Circular Tracks
Circular tracks are the most loved twist on Time-Speed-Distance in CAT. Instead of two bodies moving along a straight line that ends, they run round and round a closed loop of fixed length L, so they keep meeting again and again — and the whole game becomes about WHEN and WHERE those meetings happen. The single idea that unlocks the chapter is relative speed: when two runners move in opposite directions their speeds add, and the track of length L is covered between meetings; when they move in the same direction the faster one must gain a full lap L on the slower, so their speeds subtract. From those two facts you get the time to the first meeting, the number of distinct meeting points on the track, and the time to meet again exactly at the starting point. CAT and XAT like this topic because it punishes students who memorise without understanding — a small change of wording (same vs opposite direction, "meet anywhere" vs "meet at the start") flips the whole answer. This chapter builds the three core skills in order, with the fastest ratio-based methods, the meeting-point count rule, and the LCM trick for returning to the start.
Topics
⚡ CAT shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Read the DIRECTION first: opposite ⇒ add speeds, same ⇒ subtract. Then t(first meet) = L / relative speed.
- First meeting AT THE START is direction-independent: t = LCM(L/a, L/b) = L/HCF(a,b) for integer speeds.
- Distinct meeting points = sum of reduced ratio terms (opposite) or difference of reduced ratio terms (same).
- Always reduce the speed ratio by its HCF before counting meeting points — that is the whole trick.
- Meetings are equally spaced in time and in position, so once you have the first, every later one follows by the same gap.
- Three bodies: find each pairwise first-meeting time, then take the LCM for when all three meet together.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Confusing "first meeting anywhere" (use relative speed) with "first meeting at the start" (use LCM of lap times).
- Adding speeds for same-direction motion or subtracting for opposite-direction motion — always check direction.
- Forgetting to reduce the speed ratio by its HCF before applying the meeting-points rule (sum/difference).
- Using sum instead of difference (or vice versa) for the number of meeting points — opposite = sum, same = difference.
- Assuming the start is always a meeting point; it is only when the start itself is one of the equally spaced points.
📈 CAT exam insight & PYQ analysis
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara CAT success checklist
You have truly mastered Circular Tracks when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (3 topics) | 3/3 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 12 cards |
Formula Reference Sheet
Meeting times on a circular track of length L
| Relative speed (opposite directions) | a + b |
|---|---|
| Relative speed (same direction) | |a − b| |
| Time to first meeting (opposite) | L / (a + b) |
| Time to first meeting (same) | L / |a − b| |
| Time for one full lap | L / a and L / b |
Meeting points & return to start
| Distinct meeting points (opposite) | |a + b| / HCF(a, b) |
|---|---|
| Distinct meeting points (same) | |a − b| / HCF(a, b) |
| Time to meet again AT the start | LCM( L/a , L/b ) |
| Same start point as ratio | meet at start after each completes whole laps |
| Three bodies, first meeting | LCM of the pairwise meeting times |