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# Identification and Writing of Equivalent Rates

## Understand that equivalent rates are equivalent ratios.

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Practice Identification and Writing of Equivalent Rates
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Identification and Writing of Equivalent Rates

### Let's Think About It

Credit: Lola Audu
Source: https://www.flickr.com/photos/auduhomes/17029767859/in/
License: CC BY-NC 3.0

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 = 402mileshours=402\begin{align*}\frac{40}{2}\frac{miles}{hours}=\frac{40}{2}\end{align*}

80 miles per 4 hours = 804mileshours=804\begin{align*}\frac{80}{4}\frac{miles}{hours}=\frac{80}{4}\end{align*}

Next, reduce both fractions to their lowest terms.

402=40÷22÷2=201\begin{align*}\frac{40}{2}=\frac{40\div 2}{2\div 2}=\frac{20}{1}\end{align*}mileshour\begin{align*}\frac{miles}{hour}\end{align*}

804=80÷44÷4=201\begin{align*}\frac{80}{4}=\frac{80\div 4}{4\div 4}=\frac{20}{1}\end{align*}mileshour\begin{align*}\frac{miles}{hour}\end{align*}

Then, compare the fractions in their lowest terms. Remember to include the units.

20 miles/hour = 20 miles/hour

The answer is that the rates are equivalent. Both cars traveled at a rate of 20 miles per hour.

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.

For example:

gumdrops per person is not the same as persons per gumdrop

gumdropspersonpersonsgumdrop\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.

53meterssecond\begin{align*}\frac{5}{3}\frac{meters}{second}\end{align*} and  2018metersseconds\begin{align*}\frac{20}{18}\frac{meters}{seconds}\end{align*}

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

53=2018\begin{align*}\frac{5}{3}=\frac{20}{18}\end{align*}

Then cross multiply.

5 x 18 = 3 x 20

9060\begin{align*}90\neq 60\end{align*}

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.

39\begin{align*}\frac{3}{9}\end{align*} and 618\begin{align*}\frac{6}{18}\end{align*}

Next, determine a method. Multiply the first fraction times 2 to convert to, and work with, 18ths.

39=3×29×2=618\begin{align*}\frac{3}{9}=\frac{3\times 2}{9\times 2}=\frac{6}{18}\end{align*}

Then, compare the fractions.

618feetseconds=618feetseconds\begin{align*}\frac{6}{18}\frac{feet}{seconds}=\frac{6}{18}\frac{feet}{seconds}\end{align*}

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.

530milesminutes\begin{align*}\frac{5}{30}\frac{miles}{minutes}\end{align*}and  642milesminutes\begin{align*}\frac{6}{42}\frac{miles}{minutes}\end{align*}

Next, determine a method. Reduce both rates to their lowest terms.

530=5÷530÷5=16\begin{align*}\frac{5}{30}=\frac{5\div 5}{30\div 5}=\frac{1}{6}\end{align*}mileminutes\begin{align*}\frac{mile}{minutes}\end{align*}

642=6÷642÷6=17\begin{align*}\frac{6}{42}=\frac{6\div 6}{42\div 6}=\frac{1}{7}\end{align*}mileminutes\begin{align*}\frac{mile}{minutes}\end{align*}

Then, compare the fractions.

1617\begin{align*}\frac{1}{6}\neq \frac{1}{7}\end{align*}

The answer is that the rates are not equal.

#### Example 3

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

First, write the rates as fractions.

520poundsdollars\begin{align*}\frac{5}{20}\frac{pounds}{dollars}\end{align*}  and  832poundsdollars\begin{align*}\frac{8}{32}\frac{pounds}{dollars}\end{align*}

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

520=5÷520÷5=14\begin{align*}\frac{5}{20}=\frac{5\div 5}{20\div 5}=\frac{1}{4}\end{align*}poundsdollars\begin{align*}\frac{pounds}{dollars}\end{align*}

832=8÷832÷8=14\begin{align*}\frac{8}{32}=\frac{8\div 8}{32\div 8}=\frac{1}{4}\end{align*}poundsdollars\begin{align*}\frac{pounds}{dollars}\end{align*}

Then, compare the rates.

14=14\begin{align*}\frac{1}{4}=\frac{1}{4}\end{align*}

The answer is that the rates are equivalent.

### 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 25 balls for$10 or pay 25 cents per ba

### Vocabulary Language: English

Denominator

Denominator

The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. $\frac{5}{8}$ has denominator $8$.
Equivalent

Equivalent

Equivalent means equal in value or meaning.

1. [1]^ Credit: Lola Audu; Source: https://www.flickr.com/photos/auduhomes/17029767859/in/; License: CC BY-NC 3.0
2. [2]^ License: CC BY-NC 3.0