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Races and Games of Skill

Competitive Mathematics · Time, Work and Distance

7 Part3 worked examples18 practice questions

In a school 100 m race, if you are the fastest and your friend slower, he is allowed to start a little ahead for fairness — that is a start. If you run 100 m while he runs only 80 m, you gave him a start of 20 m. Every race question rests on one thing: how far each contestant covers in the same time. Their speeds are in the same ratio as the distances covered — as long as both ran for the same time.

The basic race rule
1

Starts and the length of the race

Full length − start

The one with a start runs a shorter race; in point games, the loser’s score.

Terms and direct questions

Start, margin, dead heat

Start, finishing point, winner, dead heat and the points game.

  • **A gives B a start of m:** A runs the full m, B only m.
  • **A beats B by m:** when A reaches the finish B is m behind — he has run m.
  • Dead heat: both finish at the same moment; the margin is zero.
  • **A game of points:** whoever scores first wins. If A scores and B scores , A can give B points.
Example 1 · Basic
In a 400 m race A gives B a start of 15 m. How far must B run?
Solution, step by step
  1. 1B’s run .
  2. 2 m
Answer 385 m
Example 2 · Basic
In a race P gives Q a start of 40 m, leaving Q a run of 260 m. What is the full length of the race?
Solution, step by step
  1. 1Q’s run full start.
  2. 2Full length m
Answer 300 m
Example 3 · Basic
In a game of 120 points A scores 120 while B scores only 85. How many points can A give B?
Solution, step by step
  1. 1A can give points.
  2. 2 points
Answer 35 points
BasicTry it yourself first
In a 400 m race A gives B a start of 15 m. How far does B have to run?
BasicTry it yourself first
In a 200 m race A gives B a start of 30 m. How far does B have to run?
StandardTry it yourself first
In a 800 m race A gives B a start of 45 m. How far does B have to run?
StandardTry it yourself first
In a 1000 m race A gives B a start of 120 m. How far does B have to run?
AdvancedTry it yourself first
In a 500 m race A gives B a start of 35 m. How far does B have to run?
BasicTry it yourself first
In a race P gives Q a start of 40 m, making the race 260 m for Q. What is the full length of the race?
BasicTry it yourself first
In a race P gives Q a start of 15 m, making the race 185 m for Q. What is the full length of the race?
StandardTry it yourself first
In a race P gives Q a start of 60 m, making the race 440 m for Q. What is the full length of the race?
AdvancedTry it yourself first
In a race P gives Q a start of 75 m, making the race 325 m for Q. What is the full length of the race?
BasicTry it yourself first
In a 400 m race M gives N some start and N's race becomes 360 m. How much start did N get?
StandardTry it yourself first
In a 500 m race M gives N some start and N's race becomes 455 m. How much start did N get?
StandardTry it yourself first
In a 1000 m race M gives N some start and N's race becomes 880 m. How much start did N get?
AdvancedTry it yourself first
In a 600 m race M gives N some start and N's race becomes 540 m. How much start did N get?
BasicTry it yourself first
In a game of 100 points A scores 100 while B scores only 72. How many points can A give B?
BasicTry it yourself first
In a game of 120 points A scores 120 while B scores only 85. How many points can A give B?
StandardTry it yourself first
In a game of 80 points A scores 80 while B scores only 53. How many points can A give B?
StandardTry it yourself first
In a game of 150 points A scores 150 while B scores only 98. How many points can A give B?
AdvancedTry it yourself first
In a game of 60 points A scores 60 while B scores only 41. How many points can A give B?
2

The margin: distance and time

Tie it to the loser’s speed

From “beat by x m or T s” find the time and speed, and win even after giving a start.

Margin, dead heat and speed ratio

B needs seconds to run m

The distance by which the loser is behind, run at his own speed, takes exactly the margin in time.

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3

Three contestants and point games

A chain of ratios

When A beats two others, their standing between themselves; races of different lengths and point games.

The A, B, C chain

While A runs

Write all three distances at one moment and form their mutual ratios.

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4

Circular tracks, overtaking and the winning post

Lead and the fairness condition

Counting overtakes on a circular track, catching up on the clock, and the distance that ends in a tie.

Mixed questions

Laps, the clock and the post

One overtake per full lap of lead; arriving together means equal times.

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5

Traps and fast tricks

Slips · tricks

Where marks go in race questions.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

A snapshot of one moment

Habits for speed.

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6

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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7

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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