A man undertakes to do a certain work in 150 days. He employs 200 men. He finds that only a quarter of the work is done in 50 days. The number of additional men that should be appointed so that the whole work will be finished in time is :

lcm of 150,200,50 = 600

lets assume total work to be 600 units

work completed in 50 days = 600*$$\frac{1}{4}$$ = 150 units

work left to be completed in 100 days = 450 units

actual work completed by per man per day = $$\frac{150}{50*200}$$

100 days of work per man = $$\frac{150*100}{50*200}$$ = 1.5 units

total men needed to complete the leftover work = $$\frac{450}{1.5}$$ = 300 men

so additional men needed = 300-200 = 100 men