Same & Opposite Direction
The number of DISTINCT points on the track where two bodies repeatedly meet is a classic CAT result. Reduce the speed ratio a:b to its lowest terms (divide by HCF). If they move in opposite directions, the number of distinct meeting points = sum of the reduced ratio terms; if same direction, it = difference of the reduced ratio terms. For example, speeds in ratio 3:5: opposite gives 3 + 5 = 8 meeting points, same gives 5 − 3 = 2. These points are equally spaced around the track. One always coincides with the start only when the start is itself a meeting point. The big CAT insight: opposite-direction motion produces MORE meeting points and earlier meetings; same-direction produces fewer points and later meetings. Combine this with the first-meeting time and you can locate every meeting precisely.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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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 |