Question

$150$ workers were engaged to finish a job in a certain number of days.$4$ workers dropped out on second day, $4$more workers dropped out on third day and so on. It took $8$ more days to finish the work. Find the number of days in which the work was completed.

Answer 1

Let $x$ be the number of days in which 150 workers finish the work.
According to the given information,
$150x=150+146+142+...(x+8)terms$
The series $150+146+142+...+(x+8)$ terms is an A.P. with first term 150, common difference -4 and number of terms as $(x+8)$
$\Rightarrow 150x=\frac{x+8}{2}\left[2\right(150)+(x+8-1\left)\right(-4\left)\right]$

$\Rightarrow 150x=(x+8)\left[\right(150)+(x+7\left)\right(-2\left)\right]\Rightarrow 150x=(x+8)(150-2x-14)$

$\Rightarrow 150x=(x+8)(136-2x)$

$\Rightarrow 75x=(x+8)(68-x)$

$\Rightarrow 75x=68x-x^2+544-8x$

$\Rightarrow x^2+75x-60x-544=0$

$\Rightarrow x^2+15x-544=0$

$\Rightarrow x^2+32x-17x-544=0$

$\Rightarrow x+(x+32)-17(x+32)=0$

$\Rightarrow (x-17)(x+32)=0$

$\Rightarrow x=17$ or $x=-32$

Since, $x$ cannot be negative . So, $x=17$

So, the number of days in which the work was to be completed by 150 workers is 17.
So, required number of days $=(17+8)=25$