Skip to content

Comparing speeds fairly

Change and motion · level 2, lesson 2 of 6

A question to start

Mia and Leo have a race, but they cannot agree on the track.

  • Mia runs 100 metres and takes 20 seconds.
  • Leo runs 50 metres and takes 9 seconds.

Who is faster? Mia ran further, so maybe Mia. But Leo took less time, so maybe Leo. Each of them has one number that looks better. You cannot just pick one number and ignore the other.

In level 1 you learned the two fair races: the same time, or the same distance. But Mia and Leo did neither. In this lesson you will learn how to make a race fair after it has been run.

The idea

Two numbers make a speed

Speed tells you how fast something moves. To know a speed, you always need two numbers:

  1. how far it went (the distance), and
  2. how long it took (the time).

One number on its own can trick you. "I ran 100 metres!" sounds great, unless it took you 5 minutes. "I ran for only 9 seconds!" sounds fast, unless you only went 1 metre.

Fair race 1: the same distance

If two movers go the same distance, the one who takes less time is faster.

Ava and Zoe both swim one length of the pool. Ava takes 30 seconds, Zoe takes 25 seconds. Zoe is faster: she did the same job in less time.

Fair race 2: the same time

If two movers go for the same time, the one who goes further is faster.

Kai and Sam each ride their bikes for 10 minutes. Kai goes 2 km, Sam goes 3 km. Sam is faster: in the same time, he covered more ground.

When neither is the same: make one the same

Back to Mia and Leo. The distances are different (100 m and 50 m), and the times are different (20 s and 9 s). The trick is to change one race so that it matches the other.

Leo ran 50 metres. 100 metres is two lots of 50 metres. If Leo kept running at the same pace, the second 50 metres would take him another 9 seconds.

Mia Leo 100 m in 20 s 50 m: 9 s 50 m: 9 s Leo: 100 m in 9 + 9 = 18 s

Now both races are 100 metres:

  • Mia: 100 metres in 20 seconds.
  • Leo: 100 metres in 9 + 9 = 18 seconds.

Same distance, and Leo takes less time. Leo is faster.

Why this works, and when it does not

We said "if Leo kept running at the same pace". That is an important "if". The trick works when a mover keeps a steady speed: every 50 metres takes the same time as the 50 metres before.

Real runners are not perfectly steady. Leo might get tired in his second 50 metres and take 11 seconds instead of 9. Then he would take 20 seconds for 100 metres, a tie! So the answer "Leo is faster" really means: Leo ran at a faster pace while he was running. To be completely sure about a 100-metre race, they would have to run one.

Good mathematicians always say what they are assuming. Here we assume a steady pace.

The other way: how far in the same time

We could also make the times match. Try "how far in one second?"

  • Mia ran 100 metres in 20 seconds. Share 100 metres equally into 20 seconds: 100 ÷ 20 = 5 metres each second. (Check: 20 lots of 5 is 100. ✓)
  • Leo ran 50 metres in 9 seconds. 50 ÷ 9 is a bit more than 5 (because 9 lots of 5 is only 45). So Leo runs a little more than 5 metres each second.

Again, Leo is faster, but only by a little. Both ways give the same answer, which is a good sign that we did it right.

"Metres each second" is a very useful way to say a speed. Grown-ups write it metres per second. "Per" just means "in each" or "for each". A car going 50 kilometres per hour would go 50 kilometres in each hour, if it kept going at that speed.

Doubling and halving to match

To make two races match, the easiest moves are doubling and halving:

  • If someone goes 30 metres in 6 seconds, then (at a steady pace) they go 60 metres in 12 seconds (double both), or 15 metres in 3 seconds (halve both).
  • Double the distance, double the time. Halve the distance, halve the time.

The distance and the time must always change together, in the same way. If you double one and not the other, you have changed the speed.

Worked examples

Example 1. Two walkers. Ana walks 6 km in 2 hours. Ben walks 9 km in 3 hours. Who walks faster?

