Question

$150$ workers were engaged to finish a piece of work in a certain number of days. $4$ workers dropped the second day, $4$ more workers dropped the third day and so on. It takes eight more days to finish the work now. The number of days in which the work was completed is

• A. $15$
• B. $20$
• C. $25$
• D. $30$

Answer 1

The correct option is C

$25$

Explanation for the correct option:

Step 1. Let the number of days be $x$ in which $150$ workers finish the work:

According to the given conditions of question, we get

$150x=150+146+142+...(x+8)terms$

The terms $150+146+142+...+(x+8)$ are in an A.P.

Where, $a=150$, $d=-4$and $t_n=(x+8)$

Step 2. Find the sum of $\left(x+8\right)$ terms of given series:
$150x=\frac{x+8}{2}\left[2\right(150)+(x+8-1\left)\right(-4\left)\right]$ $\left[\mathbf{\because }{\mathit{S}}_{\mathbf{n}}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{n}}{\mathbf{2}}\left[2a+\left(n-1\right)d\right]\right]$

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

$\Rightarrow$ $150x=(x+8)(150-2x-14)$

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

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

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

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

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

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

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

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

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

$\because x$ cannot be negative.

So, $x=17$

$\Rightarrow$the number of days required by 150 workers to complete the work is $17$.

$\therefore$required number of days $=(17+8)$

$=\mathbf{25}$

Hence, Option ‘C’ is Correct.