### Let's Think About It

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.

### Guidance

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.

###
**Guided Practice**

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.

###
**Examples**

#### Example 1

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 2

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 3**

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

### Follow Up

Remember Connie and Daniel and the cookie decorating?

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

### Video Review

### Explore More

Determine whether each is equivalent or not.

1. 60 units in 3 minutes and 80 units in 4 minutes

2. $16 for 8 pounds and $22 for 10 pounds

3. 50 kilometers in 2 hours and 75 kilometers in 3 hours

Solve each problem.

4. Max paid $45 for 15 gallons of gasoline. What was the cost per gallon of gasoline?

5. Mr. Brown paid $8.28 for 12 cans of green beans. Express this cost as a unit price.

6. A train travels 480 kilometers in 3 hours. Express this speed as a unit rate.

7. Mrs. Jenkins paid $50 for 40 square feet of carpeting. What was the cost per square foot for the carpeting?

8. 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.

9. Find the unit price for the 8-ounce package.

10. Find the unit price for the 12-ounce package.

11. 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.

12. How fast was Joe driving, in miles per hour?

13. How fast was Amy driving, in miles per hour?

14. Who was driving at a faster rate of speed?

15. How much faster was the faster person traveling?