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, $$\frac{4}{7}{\text{ }}$$ 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 ?

Explanation:-

Remaining work
= $$\left( {1 - \frac{4}{7}} \right){\text{ = }}\frac{3}{7}$$
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)
$$\left. \begin{gathered} \,\,\,\,\,{\text{Work }}\frac{4}{7}:\frac{3}{7} \hfill \\ \,\,\,\,\,\,\,{\text{Men 13}}:33 \hfill \\ {\text{Hour/day }}9:8 \hfill \\ \end{gathered} \right\}::117:x$$
$$\eqalign{ & \therefore {\text{ }}\frac{4}{7} \times 13 \times 9 \times x = \frac{3}{7} \times 33 \times 8 \times 117 \cr & \Leftrightarrow x = \frac{{\left( {3 \times 33 \times 8 \times 117} \right)}}{{\left( {4 \times 13 \times 9} \right)}} \cr & \Leftrightarrow x = 198 \cr}$$
∴ Addition men to be employed = (198 - 177) = 81