Progressions-13 (NCERT)

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?

(1)17   (2) 20   (3) 25   (4) 27   (5) 30

Solution follows here:

Solution:

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 = S_n+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

Answer (3)