5.04 Rates
Rates
A rate is a measure of how quickly one measurement changes with respect to another. Some commonly used rates in our everyday lives are speed, which measures distance per time, and the price of food, which is often measured in dollars per kilogram.
When we combine two different units into a single compound unit we call this a unit rate. We can write these compound units using a slash ( / ) between the different units, so "metres per second" becomes "\text{m/s}" and "dollars per kilogram" becomes "\text{\$/kg}". This compound unit represents the division of one measurement by another to get a rate.
Not all compound units are written using a slash and instead use the letter "p" to represent "per". For example, "beats per minute" uses the compound unit bpm and "frames per second" uses fps.
Rates have two components, the numeric value and the compound unit. The compound unit tells us which units are being measured and the numeric value tells us how quickly the numerator unit changes with respect to the denominator unit. When constructing a rate we usually start with just a fraction of measurements. Whenever we can, simplify the fraction to get a whole number value for the rate.
Examples
Example 1
Write the following as a unit rate:
91 people per 7 buses
A rate is a measure of how quickly one measurement changes with respect to another.
When we combine two different units into a single compound unit we call this a unit rate.
Apply rates
Now that we know how to make our rates, it's time to use them. Rates are very similar to ratios in that we can use them to calculate how much one measurement changes based on the change in another.
A rate of 10 metres per second (10 \text{ m/s}) is not the same as a rate of 10 seconds per metre (10\text{ s/m}).
In fact, 10 \text{ m/s}=\dfrac{1}{10} \text{ s/m}. When we flip the compound unit we also need to take the reciprocal of the numeric value.
Examples
Example 2
On a road trip, Tracy drives with an average speed of 90 \text{ km/hr}. How far does she travel in 8 hours?
Rates are very similar to ratios in that we can use them to calculate how much one measurement changes based on the change in another.
Convert rates
When applying rates it's important to make sure that we are applying the right one.
Examples
Example 3
Consider the following rate:
192 metres per 240 seconds
Express this is as a unit rate in terms of metres and seconds.
Express this is as a unit rate in terms of metres and minutes.
Before we apply rates, we should convert the units to the indicated unit rate.