Question

150 workers were engaged to finish a piece of work in a certain number of days. Four workers dropped the second day, four workers dropped the third day and so on. It takes 8 more days to finish the work now. Find the number of days in which the work was completed.

Answer 1

Hint: Here the drop in the number of workers remains constant everyday, so the concept of arithmetic progression is used to find the number of days in which the work can be completed.

Complete step-by-step answer:
Let the work finish in n days when the workers stopped dropping so that the total number of workers who worked all these days is the sum of A.P.
$\Rightarrow 150 + 146 + 142 + .......n$
We know the formula of finding the sum of first n terms
$\Rightarrow \dfrac{n}{2}[2a + (n - 1) \times d] \\\ \Rightarrow \dfrac{n}{2}[2(150) + (n - 1)( - 4)] \\\ \Rightarrow n(152 - 2n) \to (1) \\\$
If the workers had not dropped, then the work would have finished in ‘n-8’ days with 150 workers on each day i.e.., $150(n - 8) \to (2)$
Now
$n(152 - 2n) = 150(n - 8) \\\ \Rightarrow {n^2} - n - 600 = 0 \\\ \Rightarrow {n^2} - 25n + 24n - 600 = 0 \\\ \Rightarrow n(n - 25) + 24(n - 25) = 0 \\\ \Rightarrow (n - 25)(n + 24) = 0 \\\$
Number of days cannot be negative so we won’t consider n=-24.
$\Rightarrow (n - 25) = 0 \\\ \Rightarrow n = 25 \\\$
Answer is 25 days.
The work will be complete in 25 days.

Note: If the specific number of workers dropped continuously then there would be a series. Find the series and apply the sum of first n terms in A.P. series after that follow some simple logic to get the number of days in which the work will be completed.