### Let's Think About It

Keling is a good golfer. He is trying out for the golf team at school next week, so he headed out for some practice. The pro at the driving range told Keling that he could get a bucket of 45 balls for $9 or pay $1 for 4 balls at a time. Are these two prices at the same rate? If not, which rate is the better deal?

In this concept, you will learn to identify and write equivalent rates.

### Guidance

A **rate** is a ratio that compares quantities in different units. The word "per" is used when talking about rates and is sometimes abbreviated with a slash, /.

Some common rates are used all the time, like miles per hour, dollars per gallon, and days per week. Others are more specific to an occasion, like cats per household or chocolate chips per cookie.

Two rates are **equivalent** if they show the same relationship between two identical units of measure.

The same strategies used to find equivalent ratios can be used to find equivalent rates.

Here is an example.

Two cars are traveling. One car goes 40 miles in 2 hours, and the other goes 80 miles in 4 hours. Determine whether or not the rates are equivalent.

First, write the rates as fractions. Remember to make sure the terms are the same, in this case, miles per hour.

40 miles per 2 hours = \begin{align*}\frac{40}{2}\frac{miles}{hours}=\frac{40}{2}\end{align*}

80 miles per 4 hours =

Other methods that can be used to determine whether or not rates are equal include changing one or both fractions so that the denominators are equal, and cross multiplying.

Cross multiplication does not yield a rate. Only the numerical values are used as a check.

When using rates, it does not matter which unit is in the numerator and which is in the denominator, as long as they match one another.

\begin{align*}\frac{gumdrops}{person}\neq \frac{persons}{gumdrop}\end{align*}

### Guided Practice

Determine if two machines are winding thread at the same speed. One machine winds at a rate of 5 meters every 3 seconds. The other takes 18 seconds to wind 20 meters.

First, write the rates as fractions, making sure the units are identical.

\begin{align*}\frac{5}{3}\frac{meters}{second}\end{align*} and \begin{align*}\frac{20}{18}\frac{meters}{seconds}\end{align*}

Next, assume the rates are equal to one another and write an equation.

\begin{align*}\frac{5}{3}=\frac{20}{18}\end{align*}

Then cross multiply.

5 x 18 = 3 x 20

The answer is that the rates are not equivalent.

### Examples

#### Example 1

Are the rates equivalent? 3 feet in 9 seconds and 6 feet in 18 seconds

First, write the rates as fractions.

\begin{align*}\frac{3}{9}\end{align*} and \begin{align*}\frac{6}{18}\end{align*}

Next, determine a method. Multiply

Then, compare the fractions.

The answer is that the rates are equivalent.

#### Example 2

Compare the rates: 5 miles in 30 minutes and 42 minutes to go 6 miles

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

\begin{align*}\frac{5}{30}\frac{miles}{minutes}\end{align*}and

#### Example 3

Compare 5 pounds for $20.00 and 8 pounds for $32.00

First, write the rates as fractions.

\begin{align*}\frac{5}{20}\frac{pounds}{dollars}\end{align*} and \begin{align*}\frac{8}{32}\frac{pounds}{dollars}\end{align*}

Next, determine a method. Reduce each fraction to its lowest terms.

\begin{align*}\frac{pounds}{dollars}\end{align*}

\begin{align*}\frac{8}{32}=\frac{8\div 8}{32\div 8}=\frac{1}{4}\end{align*}

Then, compare the rates.

\begin{align*}\frac{1}{4}=\frac{1}{4}\end{align*}

The answer is that the rates are equivalent.

\begin{align*}\frac{1}{4}\frac{pound}{dollars}\end{align*}can also be written \begin{align*}\frac{4}{1}\frac{dollars}{pound}\end{align*}or $4 per pound.

### Follow Up

Remember Keling at the driving range? He was trying to figure out if he should get a bucket of 45 balls for $9 or pay $1 for 4 balls at a time.

### Video Review

### Explore More

Write an equivalent rate for each rate.

1. 2 for $10.00

2. 3 for $15.00

3. 5 gallons for $12.50

4. 16 pounds for $40.00

5. 18 inches for $2.00

6. 5 pounds of blueberries for $20.00

7. 40 miles in 80 minutes

8. 20 miles in 4 hours

9. 10 feet in 2 minutes

10. 12 pounds in 6 weeks

Write each rate as a fraction and simplify it.

11. 2 for $10.00

12. 3 for $15.00

13. 5 gallons for $25.00

14. 40 pounds for $80.00

15. 18 inches for $2.00