A rate is a relationship between two quantities with different units. Some common examples of rates include speed, written as kilometres per hour or km/h, and heart rate, written as beats per minute or bpm.
A rate will indicate the amount that one quantity changes relative to another quantity.
For example, speed measures how distance changes over time. A speed of $60$60 km/h indicates that in $1$1 hour, a vehicle will travel $60$60 km. In $2$2 hours, it will travel $120$120 km, and so on.
As you might expect, if we know one quantity along with the rate, then we can find the other quantity.
We can also state the value of a rate in many different ways depending on what units we use. If a car travels at a speed of $60$60 km/h, there should be a way to express this speed in terms of km/min and m/s, because these different units all measure distance over time.
Exploration
A wheel of a car rotates $9000$9000 times for every $20$20 kilometres travelled. How many times does the wheel rotate per kilometre? We can find this by dividing the number of rotations by the number of kilometres.
| Rate | $=$= | $\frac{9000}{20}$900020 rotations per kilometre | Express as a rate. |
| $=$= | $450$450 rotations per kilometre | Simplify the fraction. |
The rate of $450$450 rotations/km is what we call a unit rate, because the second quantity is $1$1 km. Unit rates give us a clear idea of how the quantities are changing, and can make calculations much easier.
So we can say that the car wheel rotates $450$450 times every kilometre it travels. But what if the car travels $10$10 km? How many times does the wheel rotate for that distance and not just $1$1 kilometre? Let's look at two approaches to solving this.
- Notice that $10$10 km is half of $20$20 km. We know the wheel rotates $9000$9000 times per $20$20 kilometres, so we would expect half the number of rotations in half the distance. In other words, the wheel rotates $\frac{9000}{2}=4500$90002=4500 times per $10$10 km. This works out nicely because $20$20 is a multiple of $10$10, but can be a little more difficult if we had chosen, say, $11$11 kilometres instead.
- The alternative method is to use the unit rate. We know that the wheel rotates $450$450 times for every $1$1 km. So we expect that over $10$10 km, the wheel will rotate $10$10 times this amount. In other words, the wheel rotates $450\times10=4500$450×10=4500 times per $10$10 km. In $11$11 kilometres, the wheel rotates $450\times11=4950$450×11=4950 times.
But what happens if we want to find the number of rotations per metre? Intuition tells us that the number of rotations over a metre should be less than in a kilometre. So we're hoping to get an answer less than $450$450 rotations per metre!
To find the number of rotations per metre, we first note that there are $1000$1000 metres in a single kilometre. So, we divide the number of rotations in a kilometre by $1000$1000 as follows:
| Rate | $=$= | $\frac{450}{1000}$4501000 rotations per metre | |
| $=$= | $0.52$0.52 rotations per metre |
This means that if the wheel turns about half way, the car travels approximately $1$1 metre.
Converting between rates
Sometimes rates can be measured using different units.
For example, speed can be measured in km/h, m/s or m/h and the units we use will depend on what is reasonable for the speed we are measuring.
A car travelling down the freeway is going to cover many kilometres each hour, so the reasonable units to use would be km/h.
A 200-metre sprinter will cover many metres over a smaller period of time, so metres/second is a more reasonable unit to describe the speed.
But if we want to compare the speed of a car travelling 40 km/h and a sprinter who runs at 10 m/s, we need to convert one of these rates to the units of the other.
| 1. | If we convert $10$10 m/s to km/h: | ||||
| Changing the denominator unit: | $10$10 metres each second | $=$= | $10\times3600$10×3600 metres each hour | ||
| Changing the numerator unit: | $36000$36000 m | $=$= | $\left(36000\div1000\right)$(36000÷1000) km | ||
| $=$= | $36000$36000 metres each hour | ||||
| $=$= | $36$36 km | ||||
| So a rate of $10$10 m/s is equivalent to a rate of $36$36 km/h. | |||||
| Now we can compare the two speeds in km/h: | |||||
| Speed of car | $=$= | $40$40 km/h | |||
| Speed of sprinter | $=$= | $36$36 km/h | |||
| 2. | If we convert $40$40 km/h to m/s: | ||||||
| Changing the denominator unit: | |||||||
| 1 hour needs to be converted to seconds: | $1$1 hour | $=$= | $60$60 minutes | $=$= | $3600$3600 seconds | ||
| $=$= | $36000$36000 metres each hour | ||||||
| $=$= | $36$36 km | ||||||
| So a rate of $10$10 m/s is equivalent to a rate of $36$36 km/h. | |||||||
| Now we can compare the two speeds in km/h: | |||||||
| Speed of car | $=$= | $40$40 km/h | |||||
| Speed of sprinter | $=$= | $36$36 km/h | |||||
Practice questions
question 1
Gwen is trying to save money and deposits $\$4.20$$4.20 into her bank account for every $7$7 coffees she purchases. What is her rate of depositing in terms of cents per coffee?
question 2
There are $224$224 tissues per box. If the manufacturer wastes $1120$1120 tissues in production, how many boxes of tissues is this?
question 3
Valerie counts $12$12 heart beats over a $10$10 second interval. What is her heart rate in beats per minute (bpm)?