  1. Find how far each walks in 1 hour.
  2. Ana: 6 km shared into 2 hours is 3 km in each hour.
  3. Ben: 9 km shared into 3 hours is 3 km in each hour.
  4. They walk at the same speed: 3 km per hour. Ben just walked for longer.

Example 2. Toy cars. A red car rolls 40 cm in 4 seconds. A blue car rolls 25 cm in 5 seconds. Which is faster?

  1. Red: 40 ÷ 4 = 10 cm each second.
  2. Blue: 25 ÷ 5 = 5 cm each second.
  3. The red car is faster, in fact twice as fast.

Example 3. Making the distances match. Noor swims 25 metres in 30 seconds. Kai swims 50 metres in 70 seconds. Who is faster?

  1. Double Noor's swim (steady pace): 50 metres in 60 seconds.
  2. Now both swim 50 metres. Noor: 60 seconds. Kai: 70 seconds.
  3. Noor is faster: less time for the same distance.

Example 4. A rover on Mars. A rover on Mars drives at about 4 cm each second at most. A child walking slowly goes about 100 cm (1 metre) each second. How far does each go in 10 seconds?

  1. Rover: 10 lots of 4 cm = 40 cm. That is about the length of your arm.
  2. Child: 10 lots of 100 cm = 1000 cm = 10 metres.
  3. In the same 10 seconds, the child goes much, much further. The child is about 25 times faster than the rover (because 25 lots of 4 is 100).

Example 5. A trap. Lily cycles 4 km in 20 minutes. Max cycles 3 km in 10 minutes. Lily says: "I went further, so I am faster." Is she right?

  1. The times are different, so we must make them match.
  2. Double Max (steady pace): 6 km in 20 minutes.
  3. Now both ride for 20 minutes: Lily 4 km, Max 6 km.
  4. Max is faster. Lily went further only because she rode for longer.

Try it

  1. Two dogs run across the same park. Rex takes 12 seconds and Bella takes 15 seconds. Which dog is faster?
  2. Two trains travel for 2 hours. One goes 300 km and the other 240 km. Which is faster?
  3. Sara runs 20 metres in 4 seconds. How many metres does she run each second?
  4. A tortoise walks 12 metres in 6 minutes. A snail crawls 2 metres in 4 minutes. How far does each go in 1 minute? Which is faster?
  5. Emma reads 10 pages in 5 minutes. Jack reads 36 pages in 12 minutes. Who reads faster?
  6. A car drives 60 km in 1 hour. A bus drives 100 km in 2 hours. Which is faster? (Assume steady speeds.)
  7. Omar runs 50 metres in 10 seconds. Priya runs 200 metres in 44 seconds. Who has the faster pace? (Hint: how long would Omar take for 200 metres?)
  8. Why do we have to assume a "steady pace" when we double or halve a race?
Answers
  1. Same distance, so less time wins: Rex (12 seconds).
  2. Same time, so further wins: the 300 km train.
  3. 20 ÷ 4 = 5 metres each second. (Check: 4 lots of 5 is 20. ✓)
  4. Tortoise: 12 ÷ 6 = 2 metres each minute. Snail: 2 ÷ 4 = half a metre each minute. The tortoise is faster (4 times as fast).
  5. Emma: 10 ÷ 5 = 2 pages each minute. Jack: 36 ÷ 12 = 3 pages each minute (check: 12 lots of 3 is 36 ✓). Jack reads faster.
  6. The bus goes 100 km in 2 hours, so 50 km in 1 hour. The car goes 60 km in 1 hour. The car is faster.
  7. 200 metres is 4 lots of 50 metres, so Omar would take 4 lots of 10 seconds = 40 seconds. Priya took 44. Omar has the faster pace.
  8. Doubling says "the second half takes the same time as the first half". That is only true if the speed stays the same. If the mover gets tired or speeds up, the second half takes a different time.

Common mistake

Looking at only one number. "I went further" or "I took less time" only decides the race if the other number is the same. If both numbers are different, first make one of them match.

Doubling only one number. If you double the distance, you must also double the time (for a steady pace). Doubling just one of them changes the speed and gives the wrong answer.

Where it leads