Rates and Unit Rates: Speed, Best Buys and Per-One Comparisons
What makes a rate different from a ratio, how to find a unit rate, the speed–distance–time relationship, and how unit prices settle which deal is better.
A ratio compares quantities of the same kind, like cups to cups. A rate compares quantities of different kinds: kilometres per hour, pounds per kilogram, words per minute, beats per second.
The most useful form is the unit rate, which tells you how much of one quantity goes with one of the other. The word to listen for is per, which means "for each".
Divide both quantities by the same number until the second one is 1.
Speed, distance and time
Speed is the rate at which distance is covered. Rearranging the same relationship gives distance and time too.
At a steady km/h, distance grows by every hour. The unit rate is the slope of this line.
Keep the units consistent. km in minutes is km/h, not km/h. Convert minutes to hours first.
Best buys
Shops sell the same thing in different sizes. Dividing price by quantity gives the unit price, and the lower unit price is the better deal.
Which is better value?
You can divide the other way, too: apples per pound. Then the higher number is the better deal. Just be clear which way round you went.
Using a unit rate
Once you have the rate for one, multiply to get any amount you like. This is often called the unitary method.
A printer prints pages in minutes. How many pages in minutes?
- 1
Unit rate.
A constant rate is the slope of a straight-line graph. When the rate changes, as with a car speeding up, calculus measures the rate at each instant.
Common questions
- What is the difference between a rate and a ratio?
A ratio compares quantities of the same kind, like boys to girls. A rate compares different kinds of quantity, like kilometres to hours, so it always has units.
- How do I find a unit rate?
Divide the first quantity by the second. for apples gives per apple.
The Rates & Unit Rates lesson has worked examples and unlimited practice that adapts to you.