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)?

• Key fact: Each runner has to cover 3000 m = 10 laps of a 300 m circular track.
• Let speeds be 3v and 2v. Time taken by the faster runner to finish = 3000/(3v) = 1000/v.
• In this time, the slower runner covers 2v × 1000/v = 2000 m = 6⅔ laps.
• Relative speed = v. One passing occurs every 300/v time units. During the race time 1000/v, number of completed passings = 1000/300 = 3⅓, so only 3 passings occur before the faster runner finishes.

Option check:

• (a) 2 — Incorrect; relative motion allows more than 2 passings before the race ends.
• (b) 3 — Correct; only 3 full overtakes happen before the faster runner completes 3 km.
• (c) 4 — Incorrect; the 4th overtake would require 1200/v time, but the race ends at 1000/v.
• (d) 5 — Incorrect; this wrongly assumes counting until the slower runner also finishes. Disclaimer: My earlier answer made this error.