_1)\),
$$ \text{In how many days can } 12 \text{ men, } 12 \text{ women and } 12 \text{ boys together finish the same work working } 8 \text{ hours per day} $$
Firstly, consider the first half of the question:
\(12\cdot M+12\cdot W+12\cdot B = 12 \cdot M + 12 \cdot \dfrac{3}{4}\cdot M + 12 \cdot \dfrac{3}{6} \cdot M = 27 \cdot M \)
whereas from equation (1), we have, \(4 \cdot W = 3 \cdot M \implies W = \dfrac{3}{4} \cdot M \)
similarly, \(B = \dfrac{3}{6} \cdot M\)
Let \(x\) be the number of days to complete the work working \(8 \) hours a day, and it was given that i.e., \((S_2)\),
$$ \text{if working } 6 \text{ hours per day } 9 \text{ men can complete a work in }6\text{ days} $$, then we can rewite this and the above statement as;
\(27\times 8\times x=9\times 6\times 6\) (same work is completed by different set of groups \(S_1 \in W_{done}, \ S_2 \in W_{done}\))
\(x=\frac{3}{2}\)
Ans A
Note: I have updated the explanation with better comments to understand the solution.
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