P and Q walk along a circular track. They start at 5:00 a.m. from the same point in opposite directions. P walks at an average speed of 5 rounds per hour and Q walks at an average speed of 3 rounds per hour. How many times will they cross each other between 5:20 a.m. and 7:00 a.m.?
Answer & explanation
Answer: (b) 13
Why not the tempting option · UPSC's key is 13. Counting the meeting exactly at 7:00 a.m. would give 14, but 'between 5:20 a.m. and 7:00 a.m.' excludes the end points, and 5:20 itself is not a meeting time. In the exam, read 'between' as excluding the end points, as UPSC does.
Walking in opposite directions, together they cover 8 rounds an hour, so they meet every 7.5 minutes. Between 5:20 and 7:00 they meet at 5:22:30, 5:30, … 6:52:30 — 13 times; the meeting exactly at 7:00 falls on the end point and is not 'between' the two times.
- In opposite directions, they close the gap at 5 + 3 = 8 rounds per hour, so they meet once every 1/8 hour = 7.5 minutes.
- Meetings happen 7.5, 15, 22.5, 30, … minutes after 5:00 a.m.
- 5:20 is 20 minutes and 7:00 is 120 minutes after 5:00. The first meeting after 5:20 is at 22.5 minutes (5:22:30).
- Meetings from 22.5 to 112.5 minutes: (112.5 − 22.5) ÷ 7.5 + 1 = 13. The next one, at 120 minutes, is exactly 7:00 a.m. — the end point, not between the two times.
- So they cross each other 13 times.
Remember · Opposite directions on a circular track: meetings per hour = sum of rounds per hour. Then check the end points: 'between' excludes them.
Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 1 Oct 2026 (how we verify). ·