Writing Rates and Calculating Unit Rates

Learning Outcomes

Translate phrases to expressions with fractions
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 $120$ miles in $2$ hours, $160$ words in $4$ minutes, and $\text{\$5}$ dollars per $64$ 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 $525$ miles in $9$ hours. Write this rate as a fraction.

Solution

	$\text{525 miles in 9 hours}$
Write as a fraction, with $525$ miles in the numerator and $9$ hours in the denominator.	${\Large\frac{\text{525 miles}}{\text{9 hours}}}$
	${\Large\frac{\text{175 miles}}{\text{3 hours}}}$

So $525$ miles in $9$ hours is equivalent to ${\Large\frac{\text{175 miles}}{\text{3 hours}}}$

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Find Unit Rates

In the last example, we calculated that Bob was driving at a rate of ${\Large\frac{\text{175 miles}}{\text{3 hours}}}$ . This tells us that every three hours, Bob will travel $175$ 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 $1$ unit is referred to as a unit rate.

Unit Rate

A unit rate is a rate with denominator of $1$ unit.

Unit rates are very common in our lives. For example, when we say that we are driving at a speed of $68$ miles per hour we mean that we travel $68$ miles in $1$ hour. We would write this rate as $68$ miles/hour (read $68$ miles per hour). The common abbreviation for this is $68$ mph. Note that when no number is written before a unit, it is assumed to be $1$ .

So $68$ miles/hour really means $\text{68 miles/1 hour.}$

Two rates we often use when driving can be written in different forms, as shown:

Example	Rate	Write	Abbreviate	Read
$68$ miles in $1$ hour	$\Large\frac{\text{68 miles}}{\text{1 hour}}$	$68$ miles/hour	$68$ mph	$\text{68 miles per hour}$
$36$ miles to $1$ gallon	$\Large\frac{\text{36 miles}}{\text{1 gallon}}$	$36$ miles/gallon	$36$ mpg	$\text{36 miles per gallon}$

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 $\text{\$12.50}$ for each hour you work, you could write that your hourly (unit) pay rate is $\text{\$12.50/hour}$ (read $\text{\$12.50}$ per hour.)

To convert a rate to a unit rate, we divide the numerator by the denominator. This gives us a denominator of $1$ .

example

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

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Solution

Start with a rate of dollars to hours. Then divide.	$\text{\$384}$ last week for $32$ hours.
Write as a rate.	${\Large\frac{$384}{\text{32 hours}}}$
Divide the numerator by the denominator.	${\Large\frac{$12}{\text{1 hour}}}$
Rewrite as a rate.	$$12/\text{hour}$

Anita’s hourly pay rate is $\text{\$12}$ per hour.

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example

Sven drives his car $455$ miles, using $14$ gallons of gasoline. How many miles per gallon does his car get?

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

	$\text{455 miles to 14 gallons of gas}$
Write as a rate.	${\Large\frac{\text{455 miles}}{\text{14 gallons}}}$
Divide 455 by 14 to get the unit rate.	${\Large\frac{\text{32.5 miles}}{\text{1 gallon}}}$

Sven’s car gets $32.5$ miles/gallon, or $32.5$ mpg.

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

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 $\text{\$3.99}$ for a case of $24$ 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.	${\Large\frac{$3.99}{\text{24 bottles}}}$
Divide to find the unit price.	${\Large\frac{$0.16625}{\text{1 bottle}}}$
Round the result to the nearest penny.	${\Large\frac{$0.17}{\text{1 bottle}}}$

The unit price is approximately $\text{\$0.17}$ per bottle. Each bottle costs about $\text{\$0.17}$ .

TRY IT

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 $\text{\$14.99}$ for $64$ loads of laundry and the same brand of powder detergent is priced at $\text{\$15.99}$ for $80$ loads.
Which is the better buy, the liquid or the powder detergent?

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

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

Now we compare the unit prices. The unit price of the liquid detergent is about $\text{\$0.23}$ per load and the unit price of the powder detergent is about $\text{\$0.20}$ 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, $\text{\$4.59}$ for $40$ count, or Brand B Storage Bags, $\text{\$3.99}$ for $30$ count

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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, $\text{\$1.89}$ for $26$ ounces, or Brand D Chicken Noodle Soup, $\text{\$0.95}$ for $10.75$ ounces

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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.

Module 6: Applications With Decimals