3.01 Unit rates

Lesson

Practice

Introduction

There are many situations where ratios can be useful in daily life. We have learned that ratios can be used to compare any two numbers. This time, we will extend our understanding of ratios as we explore using unit rates with fractions.

Ideas

Unit rates

Unit rates

Remember that 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 as a quantity of 1, such as 2 feet per second or 5 miles per hour, they are called unit rates. When we're asked to determine a rate, we are most often being asked for the unit rate.

If we are given rates that are expressed as fractions, we can calculate a unit rate in the same way that we have done in the past. For example, a recipe might call for 1\dfrac{1}{2} cups of water for every \dfrac{1}{4} cup of sugar. Let's look at how we can find the unit rate to represent the amount of water we need to use per 1 cup of sugar:

\displaystyle \text{Rate}	\displaystyle =	\displaystyle \dfrac{1\frac{1}{2}}{\frac{1}{4}}\dfrac{\text{cups}}{\text{cups}}	Write the fractions as a fraction
	\displaystyle =	\displaystyle \dfrac{\frac{3}{2}}{\frac{1}{4}} \dfrac{\text{cups}}{\text{cups}}	Convert the mixed numbers to improper fractions
	\displaystyle =	\displaystyle \dfrac{3}{2} \times \dfrac{4}{1}\dfrac{\text{cups}}{\text{cups}}	Multiply by the reciprocal
	\displaystyle =	\displaystyle \dfrac{12}{2}\dfrac{\text{cups}}{\text{cups}}	Multiply
\displaystyle \text{Rate}	\displaystyle =	\displaystyle 6	Simplify

This means that the recipe calls for 6 cups of water for every 1 cup of sugar.

Examples

Example 1

Diana uses a cake recipe which calls for 1\dfrac{3}{4} teaspoons of baking powder for every 1\dfrac{1}{2} cups of flour.

What is the unit rate of the amount of baking powder to be used per cup of flour?

Worked Solution

Create a strategy

Find the unit rate in teaspoons per cup using the fact that 1\dfrac{3}{4} teaspoons (tsp) of baking powder is to 1\dfrac{1}{2} cups of flour.

Change mixed numbers to improper fractions to make the numbers easier to work with.

Apply the idea

\displaystyle \text{Rate}	\displaystyle =	\displaystyle \dfrac{1\frac{3}{4} \text{tsp}} {1\frac{1}{2} \text{ cups}}
	\displaystyle =	\displaystyle \dfrac{\frac{7}{4} \text{ tsp}}{\frac{3}{2}\text{cups}}	Convert the mixed numbers to improper fractions
	\displaystyle =	\displaystyle \dfrac{7}{4}\times \dfrac{2}{3} \dfrac{\text{tsp}}{\text{cups}}	Mutiply the reciprocal of the divisor
	\displaystyle =	\displaystyle \dfrac{7}{6} \dfrac{\text{tsp}}{\text{cups}}	Simplify
	\displaystyle =	\displaystyle 1\dfrac{1}{6} \dfrac{\text{tsp}}{\text{cup}}	Convert to mixed number

The unit rate is 1\dfrac{1}{6} \text{ tsp/cup}. Therefore, Diana needs to use 1\dfrac{1}{6} \text{ tsp} of baking powder for every 1 cup of flour.

Idea summary

A rate is a measure of how quickly one measurement changes with respect to another.

When rates are expressed as a quantity of 1, such as 2 feet per second or 5 miles per hour, they are called unit rates.

Outcomes

7.RP.A.1

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.