150 workers were engaged to finish a job in a certain number of days.
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.

Solution:

Let x be the number of days in which 150 workers finish the work.

According to the given information,

$150 x=150+146+142+\ldots(x+8)$ terms

The series $150+146+142+\ldots .(x+8)$ terms is an A.P. with first term 150, common difference $-4$ and number of terms as $(x+8)$

$\Rightarrow 150 x=\frac{(x+8)}{2}[2(150)+(x+8-1)(-4)]$

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

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

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

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

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

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

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

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

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

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

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

However, x cannot be negative.

$\therefore x=17$

Therefore, originally, the number of days in which the work was completed is 17.

Thus, required number of days = (17 + 8) = 25

 

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