Problem

Let the number of days required to finish the job with all the 150 workers engaged all the days be ‘n’ => man-days required to complete the job = 150 n --(1) Number of workers on day-1 = 150 4 workers dropped out on second day => Number of workers on day-2 = 150-4 = 146 4 more workers dropped out on third day => Number of workers on day-3 = 146-4 = 142 In this scenario, it required to complete 8 more days => (n+8) days Arranging number of workers/ each day in an order, 150, 146, 142,…… (n+8) terms This is a decreasing A.P, with initial term = a = 150; and common difference = d = -4; Sum to (n+8) terms = Sn+8 = (n+8)/2 * (2a+(n+8-1)d) = (n+8)/2 * (300-4(n+8-1)) = (n+8)(136-2n) --(2) This is nothing but the man-days calculated in (1), => (n+8)(136-2n) = 150n => (n+8)(68-n) = 75n => 68n+544-n2-8n = 75n => n2+15n-544 = 0 => n2+32n-17n-(17 32) = 0 => n(n+32)-17(n+32)=0 => n = 17 (or) -32, we can leave out negative values as this represents number of days => n=17 Wait a minute, this is not the answer. We assumed ‘n’ to be number of days to finish the job when all 150 workers engaged for all the days. But in the actual scenario, it took (n+8) days due to gradual dropping out of workers on each day. Answer = n+8 = 17+8 = 25

We have an arithmetic progression with a first term of 150 and a common difference of -4.

Each day, we'll get a day's work from each worker who is still working. On day n, that will be:

150 - 4(n-1) = 154 - 4n workers. (This second version of the formula is equivalent and a bit more convenient for use below.)

The sum of these terms, over the first n days, is the amount of work (in worker-days) completed to that point. So in n days, the number of worker-days of work toward completion of the project is

154n - 4 (1 + 2 + 3 + ... + n) = 154n - 4n(n+1)/2

substituting the well-known formula for triangular numbers:

1 + 2 + 3 + ... + n = n (n+1) / 2

If the work is completed on the nth day, then this sum is equal to the work done on n-8 days if there had been 150 workers each day, so

154n - 4n(n+1)/2 = 150(n-8)

154n - 2n(n+1) = 150n - 1200

154n - 2n^2 - 2n = 150n - 1200

152n = 2n^2 + 150n - 1200

0 = 2n^2 - 2n - 1200

n^2 - n - 600 = 0 [dividing both sides by 2]

(n - 25) (n + 24) = 0

This has two solutions, n= -24 and n=25. A negative value for n makes no sense in the context of the problem, so the answer is that the work was completed in 25 days.