# 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.

Let total work = 1

and let total work be completed in 'n' days

Work done in 1 day = 1/n

This is the work done by 150 workers

Work done by 1 worker in one day = 1/150n

Case 1: -

No. of workers = 150

Work done per worker in 1 day = 1/150n

Total work done in 1 day = 150/150n

Case 2: -

No. of workers = 146

Work done per worker in 1 day = 1/150n

Total work done in 1 day = 146/150n

Case 3: -

No. of workers = 142

Work done per worker in 1 day = 1/150n

Total work done in 1 day = 142/150n

Given that

In this manner it took 8 more days to finish the work i.e. work finished in (n + 8) days.   …(1)

Now, is an AP

where,

first term(a) = 150

common difference(d) = 146 - 150 = - 4

we know that

Sum of n terms of AP(Sn) = (n/2)[2a + (n - 1)d]

putting n = n + 8, a = 150 & d = - 4

Sn + 8 = [(n + 8)/2] × [2(150) + (n + 8 - 1)( - 4)]

= [(n + 8)/2] × [300 + (n + 7)( - 4)]

= [(n + 8)/2] × [300 - 4n - 28]

= [(n + 8)/2] × [272 - 4n]

= (n + 8) × (136 - 2n)

= - 2n2 + 120n + 1088

From (1),

Sn + 8 = 150n

- 2n2 + 120n + 1088 = 150n

- 2n2 - 30n + 1088 = 0

- 2(n2 + 15n - 544) = 0

(n2 + 15n - 544) = 0

n2 + 32n - 17n - 544 = 0

n(n + 32) - 17(n + 32) = 0

(n - 17)(n + 32) = 0

n = 17

because n = - 32 is invalid as no. of days can't be - ve.

Hence, n =17

Thus, the work was completed in n + 8 days i.e. 17 + 8 = 25 days

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