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Study Guides > Prealgebra

Writing Rates and Calculating Unit Rates

Learning Outcomes

  • Write a rate as a fraction
  • Calculate a unit rate
  • Calculate a unit price
 

Write a Rate as a Fraction

Frequently we want to compare two different types of measurements, such as miles to gallons. To make this comparison, we use a rate. Examples of rates are 120120 miles in 22 hours, 160160 words in 44 minutes, and $5\text{\$5} dollars per 6464 ounces.

Rate

A rate compares two quantities of different units. A rate is usually written as a fraction.
When writing a fraction as a rate, we put the first given amount with its units in the numerator and the second amount with its units in the denominator. When rates are simplified, the units remain in the numerator and denominator.

example

Bob drove his car 525525 miles in 99 hours. Write this rate as a fraction. Solution
525 miles in 9 hours\text{525 miles in 9 hours}
Write as a fraction, with 525525 miles in the numerator and 99 hours in the denominator. 525 miles9 hours\frac{\text{525 miles}}{\text{9 hours}}
175 miles3 hours\frac{\text{175 miles}}{\text{3 hours}}
So 525525 miles in 99 hours is equivalent to 175 miles3 hours\frac{\text{175 miles}}{\text{3 hours}}.
 

try it

[ohm_question]146614[/ohm_question]
 

Find Unit Rates

In the last example, we calculated that Bob was driving at a rate of 175 miles3 hours\frac{\text{175 miles}}{\text{3 hours}}. This tells us that every three hours, Bob will travel 175175 miles. This is correct, but not very useful. We usually want the rate to reflect the number of miles in one hour. A rate that has a denominator of 11 unit is referred to as a unit rate.

Unit Rate

A unit rate is a rate with denominator of 11 unit.
Unit rates are very common in our lives. For example, when we say that we are driving at a speed of 6868 miles per hour we mean that we travel 6868 miles in 11 hour. We would write this rate as 6868 miles/hour (read 6868 miles per hour). The common abbreviation for this is 6868 mph. Note that when no number is written before a unit, it is assumed to be 11. So 6868 miles/hour really means 68 miles/1 hour.\text{68 miles/1 hour.} Two rates we often use when driving can be written in different forms, as shown:
Example Rate Write Abbreviate Read
6868 miles in 11 hour 68 miles1 hour\frac{\text{68 miles}}{\text{1 hour}} 6868 miles/hour 6868 mph 68 miles per hour\text{68 miles per hour}
3636 miles to 11 gallon 36 miles1 gallon\frac{\text{36 miles}}{\text{1 gallon}} 3636 miles/gallon 3636 mpg 36 miles per gallon\text{36 miles per gallon}
Another example of unit rate that you may already know about is hourly pay rate. It is usually expressed as the amount of money earned for one hour of work. For example, if you are paid $12.50\text{\$12.50} for each hour you work, you could write that your hourly (unit) pay rate is $12.50/hour\text{\$12.50/hour} (read $12.50\text{\$12.50} per hour.) To convert a rate to a unit rate, we divide the numerator by the denominator. This gives us a denominator of 11.

example

Anita was paid $384\text{\$384} last week for working 32 hours\text{32 hours}. What is Anita’s hourly pay rate?

Answer: Solution

Start with a rate of dollars to hours. Then divide. $384\text{\$384} last week for 3232 hours.
Write as a rate. \frac{$384}{\text{32 hours}}
Divide the numerator by the denominator. \frac{$12}{\text{1 hour}}
Rewrite as a rate. $12/\text{hour}
Anita’s hourly pay rate is $12\text{\$12} per hour.

 

try it

[ohm_question]146615[/ohm_question]
   

example

Sven drives his car 455455 miles, using 1414 gallons of gasoline. How many miles per gallon does his car get?

Answer: Solution Start with a rate of miles to gallons. Then divide.

