X and Y run a 3 km race along a circular course of length 300 m. Their speeds are in the ratio 3 : 2. If they start together in the same direction, how many times would the first one pass the other (the start-off is not counted as passing)?
- (a) 2
- (b) 3 ✓ UPSC's answer
- (c) 4
- (d) 5
Why the answer is (b)
• The total distance of the race is 3 km, which equals 3000 meters.
• The length of the circular course is 300 meters.
• The number of laps required to complete the race is 3000 m / 300 m = 10 laps.
• Let the speed of X be 3v and the speed of Y be 2v. The relative speed is 3v - 2v = v.
• X passes Y every time X gains one full lap (300 m) on Y. The time to gain one lap is Distance / Relative Speed = 300 / v.
• The total time for the race is Total Distance / Speed of X = 3000 / 3v = 1000 / v.
• The number of times X passes Y is Total Time / Time to gain one lap = (1000 / v) / (300 / v) = 1000 / 300 = 3.33.
• Since passing occurs in discrete events, X passes Y 3 times during the race (at the 300m, 600m, and 900m marks of relative gain, corresponding to specific points in the race).
Why the other options are wrong
- (a) 2
- Option (a) is incorrect because the calculation yields 3.33 passes, which truncates to 3, not 2.
- (c) 4
- Option (c) is incorrect because X does not complete the 4th lap gain (which would require 1200m relative distance) before the race ends at 3000m.
- (d) 5
- Option (d) is incorrect because the maximum number of passes is limited by the total race distance and relative speed, resulting in only 3 full passes.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.