How to Solve Circular Track Race Problems
Solve circular track race aptitude problems using relative speed for same and opposite directions, with worked examples and practice questions.
Expected Interview Answer
On a circular track of length L, two runners moving at speeds a and b first meet at L/(a−b) if moving in the same direction, or L/(a+b) if moving in opposite directions, and every subsequent meeting repeats after that same interval.
The key shift from linear races is that runners keep looping, so “meeting” happens repeatedly, not once. For runners moving in the same direction, the faster one must build up a full lap of relative distance L over the slower one before catching up, giving time L/(a−b); every further catch-up takes another L/(a−b). For opposite directions, their combined relative speed closes the gap of L between them, giving first-meeting time L/(a+b), and every subsequent meeting recurs every L/(a+b). To find when they meet exactly at the starting point together, take the LCM of each runner’s individual lap time.
- One relative-speed formula handles both same-direction and opposite-direction meetings
- LCM of lap times finds simultaneous returns to the start
- Meeting intervals repeat, so one calculation answers “which meeting number” questions
AI Mentor Explanation
Two fielders jogging laps around the boundary rope in the same direction only cross paths once the faster one has gained a full lap on the slower one — the gap to close is the entire boundary length L, at their speed difference, giving time L/(a−b). If they instead jog in opposite directions around the rope, they close that same length L together at their combined speed, meeting after L/(a+b), and then again every L/(a+b) as they keep circling.
Worked example
Track
- L = 300 m
Same direction
- 300/(6−4)
- = 150 s
Opposite direction
- 300/(6+4)
- = 30 s
Step-by-Step Explanation
Step 1
Identify direction of motion
Determine if runners move the same way or opposite ways around the loop.
Step 2
Compute relative speed
Same direction: subtract speeds. Opposite direction: add speeds.
Step 3
Divide track length by relative speed
First meeting time = L / (relative speed).
Step 4
Repeat for subsequent meetings or LCM cases
Later meetings recur at the same interval; simultaneous start returns use LCM of individual lap times.
What Interviewer Expects
- Correct relative-speed setup for same vs opposite direction
- Recognition that meetings repeat at a fixed interval on a loop
- Using LCM of lap times for simultaneous-start-point questions
- Careful unit consistency between track length and speed
Common Mistakes
- Adding speeds when runners move in the same direction (should subtract)
- Subtracting speeds when runners move in opposite directions (should add)
- Forgetting meetings on a loop repeat periodically, not just once
- Confusing “meeting anywhere on track” with “meeting at start point” (LCM case)
Best Answer (HR Friendly)
“On a circular track, if two people run the same way, the faster one has to gain a full lap on the slower one before catching up, so I divide the track length by the speed difference. If they run opposite ways, they are closing the gap together, so I add the speeds instead and divide the track length by that sum. Either way, once they meet, they will meet again at the same fixed time interval, so I can just add that interval repeatedly for later meetings.”
Follow-up Questions
- How do you find the number of distinct meeting points on the track for opposite-direction runners?
- How does the LCM method find when both runners return to the start together?
- How would three runners on the same circular track change the approach?
- How do you convert this into a clock-hands problem on a circular dial?
MCQ Practice
1. Track length 400m. Two runners at 5 m/s and 3 m/s start together, same direction. First time the faster laps the slower is?
Relative speed = 5−3 = 2 m/s; time = 400/2 = 200s.
2. Track length 240m. Two runners at 7 m/s and 5 m/s start together, opposite directions. They first meet after?
Relative speed = 7+5 = 12 m/s; time = 240/12 = 20s.
3. On a circular track, two runners moving the same direction meet every 60 seconds. If they meet at t=60s, the next meeting after that is at?
Meetings on a loop recur at a fixed interval — the next one is 60+60 = 120s.
Flash Cards
Same-direction first meeting time? — L / (speed of faster − speed of slower).
Opposite-direction first meeting time? — L / (sum of both speeds).
How often do subsequent meetings recur? — At the same interval as the first meeting time.
How to find when both return to start together? — LCM of each runner’s individual lap time.