In a circular race of 750 m, X and Y start from the same point and at the same time with speeds of 9 km/h and 13.5 km/h, respectively. If they are running in the same direction then, when will they meet again for the first time on the track?
- (a)500 seconds
- (b)600 seconds
- (c)900 seconds
- (d)750 seconds
Answer
Why
Correct — B. Same direction means the gap closes at the difference of the speeds, and one meeting costs a full lap of gain.
Relative speed = 13.5 − 9 = 4.5 km/h
In m/s: 4.5 × 5⁄18 = 1.25 m/s
Gain needed for a meeting = one whole lap = 750 m
Time = 750 ÷ 1.25 = 600 seconds → option (b).
Cross-check on lap times: X laps in 750 ÷ 2.5 = 300 s and Y in 750 ÷ 3.75 = 200 s, and LCM(300, 200) = 600 s — the same instant.
Why the others are wrong
- (a)500 seconds — By 500 s the faster runner has gained only 625 m. The gain accumulates at 1.25 m/s, so 500 × 1.25 = 625 — still 125 m short of the full lap a meeting needs.
- (c)900 seconds — 900 s is past the first meeting. The gain by then is 1.25 × 900 = 1125 m, one and a half laps, so they stand half a lap apart — and they had already met at 600 s.
- (d)750 seconds — 750 reuses the track length as a count of seconds, which would need the gain to run at 1 m/s. It runs at 1.25 m/s, so by 750 s the gain is 937.5 m, a quarter lap past the meeting.
Concept
Two runners on a circular track meet again when the faster has gained exactly one whole lap on the slower.
Same direction: the gap closes at the difference of the speeds. Opposite directions: it closes at the sum, so the meeting arrives far sooner on the same track.
Convert once, at the start: km/h × 5⁄18 = m/s. Here 4.5 × 5⁄18 = 1.25 m/s, and everything after that is one division.
A different question is when they next meet at the starting point. That is the LCM of the two lap times, and in general it is later than the first meeting anywhere on the track.
'Meet again for the first time on the track' means anywhere on the circle, not back at the start line.
The two answers happen to coincide on this data — both 600 s — so the distinction costs nothing here. It will not always, and the LCM answer is the larger of the two whenever they differ.
Key facts
- Same direction: relative speed is the difference of the two speeds.
- Opposite directions: relative speed is the sum of the two speeds.
- First meeting anywhere on a circular track = track length ÷ relative speed.
- First meeting back at the starting point = LCM of the two individual lap times.
Study next
Common traps
- Adding the speeds out of habit, which gives 22.5 km/h and a wrong 120 s.
- Leaving speeds in km/h while the track is in metres.
- Answering with the LCM of lap times when the question asks for a meeting anywhere on the track.
SSC reuses this stem almost word for word with the direction flipped: 18 Sep 2024, 12:30, Quant Q.12 runs 4800 m at 36 and 54 km/h in opposite directions, and 26 Sep 2024, 16:00, Quant Q.23 asks for both directions inside one option.
The same-direction version stated in m/s is at 9 Sep 2024, 09:00, Quant Q.3, and the starting-point LCM variant at 17 Sep 2024, 16:00, Quant Q.5.
Related PYQs
No directly related past PYQ was found.