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"Per": what a rate tells you

Change and motion · level 3, lesson 1 of 6

A question to start

Mia read 84 pages in 4 days. Sam read 105 pages in 5 days. Sam read more pages. But who is the faster reader?

It is hard to tell, because they read for different numbers of days. Sam had an extra day. Is it fair to compare 84 with 105? Not really. We need a way to compare that is fair, even when the times are different.

The answer is a small word you already use: per. By the end of this lesson you will see that Mia and Sam read at exactly the same speed, and you will know why.

The idea

"Per" means "for each one"

When you hear "60 kilometres per hour", the word per means "for each" or "in each one". So 60 kilometres per hour means: in each hour, 60 kilometres.

Here are some you may have met:

  • metres per second: how many metres something moves in each second (a speed);
  • pages per day: how many pages you read in each day;
  • litres per minute: how much water comes out of a tap in each minute;
  • beats per minute: how many times your heart beats in each minute;
  • dollars per kilogram: the price of each kilogram of apples.

Each one compares two different kinds of thing: metres and seconds, pages and days, litres and minutes. A number that tells you how much of one thing goes with each one of another thing is called a rate.

Finding a rate: share it out

Suppose a tap fills a 20-litre bucket in 4 minutes. How many litres per minute is that?

Think of the 20 litres being shared out fairly among the 4 minutes. Each minute gets the same amount.

5 L5 L5 L5 L minute 1minute 2minute 3minute 4 20 litres in 4 minutes

\(20 \div 4 = 5\), so the tap gives 5 litres per minute.

To find a rate, you divide: the amount of the first thing, divided by how many of the second thing.

Now the reading puzzle. Mia: \(84 \div 4 = 21\) pages per day. Sam: \(105 \div 5 = 21\) pages per day. They read at the same rate. Sam read more pages only because he read for one more day. The rate takes away the unfairness of different times, because it always talks about one day.

A rate is a "steady" way of saying it

Did Mia really read exactly 21 pages every single day? Maybe not. Perhaps she read 30 pages on Saturday and only 12 on Monday. The rate says what each day would have been if she had read the same amount every day and still finished 84 pages in 4 days.

So a rate is a kind of fair share. It is a lot like the "mean" you meet in Statistics: share the total out equally.

Using a rate: multiply

Once you know a rate, you can predict. If the tap gives 5 litres per minute, then:

  • in 2 minutes it gives \(5 \times 2 = 10\) litres;
  • in 10 minutes it gives \(5 \times 10 = 50\) litres;
  • in 30 minutes it gives \(5 \times 30 = 150\) litres.

And you can work backwards. How long to fill a 35-litre tub? Each minute gives 5 litres, so we need \(35 \div 5 = 7\) minutes.

So there are three questions you can answer with a rate:

You know You want You do
amount and time the rate amount ÷ time
rate and time the amount rate × time
amount and rate the time amount ÷ rate

Do not learn the table by heart. Think "how much in each minute?" and the right step will come to you.

The units tell the story

Always write the units of a rate: "5 litres per minute", not just "5". The number 5 alone could mean anything. The units say what is shared out (litres) and what it is shared over (minutes).

You can also write a rate short: L/min means litres per minute, km/h means kilometres per hour, m/s means metres per second. The slash / is read as "per". It is also a division sign, and that is no accident: a rate is a division.

Every rate can be turned around

The tap gives 5 litres per minute. You could also ask: how many minutes per litre? One minute gives 5 litres, so one litre takes \(\frac{1}{5}\) of a minute, which is 12 seconds.

Both are true. They say the same thing in two ways. Runners often use the turned-around rate. Instead of "12 kilometres per hour" they say "5 minutes per kilometre" (their pace), because they think about each kilometre one at a time. Check: in 60 minutes, at 5 minutes for each kilometre, a runner covers \(60 \div 5 = 12\) kilometres.

Be careful, though: when you turn a rate around, bigger is no longer better. More kilometres per hour means faster, but more minutes per kilometre means slower.

Rates of change

Many rates tell you how fast something changes: a plant grows 2 centimetres per week, a candle gets 3 centimetres shorter per hour, the water in a tank goes up 5 litres per minute. These are called rates of change, and they are what this whole topic is about. Speed is the most famous one: how fast your position changes.

