150 workers were engaged to finish a piece of work in a certain number of days. Four workers dropped on the second day, four more workers dropped on the third day and so on. It takes 8 more days to finish the work now. Find the number of days in which the work was completed. 

Solution:

Let X be the amount of days taken by 150 men to complete the work.

Number of days taken by 1 man to complete the work = 150X

Work done by 1 man in 1 day = 1/150X

On 1st day the number of workers = 150

On 2nd day the number of workers = 146

On nth day the number of workers = 150 – 4(n – 1) = 154 – 4n

According to the statement we know in this way the time taken to finish the work = (X + 8) days

1/150X × å(154 – 4n) = 1 (the summation is from 1 to (X + 8) days)

\(\frac{1}{{150X}} \times \left( {154 \times \left( {X + 8} \right)-4 \times \left( {X + 8} \right) \times \frac{{X\; + \;9}}{2}} \right) = 1\)

154X + 1232 – 2X2 – 34X – 144 = 150X

X2 + 15X – 544 = 0

(X – 17) (X + 32) = 0

Therefore X = 17, -32

since the number of days cannot be negative therefore the X = 17 days