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1. 12 men and 18 boys, working $7 \frac{1}{2}$ hours a day, can do a piece of work in 60 days. If a man works equal to 2 boys, then how many boys will be required to help 21 men to do twice the work in 50 days, working 9 hours a day ?

A. 30

B. 42

C. 48

D. 90

Solution:

1 man ≡ 2 boys ⇔ (12 men + 18 boys)
≡ (12 × 2 ×18) boys = 42 boys
Let required number of boys = x
⇒ (21 men + x boys) ≡ (21 × 2 × x) boys = (42 + x) boys
Less days, More boys (Indirect proportion)
More hours per day, Less boys (Indirect proportion)
More work, More boys (Direct proportion)

\begin{matrix} Days 50 : 60 \\ Hours per day 9 : \frac{15}{2} \\ Work 1 : 2 \end{matrix}} :: 42 : (42 + x)

∴ [50 \times 9 \times 1 \times (42 + x)] =

(60 \times \frac{15}{2} \times 2 \times 42)

\begin{aligned} \Leftrightarrow (42 + x) = \frac{37800}{450} \\ \Leftrightarrow 42 + x = 84 \\ \Leftrightarrow x = 42 \end{aligned}

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