Question

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.

Answer 1

Let x be the number of days in which 150 workers finish the work.
According to the given information,
150x = 150 + 146 + 142 + …. (x + 8) terms
The series 150 + 146 + 142 + …. (x + 8) terms is an A.P. with first term 146, common difference –4 and number of terms as (x + 8)
\(⇒ 150x = \frac{(x + 8) }{ 2} [2(150) + (x + 8 - 1)(- 4)]\)
\(⇒ 150x = (x + 8) [150 + (x + 7)(-2)]\)
\(⇒ 150x = (x + 8) (150 - 2x - 14)\)
\(⇒ 150x = (x + 8) (136 - 2x)\)
\(⇒ 75x = (x + 8) (68 - x)\)
\(⇒ 75x = 68x - x^2 + 544 - 8x\)
\(⇒ x^2 + 75x - 60x - 544 = 0\)
\(⇒ x^2 + 15x - 544 = 0\)
\(⇒ x^2 + 32x - 7x -544 = 0\)
\(⇒ x (x + 32) - 17 (x + 32) = 0\)
\(⇒ (x - 17)(x + 32) = 0\)
\(⇒ x = 17 or x = -32\)

However, x cannot be negative.
\(∴x = 17\)
Therefore, originally, the number of days in which the work was completed is 17.
Thus, required number of days = (17 + 8) = 25