Races and games

The essentials

In a race, the contestants who run for the same time cover distances in the ratio of their speeds, and over the same distance their times are in the inverse ratio of their speeds.

Common phrases:

• "A beats B by x metres" in a race of d metres means that when A finishes d metres, B has covered d − x metres.
• "A beats B by t seconds" means B finishes t seconds after A.
• "A gives B a start of x metres" means B starts x metres ahead of A, so B runs only d − x metres.
• In a game of 100 points, "A can give B 20 points" means that when A scores 100, B scores 80.

Key points

• Speed ratio A : B = (distance A runs) : (distance B runs) in the same time.
• Chain comparisons multiply: if, over 1000 m, A : B = 1000 : 900 and B : C = 1000 : 900, then A : C = 1000 : 810, so A beats C by 190 m.
• A dead heat means both reach the finishing point at the same time.
• Converting a time margin into a distance margin: distance = (slower runner's speed) × (time margin).

Exam tips

• Translate every phrase into "when A covers this, B covers that" before calculating.
• For starts and margins together, subtract both from the race length to get B's distance.

Common mistakes

• Adding the two margins in a chain comparison (100 + 100 = 200 m) instead of multiplying ratios.
• Using the faster runner's speed to convert a time margin into metres.