A contract is to be finished in 46 days and 117 men involved in it, each working 8 hours per day. After 33 days, 4/7 of the work is finished, how many additional men may be employed so that it may be completed in time, each man now working 9 hours a day?

Question: A contract is to be finished in 46 days and 117 men involved in it, each working 8 hours per day. After 33 days, 4/7 of the work is finished, how many additional men may be employed so that it may be completed in time, each man now working 9 hours a day?

Solution:
Remaining work after 33 days = 1 - 4/7 = 3/7
Remaining period = 46 - 33 = 13 days
Now, we have
Less work, less man (directly proportion)
Less days, more men (inverse proportion)
More hours/days, less man (inverse proportion)
Now,
we can say that work = 4/7 : 3/7
Therefore, days = 13 : 33
And, hours/day = 9 : 8

(4/7) × 13 × 9 : (3/7) × 33 × 8 = 117 : x [Where, x is total number of men after 33 days.]
⇒ (4/7) × 13 × 9 × x = (3/7) × 33 × 8 × 117
⇒ x = (3 × 33 × 8 × 117)/(4 × 13 × 9)
∴ x = 198

Therefore, extra men to be employed = 198 - 117 = 81