4.05 Find the unit rate

A rate is a measure of how quickly one measurement changes with respect to another. A commonly used rate in our everyday lives is speed, which is measured in distance over time.

Rates are measured by combining two different units into a single compound unit. We can write these compound units using a slash ( / ) between the different units, so "meters per second" becomes "\text{m/s}".

This compound unit represents the division of one measurement by another to get a rate. When rates are expressed such that the quantity of the denominator is 1, such as 2 feet per 1 second or 5 miles per 1 hour, they are called unit rates. We usually shorten this by not writing the number '1', such as 2 feet per second or 5 miles per hour. When we're asked to determine a rate, we are most often being asked for the unit rate.

Consider an Olympic sprinter who runs 100 meters in 10 seconds. Let's represent that as a rate:

\text{Sprinter's speed}= 100 \text{ m}/ 10 \text{ s}

We want to know how fast he can run in a single second. We can find how far the sprinter runs in 1 second by dividing the 100 meters evenly between the 10 seconds.

\text{Sprinter's speed}=\dfrac{100}{10} \text{m/s}

This calculation tells us that the sprinter runs 10 meters in one second.

We can write this as a unit rate for the sprinter's speed in meters per second using the compound unit m/s to give us:\text{Sprinter's speed}=10\text{ m/s}

Whenever we can, simplify the fraction to get the unit rate. This is much nicer to work with as we can now say that speed is 10 meters per second, rather than 100 meters per 10 seconds.

We can simplify the fraction to get the unit rate.

Let's look at the following examples on how to find the unit rate.

Examples