Worked examples

1. A bike ride. A cyclist rides 36 kilometres in 3 hours. What is her rate in kilometres per hour?

Share 36 km among 3 hours: \(36 \div 3 = 12\). She rides 12 kilometres per hour (12 km/h).

2. Heartbeats. A nurse counts 18 heartbeats in 15 seconds. What is the heart rate in beats per minute?

A minute has 60 seconds, and \(60 \div 15 = 4\), so a minute is 4 lots of 15 seconds. Each lot has 18 beats, so a minute has \(18 \times 4 = 72\) beats. The heart rate is 72 beats per minute.

(Here we did not divide by 15 first. We scaled up straight to one minute. Both ways are fine.)

3. Filling a bath. A tap pours 6 litres per minute. The bath holds 150 litres. How long will it take to fill?

Each minute adds 6 litres. How many 6s make 150? \(150 \div 6 = 25\). It takes 25 minutes.

4. Which is the better buy? Shop A sells 6 apples for 3 dollars. Shop B sells 8 apples for 4.40 dollars.

  • Shop A: \(3 \div 6 = 0.50\) dollars per apple (50 cents each).
  • Shop B: \(4.40 \div 8 = 0.55\) dollars per apple (55 cents each).

Shop A is cheaper per apple. Without the rate, you could not compare, because the bags have different numbers of apples.

5. A rover on Mars. The Curiosity rover on Mars drives very slowly, about 4 centimetres per second on flat ground. How far does it go in one minute? In one hour, if it never stops?

  • One minute is 60 seconds: \(4 \times 60 = 240\) cm, which is 2.4 metres.
  • One hour is 60 minutes: \(240 \times 60 = 14\,400\) cm, which is 144 metres.

A person walking for an hour goes about 5 kilometres, which is about 35 times further. The rover is slow on purpose: nobody can come and fix it if it tips over.

Try it

  1. A car goes 40 kilometres in 2 hours. What is its rate in kilometres per hour?
  2. Lee reads 120 pages in 6 days. How many pages per day is that?
  3. A tap pours 8 litres per minute. How long does it take to fill a 96-litre tank?
  4. You count 25 heartbeats in 20 seconds. What is your heart rate in beats per minute?
  5. Which is faster: a dog that runs 150 metres in 30 seconds, or a bike that goes 200 metres in 50 seconds?
  6. A printer prints 45 pages in 3 minutes. How many pages does it print in 10 minutes?
  7. A snail crawls 30 centimetres in 5 minutes. At the same rate, how far does it crawl in one hour? Give your answer in metres.
  8. A hose fills 12 litres in 4 minutes. Write its rate in litres per minute, and then turned around, in seconds per litre.
Answers
  1. \(40 \div 2 = 20\) kilometres per hour.
  2. \(120 \div 6 = 20\) pages per day.
  3. \(96 \div 8 = 12\) minutes.
  4. A minute is 3 lots of 20 seconds (\(60 \div 20 = 3\)), so \(25 \times 3 = 75\) beats per minute.
  5. The dog: \(150 \div 30 = 5\) metres per second. The bike: \(200 \div 50 = 4\) metres per second. The dog is faster, even though the bike went further, because the bike took much longer.
  6. Rate: \(45 \div 3 = 15\) pages per minute. In 10 minutes: \(15 \times 10 = 150\) pages.
  7. Rate: \(30 \div 5 = 6\) centimetres per minute. One hour is 60 minutes: \(6 \times 60 = 360\) centimetres, which is 3.6 metres.
  8. \(12 \div 4 = 3\) litres per minute. Turned around: one litre takes \(\frac{1}{3}\) of a minute, and a third of 60 seconds is 20 seconds per litre. Check: 3 litres at 20 seconds each is 60 seconds, one minute. ✓

Common mistake

Dividing the wrong way round. For "36 kilometres in 3 hours", \(36 \div 3 = 12\) gives kilometres per hour. If you do \(3 \div 36\) you get hours per kilometre, a much smaller number that means something quite different. Say the units out loud: "kilometres per hour" means kilometres on top, hours underneath.

Mixing units. If one tap gives "2 litres per minute" and another gives "100 litres per hour", you cannot just compare 2 with 100. Change one of them first: 2 litres per minute is \(2 \times 60 = 120\) litres per hour, so the first tap is actually faster.

Where it leads