Races and circular track problems use the same speed-distance-time fundamentals, but with a specific vocabulary and some neat patterns. Once we learn the language of races (“A beats B by 20 meters,” “A gives B a 5 second head start”), the math is straightforward.
Race Terminology
Let’s get the vocabulary down first.
Linear Race Problems
Example 1: Basic race
In a 100 m race, A beats B by 20 m. What is the ratio of their speeds?
When A finishes 100 m, B has run only 80 m. They ran for the same time.
Speed ratio = Distance ratio = A : B = 100 : 80 = 5 : 4
Example 2: Chaining results
In a 100 m race, A beats B by 10 m and B beats C by 20 m. By how much does A beat C?
When A runs 100 m, B runs 90 m. When B runs 100 m, C runs 80 m.
When B runs 90 m, C runs? → 80/100 × 90 = 72 m.
So when A finishes 100 m, C has run 72 m. A beats C by 28 m.
The key insight: we can’t just add 10 + 20 = 30. We have to scale B’s advantage over C to the actual distance B ran.
Example 3: Head start to make it a dead heat
A beats B by 30 m in a 200 m race. What head start should A give B so they finish together?
Speed ratio = 200 : 170 = 20 : 17.
For a dead heat, B needs to cover less distance. If A gives B a head start of x meters, then A runs 200 m while B runs (200 - x) m.
For them to finish together: time must be equal. 200/Speed_A = (200 - x)/Speed_B Since speed ratio is 20:17: 200/20 = (200 - x)/17 10 = (200 - x)/17 200 - x = 170 x = 30 m head start
This makes sense — A beats B by exactly 30 m, so giving B a 30 m head start makes it a tie.
Example 4: Time-based head start
A runs at 10 m/s and B runs at 8 m/s. In a 200 m race, how much head start in time should A give B for a dead heat?
A’s time = 200/10 = 20 seconds. B’s time = 200/8 = 25 seconds.
A needs to give B a head start of 25 - 20 = 5 seconds.
If B starts 5 seconds early, both finish at the same time.
Circular Track Problems
Circular tracks add a new dimension: the runners go around and around, and we need to figure out when and where they meet. This is where relative speed and LCM become our best friends.
Same Direction — The Faster Catches the Slower
In simple language, when two people run in the same direction on a circular track, the faster person “laps” the slower one. The first meeting happens when the faster runner gains exactly one full lap on the slower runner.
Example 5: Same direction on circular track
A and B start at the same point on a 400 m circular track. A runs at 5 m/s and B at 3 m/s, both in the same direction. When do they first meet?
Relative speed = 5 - 3 = 2 m/s.
A needs to gain 400 m on B (one full lap) to meet again.
Time = 400/2 = 200 seconds
Opposite Direction — They Run Toward Each Other
When running in opposite directions, they’re essentially approaching each other on the track. The first meeting happens when they’ve collectively covered one full lap.
Example 6: Opposite direction on circular track
Same setup, but A and B run in opposite directions. When do they first meet?
Relative speed = 5 + 3 = 8 m/s.
Time = 400/8 = 50 seconds
Much faster meeting time — makes sense since they’re running toward each other.
Meeting at the Starting Point
This is different from “when do they meet?” — this asks when both are simultaneously back at the starting point.
Each person returns to the start after completing full laps. A returns every 400/5 = 80 seconds. B returns every 400/3 = 133.33 seconds.
They’re both at the start at time = LCM(80, 400/3).
For clean numbers, let’s use a simpler example.
Example 7: Meeting at starting point
A and B run around a 600 m circular track at 10 m/s and 8 m/s. When do they first meet at the starting point?
Time for A to complete one lap = 600/10 = 60 seconds. Time for B to complete one lap = 600/8 = 75 seconds.
First time both are at the start = LCM(60, 75) = 300 seconds = 5 minutes.
At 300 seconds: A has done 300/60 = 5 laps, B has done 300/75 = 4 laps. Both are at the start!
Three or More Runners
For three runners A, B, C:
First meeting of all three at the same point on the track: Find when each pair meets, then take the LCM of those times. Or more practically, find LCM of individual lap times for “meeting at starting point.”
First time all three meet anywhere: Find when A&B meet (time T₁) and when A&C meet (time T₂). All three meet at LCM(T₁, T₂).
Example 8: Three runners
A, B, C run on a 300 m track at 6 m/s, 4 m/s, and 3 m/s in the same direction. When do all three first meet?
A&B first meet: 300/(6-4) = 150 seconds. A&C first meet: 300/(6-3) = 100 seconds.
All three meet: LCM(150, 100) = 300 seconds.
Verify: In 300 sec, A runs 1800 m (6 laps), B runs 1200 m (4 laps), C runs 900 m (3 laps). A is 2 laps ahead of B (same spot) and 3 laps ahead of C (same spot). All at the same point!
Number of Distinct Meeting Points
When two runners go in opposite directions with speed ratio a:b (in simplest form), they meet at (a + b) distinct points on the track as they keep running.
When going in the same direction with speed ratio a:b, they meet at |a - b| distinct points.
Common Exam Variations
- “A beats B by x meters” — find speed ratio.
- Chaining: A beats B, B beats C, how much does A beat C?
- Head start (time or distance) for a dead heat.
- Circular track: first meeting time (same/opposite direction).
- Circular track: meeting at starting point (LCM method).
- Three runners on a circular track.
- Number of distinct meeting points.
Practice Problems
Problem 1: In a 1 km race, A beats B by 50 m and B beats C by 80 m. By how many meters does A beat C?
Problem 2: Two runners start from the same point on a 500 m circular track running in opposite directions at 4 m/s and 6 m/s. After how many seconds do they meet for the 3rd time?
Problem 3: A and B run around a 400 m track at 8 m/s and 5 m/s. When do they first meet at the starting point together?
Answers
Problem 1: When A finishes 1000 m, B has run 950 m. When B runs 1000 m, C runs 920 m. When B runs 950 m, C runs 920/1000 × 950 = 874 m. A beats C by 1000 - 874 = 126 m.
Problem 2: Relative speed = 4 + 6 = 10 m/s. Time for each meeting = 500/10 = 50 seconds. 3rd meeting = 3 × 50 = 150 seconds.
Problem 3: A’s lap time = 400/8 = 50 seconds. B’s lap time = 400/5 = 80 seconds. LCM(50, 80) = 400 seconds. (A completes 8 laps, B completes 5 laps.)