# Comparison of Unit Rates

## Use <, > or = to compare unit rates.

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Comparison of Unit Rates

Mark adopted a dog today and is at the pet store looking for dog food. He researched the different brands before coming to the store and now he just needs to decide between two brands. Doggie Delights costs $46 for a 20 pound bag. Nature’s Choice costs$55 for a 25 pound bag. Mark wants to compare the cost of dog food per pound to find the better deal. Which bag should Mark buy?

In this concept, you will learn to write and compare unit rates.

### Comparing Unit Rates

Equivalent rates can be used to compare different sets of quantities that have the same value. A rate that compares a quantity to one is called a unit rate. The unit rate has a denominator equal to one when written as a fraction. Unit rates can be used to find larger equivalent rates.

Let’s use a unit rate to solve a problem.

Mrs. Harris’ class went apple picking. Each student picked 8 apples. At this rate, how many apples were picked by 7 students?

The problem gives the unit rate of 8 apples per student. Solve by finding the equivalent rate where the number of students is 7.

First, write the unit rate as a fraction.

\begin{align*}\frac{8\text{ apples}}{1 \text{ student}}\end{align*}

Next, multiply the numerator and denominator by 7 to find the equivalent rate for 7 students.

\begin{align*}\frac{8\text{ apples} \times 7}{1 \text{ student} \times 7} = \frac{56\text{ apples}}{7 \text{ students}}\end{align*}

There were 56 apples picked by 7 students.

Now let’s solve to find the unit rate.

Laquita picked 12 peaches in 6 minutes. How many peaches did Laquita pick per minute?

The problem gives the larger rate of 12 peaches per 6 minutes. Simplify the rate so the denominator is equal to 1 to find the unit rate.

First, write the rate as a fraction.

\begin{align*}\frac{12\text{ peaches}}{6 \text{ minutes}}\end{align*}

Next, simplify the denominator to equal 1. Divide the numerator and denominator by 6.

\begin{align*}\frac{12\text{ peaches} \div 6}{6 \text{ minutes} \div 6} = \frac{2\text{ peaches}}{1 \text{ minute}}\end{align*}

Laquita picked 2 peaches per minute.

### Examples

#### Example 1

Earlier, you were given a problem about Mark at the pet store.

Mark is trying to make a choice. Doggie Delights costs $46 for a 20 pound bag and Nature’s Choice costs$55 for a 25 pound bag. To find the better deal, Mark must compare the price per pound.

First, find the unit price for each brand.

\begin{align*}\begin{array}{rcl} \text{Doggie Delights} & = & \frac{\46}{20 \text{ pound}} = \frac{46 \div 20}{20 \div 20}=\frac{2.30}{1}\\ \text{Nature’s Choice} & = & \frac{\55}{25 \text{ pound}} = \frac{55 \div 25}{25 \div 25}=\frac{2.20}{1} \end{array}\end{align*}

Next, compare the price per pound. Doggie Delights is $2.30 per pound. Nature’s Choice is$2.20 per pound.

\begin{align*}\ 2.30 > \ 2.20\end{align*}

Nature’s Choice is cheaper than Doggie Delights.

Mark decides to buy Nature’s Choice.

#### Example 2

Find the unit rate for the following problem: Harold cuts 7 lawns in 4 hours. How many lawns does Harold cut per hour?

First, write the rate as a fraction.

\begin{align*}\frac{7\text{ lawns}}{6 \text{ hours}}\end{align*}

Next, simplify the denominator to equal 1. Divide the numerator and denominator by 4.

\begin{align*}\frac{7\text{ lawns} \div 4}{4 \text{ hours} \div 4} = \frac{1.75\text{ lawns}}{1 \text{ hour}}\end{align*}

Harold cuts 1.75 lawns per hour.

#### Example 3

Find the unit rate for the following problem: 24 buttons on 4 shirts.

First, write the rate as a fraction.

\begin{align*}\frac{24\text{ buttons}}{4 \text{ shirts}}\end{align*}

Next, divide the numerator and denominator by 4.

\begin{align*}\frac{24\text{ buttons} \div 4}{4 \text{ shirts} \div 4} = \frac{6\text{ buttons}}{1 \text{ shirt}}\end{align*}

The unit rate is 6 buttons per shirt.

#### Example 4

Find the unit rate for the following problem: 4 ice cream cones for 2 people.

First, write the rate as a fraction.

\begin{align*}\frac{4\text{ ice cream cones}}{2 \text{ people}}\end{align*}

Next, divide the numerator and denominator by 2.

\begin{align*}\frac{4\text{ ice cream cones} \div 2}{2 \text{ people} \div 2}=\frac{2\text{ ice cream cones}}{1 \text{ person}}\end{align*}

The unit rate is 2 ice cream cones per person.

#### Example 5

Find the unit rate for the following problem: 45 gallons in 3 miles.

First, write the rate as a fraction.

\begin{align*}\frac{45 \text{ miles}}{3 \text{ gallons}}\end{align*}

Next, divide the numerator and denominator by 3.

\begin{align*}\frac{45\text{ miles} \div 3}{3 \text{ gallons} \div 3}=\frac{15\text{ miles}}{1 \text{ gallon}}\end{align*}

The unit rate is 15 miles per gallon.

### Review

Find the unit rate.

1. Fourteen apples in two barrels
2. Thirty-two crayons in two boxes
3. Eighteen bottles in three carriers
4. Twenty students on four teams
5. Twenty-five students on five teams
6. Fifty students in two classes
7. Ninety students on three buses
8. Thirteen students ate twenty-six cupcakes
9. Twelve campers in two tents
10. Forty- eight hikers on two trails
11. 16 hikers on 4 trails
12. 72 bikes on 6 racks
13. 15 backpacks for 5 children
14. 28 slices of pizza for 7 teenagers
15. \$24.00 for 2 people

To see the Review answers, open this PDF file and look for section 8.5.

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### Vocabulary Language: English

TermDefinition
Rate A rate is a special kind of ratio that compares two quantities.
Unit Rate A unit rate is a ratio that compares a quantity to one. The word “per” is a key word with unit rates.