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. ?
- (a) 12
- (b) 13 ✓ UPSC's answer
- (c) 14
- (d) 15
Why the answer is (b)
• The relative speed of P and Q walking in opposite directions is the sum of their individual speeds: 5 + 3 = 8 rounds per hour.
• They meet every time their combined distance covered equals 1 full round, so the time interval between consecutive meetings is 1/8 hour, which is 7.5 minutes.
• The first meeting after starting at 5:00 a.m. occurs at 5:07:30 a.m.
• The time window of interest is from 5:20 a.m. to 7:00 a.m., which is a duration of 100 minutes.
• The meetings occurring within this window are at 5:22:30, 5:30:00, 5:37:30, 5:45:00, 5:52:30, 6:00:00, 6:07:30, 6:15:00, 6:22:30, 6:30:00, 6:37:30, 6:45:00, and 6:52:30.
• Counting these instances gives a total of 13 meetings, which corresponds to option (b).
Why the other options are wrong
- (a) 12
- Option (a) is incorrect because it undercounts the number of meetings by 1, likely due to excluding the meeting at 6:52:30 or miscalculating the start of the interval.
- (c) 14
- Option (c) is incorrect because it overcounts the number of meetings by 1, likely by including a meeting that occurs after 7:00 a.m. or miscalculating the frequency.
- (d) 15
- Option (c) is incorrect because it overcounts the number of meetings by 2, likely due to a significant error in calculating the relative speed or the time interval between meetings.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2025, held on 25 May 2025. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.