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?
Answer & explanation
Answer: (a) 35
On a circular track, runners meet once for every full round of difference (same direction) or every full round of combined distance (opposite directions). That gives 2 meetings in the first phase and 34 in the second, but the last one happens exactly at the stop, so 35 are counted.
- Speeds are in the ratio 7 : 5. Phase 1 (same direction): X runs 7 rounds, Y runs 5, so X gains 2 rounds → 2 meetings (the second at the start point when Y turns).
- Phase 2: X runs 21 − 7 = 14 more rounds. In the same time Y, at double speed (10 per 7 of X), runs 20 rounds the other way.
- Opposite directions: they meet once per round of combined distance: 14 + 20 = 34 meetings.
- The 34th meeting is exactly when they stop (X at 21 rounds, Y at 25 — both at the start point), which is not 'before they finally stopped'.
- Meetings counted = 2 + 34 − 1 = 35.
Remember · Circular track meetings = relative rounds covered (difference if same direction, sum if opposite). Check whether the final meeting falls at the stop.
Question and answer: UPSC's provisional GS Paper II (2026, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·