Arvind and Chetan started running simultaneously from the same point in the same direction on a circular track of length 210 m. If Arvind takes 180 seconds to complete one round, and Chetan takes 420 seconds to complete one round, after how much time will they meet for the first time at the starting point on the track?

Correct — D. Meeting at the starting point means both runners have completed whole laps, so the time must be a common multiple of the two lap times.

Take the LCM of 180 s and 420 s:
180 = 2² × 3² × 5
420 = 2² × 3 × 5 × 7
LCM = 2² × 3² × 5 × 7 = 1260 s

1260 ÷ 60 = 21 minutes → option (d).

Why the others are wrong

• (a)
  28 minutes — 28 minutes is 1680 s — 4 laps for Chetan but 9⅓ laps for Arvind. It is a multiple of 420 alone, so Arvind is not at the start line.
• (b)
  15 minutes — 15 minutes is 900 s — 5 laps for Arvind but 2⅐ laps for Chetan. 900 is not divisible by 420, so Chetan is mid-lap.
• (c)
  10 minutes — 10 minutes is 600 s, a multiple of neither 180 nor 420. Both runners are away from the start line, so they cannot meet there.

Two different questions hide behind the words when will they meet.

At the starting point: each runner must have finished whole laps, so the answer is the LCM of the lap times — 1260 s here.

Anywhere on the track: only the gap between them matters, so the answer uses relative speed. Arvind covers 210 m in 180 s and Chetan in 420 s, a difference of 2⁄3 m/s, giving 210 ÷ 2⁄3 = 315 s.

The track length is needed for the second reading and not for the first.

Key facts

• Meeting at the starting point takes the LCM of the individual lap times.
• LCM(180, 420) = 1260 seconds, which is 21 minutes.
• Meeting anywhere on this track uses relative speed and takes 315 seconds.

Common traps

• Using relative speed and answering the meet-anywhere time of 315 seconds
• Leaving 1260 in seconds when every option is in minutes

No directly related past PYQ was found.