455 miles to 14 gallons of gas\text{455 miles to 14 gallons of gas}
Write as a rate. 455 miles14 gallons\frac{\text{455 miles}}{\text{14 gallons}}
Divide 455 by 14 to get the unit rate. 32.5 miles1 gallon\frac{\text{32.5 miles}}{\text{1 gallon}}
Sven’s car gets 32.532.5 miles/gallon, or 32.532.5 mpg.

 

try it

[ohm_question]146616[/ohm_question]
 

The next video shows more examples of how to find rates and unit rates.

https://youtu.be/jlEJU-l5DWw

Calculating Unit Price

Sometimes we buy common household items ‘in bulk’, where several items are packaged together and sold for one price. To compare the prices of different sized packages, we need to find the unit price. To find the unit price, divide the total price by the number of items. A unit price is a unit rate for one item.

Unit price

A unit price is a unit rate that gives the price of one item.
 

example

The grocery store charges $3.99\text{\$3.99} for a case of 2424 bottles of water. What is the unit price? Solution What are we asked to find? We are asked to find the unit price, which is the price per bottle.
Write as a rate. \frac{$3.99}{\text{24 bottles}}
Divide to find the unit price. \frac{$0.16625}{\text{1 bottle}}
Round the result to the nearest penny. \frac{$0.17}{\text{1 bottle}}
The unit price is approximately $0.17\text{\$0.17} per bottle. Each bottle costs about $0.17\text{\$0.17}.
   

TRY IT

[ohm_question]146617[/ohm_question]
  Unit prices are very useful if you comparison shop. The better buy is the item with the lower unit price. Most grocery stores list the unit price of each item on the shelves.

example

Paul is shopping for laundry detergent. At the grocery store, the liquid detergent is priced at $14.99\text{\$14.99} for 6464 loads of laundry and the same brand of powder detergent is priced at $15.99\text{\$15.99} for 8080 loads. Which is the better buy, the liquid or the powder detergent?

Answer: Solution To compare the prices, we first find the unit price for each type of detergent.

Liquid Powder
Write as a rate. $14.9964 loads\frac{\text{\$14.99}}{\text{64 loads}} $15.9980 loads\frac{\text{\$15.99}}{\text{80 loads}}
Find the unit price. $0.234…1 load\frac{\text{\$0.234\ldots }}{\text{1 load}} $0.199…1 load\frac{\text{\$0.199\ldots }}{\text{1 load}}
Round to the nearest cent. $0.23/load(23 cents per load.)\begin{array}{c}\text{\$0.23/load}\hfill \\ \text{(23 cents per load.)}\hfill \end{array} $0.20/load(20 cents per load)\begin{array}{c}\text{\$0.20/load}\hfill \\ \text{(20 cents per load)}\hfill \end{array}

Now we compare the unit prices. The unit price of the liquid detergent is about $0.23\text{\$0.23} per load and the unit price of the powder detergent is about $0.20\text{\$0.20} per load. The powder is the better buy.   Notice in the example above that we rounded the unit price to the nearest cent. Sometimes we may need to carry the division to one more place to see the difference between the unit prices.

Example

Find each unit price and then determine the better buy. Round to the nearest cent if necessary. Brand A Storage Bags, $4.59\text{\$4.59} for 4040 count, or Brand B Storage Bags, $3.99\text{\$3.99} for 3030 count

Answer: Brand A costs $0.12 per bag. Brand B costs $0.13 per bag. Brand A is the better buy.

  Find each unit price and then determine the better buy. Round to the nearest cent if necessary. Brand C Chicken Noodle Soup, $1.89\text{\$1.89} for 2626 ounces, or Brand D Chicken Noodle Soup, $0.95\text{\$0.95} for 10.7510.75 ounces

Answer: Brand C costs $0.07 per ounce. Brand D costs $0.09 per ounce. Brand C is the better buy.

The follwoing video shows another example of how you can use unit price to compare the value of two products. https://youtu.be/ZI4WaviYNsk

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