Connie is decorating cookies for the school bake sale. She has worked for 3 hours and completed 150 cookies. Her friend, Daniel, is also participating in the sale. He decorated 98 cookies in 2 hours. Which cookie decorator was faster, Connie or Daniel?

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

### Comparing Unit Rates

A **unit rate** is a comparison of two measurements, one of which has a value of 1. Given the number of units, the unit rate can be used to calculate a total rate. Unit rates can be used for comparison purposes.

Here is an example.

Felicia needs to buy sugar. She could buy a 16-ounce box of sugar for $1.12, or she could buy a 24-ounce box of sugar for $1.44. Which is the better buy? How much cheaper, in cents per ounce, is that buy?

First, find the unit price for the 16-oz box by writing a fraction.

\begin{align*}\frac{16}{1.12}\frac{ounces}{dollars}\end{align*}

Next, recognize the answer must be per ounce and reduce the fraction to its lowest terms by reducing ounces to 1.

Then, find the unit price per ounce for the 24-ounce box. Remember to keep the units consistent. In this case, ounces in the numerator and dollars in the denominator.

\begin{align*}\frac{24}{1.44}\frac{ounces}{dollars}\end{align*}

Next, reduce this fraction to its lowest terms.

\begin{align*}\frac{24\div 24}{1.44\div 24}=\frac{1}{.06}\frac{ounce}{dollars}\end{align*}

Then, compare the two unit rates.

1 ounce for $0.07 \begin{align*}>\end{align*} 1 ounce for $0.06

The 16-oz box is more expensive than the 24-oz box.

Next, subtract one unit price from the other to find the difference in price.

.07 - .06 = .01

The answer is that the best buy is the 24-oz box because it is $0.01 per ounce cheaper than the 16-oz box.

**Examples**

#### Example 1

Earlier, you were given a problem about Connie and Daniel, who are decorating cookies.

Connie completed 150 cookies in three hours, and it took Daniel 2 hours to decorate 98 cookies. Who is the faster cookie decorator?

First, write the fractions with consistent units.

**Connie**

**Daniel**

Next, reduce to lowest terms.

Connie decorated 50 cookies per hour.

\begin{align*}\frac{98}{2}=49\end{align*}

Daniel decorated 49 cookies per hour.

Then, compare.

50

#### Example 2

Compare: 3:1 and 6 to 2.

First, write 3:1 as a fraction.

Next, reduce to lowest terms.

\begin{align*}\frac{3}{1}=3\end{align*}

Then, write 6 to 2 as a fraction and reduce to lowest terms.

Next, compare the values.

3 = 3

The answer is that the rates are equal.

#### Example 3

Which rate is higher? 4 to 5 or \begin{align*}\frac{16}{20}\end{align*}

First, write the fractions,and reduce.

\begin{align*}\frac{4}{5}\end{align*}

\begin{align*}\frac{16}{20}=\frac{16\div 4}{20\div 4}=\frac{4}{5}\end{align*}

Next, compare.

The answer is that the rates are equal.

#### Example 4

Matt can melt 18 m&ms in his mouth without chewing in 3 minutes. It takes Kelly 5 minutes to melt 15. Who melts the m&ms faster, Matt or Kelly?

First, write the fractions.

**Matt **

**Kelly \begin{align*}\frac{15}{5}\frac{m\&ms}{minutes}\end{align*}**

Next, calculate each of the unit rates recognizing that the word "faster" applies to minutes and reducing that time to 1.

**Matt** \begin{align*}\frac{18}{3}=\frac{6}{1}\end{align*}

Matt can melt 6 m&ms in 1 minute.

**Kelly** \begin{align*}\frac{15}{5}=\frac{3}{1}\end{align*}

Kelly can melt 3 m&ms in 1 minute.

Then, compare the unit rates.

6

The answer is that Matt melts m&ms faster.

#### Example 5

Frank read 8 books in three weeks. It took Bonnie 4 weeks to read 10 books. Who is the faster reader?

First, write the fractions being sure to keep the units consistent.

**Frank** \begin{align*}\frac{8}{3}\frac{books}{weeks}\end{align*}

**Bonnie**

Next, reduce to unit rates by reducing the times to 1.Faster can refer to time, not books.

**Frank** \begin{align*}\frac{8}{3}=2.67\end{align*}

Frank read 2.67, or 2

### Review

Determine whether each is equivalent or not.

- 60 units in 3 minutes and 80 units in 4 minutes
- $16 for 8 pounds and $22 for 10 pounds
- 50 kilometers in 2 hours and 75 kilometers in 3 hours

Solve each problem.

- Max paid $45 for 15 gallons of gasoline. What was the cost per gallon of gasoline?
- Mr. Brown paid $8.28 for 12 cans of green beans. Express this cost as a unit price.
- A train travels 480 kilometers in 3 hours. Express this speed as a unit rate.
- Mrs. Jenkins paid $50 for 40 square feet of carpeting. What was the cost per square foot for the carpeting?
- A copy machine can produce 310 copies in 5 minute. How many copies can the machine produce per minute?

Nadia needs to buy some cheddar cheese. An 8-ounce package of cheddar cheese costs $2.40. A 12-ounce package of cheddar cheese costs $3.36.

- Find the unit price for the 8-ounce package.
- Find the unit price for the 12-ounce package.
- Which is the better buy? How many cents per ounce cheaper is the better buy?

Joe drove 141 miles in 3 hours. His cousin Amy drove 102 miles in 2 hours. Assume both cousins were driving at constant speeds.

- How fast was Joe driving, in miles per hour?
- How fast was Amy driving, in miles per hour?
- Who was driving at a faster rate of speed?
- How much faster was the faster person traveling?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.6.