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1. A contract is to be complete in 46 days and 117 men were set to work, each working 8 hours a day. After 33 days, $\frac{4}{7}$ of the work is completed. How many additional men may be employed so that the work may be completed in time, each man now working 9 hours a day ?

A. 80

B. 81

C. 82

D. 83

Solution:

Remaining work
=

(1 - \frac{4}{7}) = \frac{3}{7}

Remaining period = (46 - 33) = 13 days
Let the total men working at it be x
Less work, Less men (Direct proportion)
Less days, More men (Indirect proportion)
More hour/day, Less men (Indirect proportion)

\begin{matrix} Work \frac{4}{7} : \frac{3}{7} \\ Men 13 : 33 \\ Hour/day 9 : 8 \end{matrix}} :: 117 : x

\begin{aligned} ∴ \frac{4}{7} \times 13 \times 9 \times x = \frac{3}{7} \times 33 \times 8 \times 117 \\ \Leftrightarrow x = \frac{(3 \times 33 \times 8 \times 117)}{(4 \times 13 \times 9)} \\ \Leftrightarrow x = 198 \end{aligned}

∴ Addition men to be employed = (198 - 117) = 81

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