150 workers were engaged to finish a piece of work in a certain number of days. Four workers dropped the second day, four more workers dropped the third day, and so on. It takes 8 more days to finish work now. Find the number of days in which the work was completed.
Given:
Initially let the work can be completed in n days when 150 workers work on every day.
However every day 4 workers are being dropped from the work so that work took 8 more days to be finished.
Finally, it takes $(n+8)$ days to finish the works.
Work equivalent when 150 workers work without being dropped $=150 \times n$
Work equivalent when workers are dropped day by day $=150+(150-4)+(150-8)+$ $\ldots \ldots+(150-4(n+8))$
So,
$150 \times n=150+(150-4)+\ldots \ldots . .+(150-4 \times(n+8))$
$150 \times n=150 \times n+150 \times 8-4 \times(1+2+3+\ldots \ldots+(n+8))$
$(n+8)(n+9)=600$
$n^{2}+17 n-528=0$
$n=-33$ or $n=16$
Since the number of days cannot be negative, $n=16$.
$\therefore$ In 24 days the work is completed.