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ABHYAS PREP: RACES, DISCOUNTS, STOCKS AND HEIGHTS · RACES & GAMES

Races: beating, giving a start and dead heats

In the time the winner runs the whole race, the loser runs the race minus the margin. Speeds are in the ratio of those two distances.

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Worked examples

Example 1 · level 1

In a 1000 m race, Tenzin can give Aarav a start of 380 m and the race ends in a dead heat. What is the ratio of Tenzin’s speed to Aarav’s speed?

  1. 31 : 50
  2. 50 : 31
  3. 50 : 19
  4. 69 : 50
Show solution
  1. In the time Tenzin runs 1000 m, Aarav runs 620 m.
  2. Same time, so speed ratio = distance ratio = 1000 : 620.
  3. In lowest terms: 50 : 31.

Answer: 50 : 31

Example 2 · level 2

In a 400 m race, Saanvi beats Ananya by 30 m or by 3 seconds. How long does Saanvi take to run the 400 m?

  1. 0.24 seconds
  2. 37 seconds
  3. 43 seconds
  4. 40 seconds
Show solution
  1. Ananya runs 30 m in 3 s, so Ananya’s speed = 10 m/s.
  2. Ananya’s time for 400 m = 400/10 = 40 s.
  3. Saanvi finishes 3 s earlier: 40 − 3 = 37 s.

Answer: 37 seconds

Example 3 · level 3

In a 500 m race, Aditya can give Harpreet a start of 40 m, and Harpreet can give Fatima a start of 25 m. By how many metres will Aditya beat Fatima in a 500 m race?

  1. 63 m
  2. 437 m
  3. 65 m
  4. 67 m
Show solution
  1. Aditya : Harpreet = 500 : 460 and Harpreet : Fatima = 500 : 475.
  2. When Aditya runs 500 m, Fatima runs 460 × 475/500 = 437 m.
  3. Aditya beats Fatima by 500 − 437 = 63 m.

Answer: 63 m

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