Time, Speed & Distance • Topic 4 of 4

Pursuit & Races

Races translate speed ratios into distances and head starts. "A beats B by 20 m in a 100 m race" means when A finishes the 100 m, B has covered only 80 m — so in the SAME time their distances are 100 : 80, which equals their speed ratio 5 : 4. "A beats B by 5 seconds" means B needs 5 s more than A to finish the same distance. A head start ("A gives B a start of 10 m") shortens B’s race to 90 m. The standard method: convert every "beat" into a speed ratio (equal-time legs give distance ratios), then apply it to the leg each runner actually runs. Dead heat means they finish together. Pursuit problems (one chases another with a head start) are just same-direction relative speed: time to catch = lead distance ÷ relative speed. Set up the ratio cleanly and most race questions collapse to one line.

✅ Solved examples

1. In a 100 m race A beats B by 20 m. What is the ratio of their speeds?
Same time: A runs 100, B runs 80 ⇒ speed ratio = 100 : 80 = 5 : 4.
2. A beats B by 20 m in a 100 m race; B beats C by 25 m in the same race. By how much does A beat C in a 100 m race?
A:B = 100:80, B:C = 100:75. When A runs 100, B runs 80, and C runs 80×(75/100) = 60 ⇒ A beats C by 40 m.
3. A runs at 8 m/s, B at 6 m/s. B gets a 50 m head start in a 200 m race. Who wins?
A needs 200/8 = 25 s. B runs 150 m at 6 m/s = 25 s. Both take 25 s ⇒ dead heat (a tie).
4. A is twice as fast as B. In a race A gives B a start of 60 m and still beats him by 20 m. Find the race length.
Let length L. A runs L while B runs L − 60 − 20 = L − 80. Speed ratio 2:1 ⇒ L/(L−80) = 2 ⇒ L = 160 m.

✏️ Practice — try these, take hints as needed

1. In a 200 m race A beats B by 40 m. Ratio of their speeds?
Same time.
A 200, B 160.
Reduce 200:160.
5 : 4
2. A can give B a 20 m start and C a 28 m start in a 100 m race. How much start can B give C?
A:B = 100:80, A:C = 100:72.
When B runs 80, C runs 72.
Scale B to 100: C = 90.
10 m
3. A beats B by 5 s in a 100 m race; A’s speed is 5 m/s. Find B’s time for 100 m.
A’s time = 100/5 = 20 s.
B is 5 s slower.
20 + 5.
25 s
4. A runs at 10 m/s, B at 8 m/s. B has a 40 m head start in a 200 m race. Who wins?
A: 200/10 = 20 s.
B runs 160 m at 8 m/s.
160/8 = 20 s.
Dead heat (tie)
5. A is 25% faster than B. In a 500 m race, what start (in metres) can A give B for a dead heat?
Speed ratio 5:4.
A runs 500, B runs 400 in same time.
Start = 500 − 400.
100 m

📝 Topic test — 8 questions

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