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(A) Let $x$ be the number of days in which $150$ workers would finish the work.The total work is $150x$.According to the given information,the work is

Vedclass · Accepted Answer

$25$

Vedclass · Answer

(A) Let $x$ be the number of days in which $150$ workers would finish the work.The total work is $150x$.According to the given information,the work is completed in $(x+8)$ days with a decreasing number of workers:$150x = 150 + 146 + 142 + \dots$ up to $(x+8)$ terms.The series $150 + 146 + 142 + \dots$ is an $A.P.$ with first term $a = 150$,common difference $d = -4$,and number of terms $n = x+8$.The sum of $n$ terms of an $A.P.$ is $S_n = \frac{n}{2}[2a + (n-1)d]$.$150x = \frac{x+8}{2}[2(150) + (x+8-1)(-4)]$$150x = \frac{x+8}{2}[300 + (x+7)(-4)]$$150x = (x+8)[150 + (x+7)(-2)]$$150x = (x+8)(150 - 2x - 14)$$150x = (x+8)(136 - 2x)$$75x = (x+8)(68 - x)$$75x = 68x - x^2 + 544 - 8x$$x^2 + 75x - 60x - 544 = 0$$x^2 + 15x - 544 = 0$$(x + 32)(x - 17) = 0$Since $x > 0$,we have $x = 17$.The total number of days taken is $x + 8 = 17 + 8 = 25$ days.

$150$ workers were engaged to finish a job in a certain number of days. $4$ workers dropped out on the second day,$4$ more workers dropped out on the 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.

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