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5.5: Unit Rates

Difficulty Level: At Grade Created by: CK-12
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Hannah is a runner on her school's track team. In order to perform well in races, she trains nearly every day. The running coach requires that all athletes run at least 35 miles per week. So far this week, Hannah ran 3 miles on Sunday, 6 miles on Monday, and 5 miles on Tuesday. At her current rate, will Hannah meet the requirement?

In this concept, you will learn how to work with unit rates. 

Working with Unit Rates

A unit rate is a comparison of two measurements, one of which has a value of 1. 

For example:

The cost for 1 pound of apples is $1.50

This can be written as  \begin{align*}\frac{1}{1.50}\frac{pound}{dollars}\end{align*}11.50pounddollars

 Based on this unit rate, the cost for any number of pounds of apples can be calculated. Given the number of units, the unit rate can be used to calculate a total rate. The unit rate can be calculated by dividing one term of a fraction by the other, and reducing the desired term to 1 . 

Here is an example. 

A cyclist pedaled 36 miles in 2 hours. What is her unit rate? In other words, how many miles did she pedal in 1 hour?

First, write the rate as a fraction. The term that needs to be reduced to 1 is hours in order to get a distance per 1 hour. In this case, the term to be reduced to 1 is the denominator.

\begin{align*}\frac{36}{2}\frac{miles}{hours}\end{align*}362mileshours

Next, divide both the numerator and denominator by 2 to get a denominator of 1.

A unit rate must have 1 in one of its terms. It can be either the numerator or the denominator.

 \begin{align*}\frac{36\div 2}{2\div 2}=\frac{18}{1}\frac{miles}{hour}\end{align*}36÷22÷2=181mileshour

 The answer is 18 miles/hour.

Examples

Example 1

Earlier, you were given a problem about Hannah, who is on the school's track team.

The coach requires all runners to complete 35 miles per week. So far Hannah has run 4 miles on Sunday, 6 miles on Monday, and 5 miles on Tuesday. At her current rate, will Hannah meet the requirement?

First, recognize that the unit rate (miles per day) needs to be found first in order to determine whether or not Hannah will meet her goal.

Next, add the number of miles together that Hannah has already run.

4 + 6 + 5 = 15

Hannah has run a total of 15 miles in 3 days this week.

Then, write a fraction that compares the number of miles Hannah has already run to the number of days she ran to get her unit rate per day.

\begin{align*}\frac{15}{3}\frac{miles}{days}\end{align*}153milesdays

Next, reduce to lowest terms.

\begin{align*}\frac{15\div 3}{3\div 3}=\frac{5}{1}\end{align*}15÷33÷3=51

Hannah has run 5 miles per day.

Then, think. If Hannah runs 5 miles per day, how many miles will she run in a week?

\begin{align*}\frac{5}{1}\frac{miles}{day}\times \frac{7}{1}\frac{days}{week}=\frac{35}{1}\frac{miles}{week}\end{align*}51milesday×71daysweek=351milesweek

The answer is that at her current rate of 5 miles per day, Hannah will run 35 miles per week and will meet the coach's requirements.

Example 2

Thomas has the record among all his friends for being the fastest texter. Yesterday, the boys decided to find out just how fast. Irwin gave him a message with 240 characters in it, and set a timer. Thomas completed the text in exactly 3 minutes. What is his texting rate per minute?

First, write a fraction.

 \begin{align*}\frac{240}{3}\frac{characters}{minutes}\end{align*}2403charactersminutes

Next, recognize that the problem is asking for a rate per minute and reduce the fraction to its lowest terms by reducing the number of minutes to 1.

 \begin{align*}\frac{240\div 3}{3\div 3}=\frac{80}{1}\end{align*}240÷33÷3=801

The answer is 80 characters per minute. 

Example 3

Nathaniel's car gets 45 miles for 3 gallons of gasoline. How many miles does he get per gallon?

First, write a fraction. It does not matter which value is on top.

\begin{align*}\frac{45}{3}\frac{miles}{gallons}\end{align*}453milesgallons

Next, recognize which value needs to be reduced to 1. In this case, "per gallon" signals that gallons must be reduced to 1.  3 gallons must be divided by 3 to get 1.

\begin{align*}\frac{45\div 3}{3\div 3}=\frac{15}{1}\end{align*}45÷33÷3=151

The answer is 15 miles per gallon.

Example 4

Cassandra has a part time job after school in her mom's office. She spent 6 hours this week filing a stack of 850 folders. How many folders did she file each hour?

First, write a fraction showing the relationship between hours and folders. Remember, it does not matter which is in the numerator or which is in the denominator.

 \begin{align*}\frac{6}{850}\frac{hours}{folders}\end{align*}6850hoursfolders

Next, reduce to lowest terms based on which value the problem calls for. In this problem, it is per hour.

 \begin{align*}\frac{6\div 6}{850\div 6}=\frac{1}{142}\end{align*}6÷6850÷6=1142

The answer is 142 folders per hour.

Example 5

Find the unit rate for 1 minute: 18 inches in 2 minutes

First, write a fraction.

\begin{align*}\frac{18}{2}\frac{inches}{minutes}\end{align*}182inchesminutes

Next, reduce to lowest terms.

 \begin{align*}\frac{18\div 2}{2\div 2}=\frac{9}{1}\end{align*}18÷22÷2=91

The answer is 9 inches per minute. 

Review

Write a unit rate for each ratio.

  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
  11. 14 pounds for $7.00
  12. 18 miles in 3 hours
  13. 21 inches of cloth costs $7.00
  14. 45 miles on 3 gallons of gasoline
  15. 200 miles in four hours

Review (Answers)

To see the Review answers, open this PDF file and look for section 5.5.

Resources

 

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Difficulty Level:
At Grade

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Date Created:
Aug 10, 2015
Last Modified:
Mar 23, 2016
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