150 workers were engaged to finish a piece of work in a certain number of days. Four workers dropped on the second day, four more workers dropped on third day and so on. It takes 8 more days to finish the work no. Find the number of days in which the work was completed ?

Let 1 worker does 1 unit work in a day.

Let 150 workers can finish the work in $$(n-8)$$ days, if all workers work all the days.

Then, total work $= \$\$150(n-8)\$\$------------(i)$

Also, 150 workers work on day 1, 146 workers work on day 2, ... and so on. Work is completed in $$n$$ days. 

Thus, total work = $$150+146+....(n$$ terms$$)$$

This is an arithmetic progression with first term, $$a=150$$, $$d=-4$$. 

Thus, total work = $$\frac{n}{2}[2a+(n-1)d]$$

= $$\frac{n}{2}[2(150)+(n-1)(-4)]$$

= $$\frac{n}{2}[300-4n+4]$$

= $$\frac{n}{2}[304-4n]=n(152-2n)$$ -------------(ii)

Comparing equations (i) and (ii), 

=> $$150(n-8)=n(152-2n)$$

=> $$75(n-8)=n(76-n)$$

=> $$75n-600=76n-n^2$$

=> $$n^2-n-600=0$$

=> $$(n-25)(n+24)=0$$

=> $$n=25,-24$$

$$\because n$$ cannot be negative, => $$n=25$$

=> Number of days in which the work was completed = 25

=> Ans - (D)