A group of men decided to do a job in 8 days. But since 10 men dropped out…
2025
A group of men decided to do a job in 8 days. But since 10 men dropped out every day, the job got completed at the end of the 12th day. How many men were there at the beginning?
- A.
150
- B.
165
- C.
175
- D.
180
Attempted by 3 students.
Show answer & explanation
Correct answer: B
Concept: The total work for a job, measured in man-days, is fixed. When the workforce does not stay constant, total work equals the sum of however many men actually work each day. If that number falls by a fixed amount every day, the daily workforce forms an arithmetic progression, and the total work is the sum of that progression.
Application:
Let x be the number of men at the start.
Had the job been done at full strength, it would have needed x men for 8 days, so the total work is 8x man-days.
Since 10 men leave every day, the crew is x on day 1, x-10 on day 2, and so on down to x-110 on day 12 — an arithmetic progression of 12 terms with common difference -10.
The sum of this progression is 12x − 10(0+1+2+...+11) = 12x − 10(66) = 12x − 660 man-days.
Both expressions total the same job, so 8x = 12x − 660.
Solving gives 660 = 4x, so x = 165.
Cross-check: With x = 165, the crew runs from 165 men on day 1 down to 55 men on day 12; the sum of this 12-term progression is (12/2) × (165 + 55) = 1320 man-days, which exactly equals 8 × 165 = 1320 — confirming the value.
