X and Y are two runners who run for the same duration of time on the same circular track. They started running at the same time in the same direction with uniform speeds. When X completed 7 rounds, Y did exactly 5. After completing 5 rounds, Y changed his direction and started running in the opposite direction with speed which is double of his earlier speed. On the other hand, X continued to run with the same speed. They stopped running when X completed exactly 21 rounds. How many times did X and Y meet after they had started and before they finally stopped?
- (a) 35 ✓ UPSC's answer
- (b) 34
- (c) 31
- (d) 29
Why the answer is (a)
• Take the time X needs for 7 rounds as 1 unit: X runs 7 rounds and Y 5 rounds per unit.
• Phase 1, same direction, 1 unit: X gains 7 − 5 = 2 rounds, so they meet 2 times (the second at the starting point, just as Y finishes 5 rounds).
• Phase 2: Y turns back at 10 rounds per unit. X needs 14 more rounds to reach 21, i.e. 2 units. Running in opposite directions they close 7 + 10 = 17 rounds per unit, so they meet 34 times in 2 units, the 34th meeting coming exactly at the moment they stop.
• Meetings before they stop: 2 + 34 − 1 = 35: (a).
Why the other options are wrong
- (b) 34
- 34 omits one meeting from the same-direction phase.
- (c) 31
- 31 undercounts the 17-per-unit meetings of the second phase.
- (d) 29
- 29 undercounts both phases.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2026, held on 24 May 2026. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.