About Races and Games
A race question gives you two runners over the same distance. In the time the winner covers the course, the loser covers less, and the ratio of those two distances is the ratio of their speeds. Almost every question in the topic is that one idea applied once.
What you need to understand
- When the winner finishes, the loser has run for exactly the same length of time.
- The distances covered in equal times are in the same ratio as the speeds.
- The margin, or the distance by which the winner beats the loser, is what the loser still had to run when the winner finished.
- A start of a certain number of metres is the same thing as a margin, viewed from the other side.
- A dead heat means the two runners finish together, so their speeds must be equal over that distance.
- Doubling the distance doubles the margin only if the ratio of the speeds stays the same.
Formulas to remember
How to work through these questions
- Work out how far the loser gets in the winner time before doing anything else.
- Keep the two times straight, because the winner time is the one that fixes the moment of comparison.
- For a question about giving a start, treat the start as a margin and use the same relations.
- For a speed ratio question, remember that the distances in equal times are the speeds.
- Check that the margin is smaller than the race distance and that the loser covers a sensible fraction of it.
Mistakes that cost marks
- Using the loser time as though it were the winner time, which reverses the ratio.
- Giving the distance the loser covered when the question asked for the margin.
- Subtracting the two times and treating the result as a distance.
- Inverting the speed ratio, which puts the slower runner first.
- Adding the margin to the race distance instead of subtracting it.
Worked example
A beats B by 50 metres in a race of 100 metres. If A finishes in 10 seconds, how long does B take to run the full 100 metres?
- While A runs 100 metres in 10 seconds, B runs 100 - 50 = 50 metres in the same 10 seconds.
- So B covers 50 metres in 10 seconds.
- B therefore needs 10 x 100 / 50 = 20 seconds for the full 100 metres.
- Check: B is half as fast as A, so B must take twice as long, and 20 seconds is twice 10 seconds.
Answer: 20 seconds
Practice questions with answers
A few Races and Games questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
A beats B by 400 m in a race of 600 m. If A finishes the race in 16 seconds, how long does B take to finish?
Answer: Option C — with explanation
While A runs 600 m, B runs 200 m in the same 16 seconds.
B therefore runs 200 m in 16 seconds, so the full 600 m takes 16 x 600/200 = 48 seconds.
Common mistakes
- repeated the time taken by A: 16
- added the margin to the time: 416
- added the two times together: 64
Question 2
A can run 1000 m in 13 seconds and B can run the same distance in 26 seconds. By how many metres does A beat B in a 1000 m race?
Answer: Option C — with explanation
When A finishes after 13 seconds, B has run for 13 seconds as well.
26 seconds is enough for B to cover all 1000 m, so in 13 seconds B covers 1000 x 13/26 = 500 m.
A therefore beats B by 1000 - 500 = 500 m.
Common mistakes
- doubled the margin: 1000
- subtracted the two times and called it a distance: 13
- an arithmetic slip in the final step: 497
Question 3
A beats B by 200 m in a race of 400 m. If A finishes the race in 23 seconds, how long does B take to finish?
Answer: Option C — with explanation
While A runs 400 m, B runs 200 m in the same 23 seconds.
B therefore runs 200 m in 23 seconds, so the full 400 m takes 23 x 400/200 = 46 seconds.
Common mistakes
- added the margin to the time: 223
- repeated the time taken by A: 23
- added the two times together: 69
Question 4
A covers 100 m in 10 seconds while B takes 40 seconds for the same distance. How far behind is B when A finishes a 100 m race?
Answer: Option D — with explanation
When A finishes after 10 seconds, B has run for 10 seconds as well.
40 seconds is enough for B to cover all 100 m, so in 10 seconds B covers 100 x 10/40 = 25 m.
A therefore beats B by 100 - 25 = 75 m.
Common mistakes
- doubled the margin: 150
- gave the distance B had covered instead of the margin: 25
- subtracted the two times and called it a distance: 30
Question 5
A beats B by 400 m in a race of 600 m. If A finishes the race in 28 seconds, how long does B take to finish?
Answer: Option C — with explanation
While A runs 600 m, B runs 200 m in the same 28 seconds.
B therefore runs 200 m in 28 seconds, so the full 600 m takes 28 x 600/200 = 84 seconds.
Common mistakes
- added the two times together: 112
- repeated the time taken by A: 28
- added the margin to the time: 428
Frequently asked questions
Why are the distances proportional to the speeds?
Because the two runners are being compared over the same interval of time. Distance is speed multiplied by time, and the time is common to both.
What does giving a start mean?
One runner begins some metres ahead of the other, which is exactly the same as saying that runner is being given that distance as a margin.
How do I find the ratio of the speeds?
Compare the distances covered in the same time. If the winner beats the loser by x metres in a race of D metres, the ratio is D to D minus x.
What is a dead heat?
A race in which the two runners finish together. Their speeds must therefore be in the ratio of the distances they were required to run, which allows a start to be calculated.
