Fundamentals of Business Mathematics and Statistics · Time and Distance

Races, Circular Tracks and Clocks-Style Problems in Time and Distance

Updated 10 October 2026 · Fact-checked

Race problems compare how far two runners go in the same time. "A beats B by d metres" means B is d metres short when A finishes. On a circular track, runners meet when their gap equals one full lap, so meeting time is track length ÷ relative speed.

Understand Races, Circular Tracks and Clocks-Style Problems

A race is a time and distance problem with one key idea: both runners run for the same time. So the ratio of distances covered equals the ratio of their speeds. Everything else is reading the words correctly.

Two phrases cause most trouble. A beats B by 20 m means that when A reaches the finish line, B is still 20 m away. A beats B by 5 seconds means B reaches the finish 5 seconds after A. A head start means one runner starts ahead, or the other starts late, so the distance to run is different for each.

On a circular track, runners keep going round. If they run in the same direction, the faster one gains on the slower one, so you use the difference of speeds. If they run in opposite directions, they close the gap faster, so you use the sum of speeds. They first meet when the faster has gained one full lap (same direction) or when together they have covered one full lap (opposite direction).

The same logic works for clock-style problems. The minute hand and hour hand are two runners on a circular track of 360 degrees, moving at different speeds. You treat them exactly like runners and use relative speed.

Key formulas to remember

Distance ratio in a race: Distance of A ÷ Distance of B = Speed of A ÷ Speed of B; Valid because both run for the same time.
A beats B by d metres in a race of L metres: When A runs L, B runs (L − d). So Speed A : Speed B = L : (L − d); B is d metres behind at the moment A finishes.
A beats B by t seconds: Distance B covers in t seconds = distance by which A beats B; Use B's speed to convert time to distance, or distance to time.
Head start: Distance run by the runner with the head start = race length − head start; the other runner runs the full race length; The head start is measured from the start line, so the runner with it runs a shorter distance.
Circular track, same direction: Time to meet = track length ÷ (faster speed − slower speed); Applies when both start together from the same point.
Circular track, opposite directions: Time to meet = track length ÷ (sum of speeds); Applies when both start together from the same point.
Meeting again at the starting point: Time = LCM of the times each runner takes for one lap; Use when the question asks when both are back at the start together.
Unit conversion: km/h × 5/18 = m/s; Convert before using lap length in metres.

How to solve Races, Circular Tracks and Clocks-Style Problems questions

Use this method for any race or circular track question.

1Read the wording and note who finishes first and what is left for the other runner.
2Convert all speeds and distances to the same units, usually metres and seconds.
3For a race, write the distances each runner covers in the same time and form the ratio of speeds.
4For a head start or beat by time, convert time to distance using the loser's speed (the runner who finishes later).
5For a circular track, decide the direction: same direction uses difference of speeds, opposite uses sum.
6Treat the track length as the distance to be gained or covered, then divide by the relative speed.
7If the question asks for the meeting at the starting point, use the LCM of the lap times.
8Check that the answer is sensible: the faster runner covers more distance, and time is positive.

Quickest way: Ratio shortcut for races and relative speed for tracks

When to use it: Use for most MCQs where speeds or race results are given as ratios or simple numbers.

For "A beats B by d in L", write the speed ratio directly as L : (L − d).
Use this ratio to find the result in a different race length by simple proportion.
For a track, jump straight to track length ÷ relative speed.
Pick the answer option that matches and eliminate options with the wrong direction logic, such as the sum when the runners go the same way.

Common mistakes in Races, Circular Tracks and Clocks-Style Problems

Using the beat distance as B's total distance.
The phrase "beats by 20 m" is read as B running 20 m.
Fix: B runs L − d when A runs L. The beat is only the gap left.
Adding speeds for runners moving in the same direction.
Students remember "add" and apply it blindly.
Fix: Same direction means subtract speeds. Opposite direction means add.
Mixing km/h with metres and seconds.
Speeds are given in km/h but the track is in metres.
Fix: Convert km/h to m/s by multiplying by 5/18 before dividing.
Treating a beat by time as a beat by distance.
The two phrases look similar.
Fix: For a beat by t seconds, find the distance the loser covers in t seconds at the loser's speed.
Forgetting that a head start reduces the distance for the runner who has it.
Students give the same race length to both runners.
Fix: Draw a quick line: the runner with the head start runs the race length minus the head start.

Worked examples

Example 1

In a 100 m race, A beats B by 20 m. In a race of 400 m, by how many metres will A beat B if both run at the same speeds as before?

Show the solution

When A runs 100 m, B runs 100 − 20 = 80 m.
Speed ratio A : B = 100 : 80 = 5 : 4.
In a 400 m race, when A runs 400 m, B runs 400 × 4/5 = 320 m.
A beats B by 400 − 320 = 80 m.

Answer: 80 m

Example 2

Two runners start together from the same point of a circular track 400 m long. Their speeds are 5 m/s and 3 m/s. After how many seconds will they first meet if they run in opposite directions, and if they run in the same direction?

Show the solution

Opposite directions: relative speed = 5 + 3 = 8 m/s.
Time to meet = 400 ÷ 8 = 50 seconds.
Same direction: relative speed = 5 − 3 = 2 m/s.
Time to meet = 400 ÷ 2 = 200 seconds.

Answer: 50 seconds in opposite directions and 200 seconds in the same direction

Exam tips

Write the speed ratio first. Most race questions become simple proportion after that.
Underline the words "beats by", "head start" and "same/opposite direction" before calculating.
Check units. A track in metres with speed in km/h needs conversion.
With no negative marking, always mark an answer. Eliminate options that break the logic: for same-direction runners, the meeting time must be longer than for opposite directions, since the relative speed is smaller.

Practice questions from Time and Distance

Races, Circular Tracks and Clocks-Style Problems: frequently asked questions

What does A beats B by 10 metres mean?

It means that when A reaches the finish line, B is 10 metres short of it. So in a 100 m race, A runs 100 m while B runs 90 m.

What is the formula for meeting time on a circular track?

Divide the track length by the relative speed. Use the sum of speeds if the runners go in opposite directions and the difference if they go the same way.

How do I solve a race problem with a head start?

Reduce the distance of the runner who has the head start by the head start amount. Then use equal time for both runners and compare distances to get the speed ratio.

How do I find when two runners meet at the starting point again?

Find the time each takes to complete one lap. The LCM of these lap times gives the first time both are at the start together.