8.2: Rates
Introduction
The Measuring Dilemma
Kiley is enjoying her work in the supermarket. Today, while she was working in the section of the supermarket that has nuts and other bulk items, a customer needed her help. This customer was trying to figure out a couple of different prices for almonds and cashews.
The customer had measured out three pounds of almonds. When she weighed the almonds and printed her price ticket, the price read “$8.97.”
“How much are these per pound?” the customer asked Kiley.
Kiley looked at the bin, but the label had become worn and she could not see the actual ticket. For Kiley to figure this out, she is going to have to use her arithmetic skills. How much are the almonds per pound?
The same customer weighed out four pounds of cashews. The cashews are $3.29 per pound. Given this information, how much did the four pounds of cashews cost?
To help Kiley with this arithmetic, you will need to learn about rates. Supermarkets are a great place to learn about rates because there are many different rates depending on what you are purchasing at the store.
Pay close attention and you can help Kiley with this work at the end of the lesson.
What You Will Learn
In this lesson, you will learn the following skills:
- Identify and write equivalent rates.
- Write and compare unit rates.
- Solve real-world problems involving rates.
Teaching Time
In the world around us there are many times when we need to use a rate. We use rates when we think about how many miles a car can travel on a gallon of gasoline. We use a rate when we think about how fast or slow something or someone goes-that is a rate of travel, commonly called speed. You may be familiar with many different rates, but that doesn’t help us to understand exactly what a rate is. This lesson will explain all about rates.
What is a rate?
A rate is special ratio that represents an amount in terms of a single unit of time or some other quantity.
We know that we are working with a rate when we see the key word PER.
Example
The car gets 15 miles per gallon.
Here we are comparing the number of miles to one gallon. This is a rate. It is the rate of miles per gallon of gasoline.
Rates can take a different form too. Sometimes, a rate isn’t compared to one, but it is still a rate.
Example
John ran three miles in twenty-one minutes.
What is being compared here? Three miles is being compared to seven minutes. This is the rate. We could use the word “per” in this sentence and it would make perfect sense. When this happens, you know that you are looking at a rate.
How do we write a rate in ratio form?
Once you understand how to identify a rate, you need to know how to write the rate as a ratio since a rate is a special type of ratio.
We can write the example we just worked with as a ratio.
Example
John ran three miles in twenty-one minutes.
To write this as a ratio, we are comparing three miles to twenty-one minutes. The three miles becomes our numerator and the twenty-one minutes our denominator
\begin{align*}\frac{3\ miles}{21\ minutes}\end{align*}
Example
The apples are $.99 per pound.
To write this as a ratio it may help to first see that it is a rate. We are comparing the price of apples to the number of pounds. Our key word here is the word “per” and that lets us know that we are comparing to one.
Next, we write it as a ratio.
Our money amount is our numerator. The number of pounds is our denominator.
\begin{align*}\frac{Price\ of\ apples}{number\ of\ pounds} = \frac{.99}{1}\end{align*}
When a rate is compared to one-it is called a unit rate.
Unit rates and rates can be equivalent to each other.
How do we write equivalent rates?
Writing an equivalent rate can be done in a couple of different ways. First, we can take a rate, write it as a ratio and simplify it to a unit rate. Then the two rates will be equivalent.
Example
Karen ran four miles in 20 minutes.
First, we write it as a ratio. We are comparing four miles to twenty minutes.
\begin{align*}\frac{20\ minutes}{4\ miles}\end{align*}
Next, we simplify the ratio to a unit rate. That means we are comparing to one. We simplify using the greatest common factor of the numerator and the denominator.
\begin{align*}\frac{20\ minutes}{4\ miles} = \frac{5\ minutes}{1\ mile}\end{align*}
These two rates are equivalent. The unit rate is that it took Karen five minutes per mile.
We can also write an equivalent rate the other way around. Let’s take a unit rate and expand it.
Example
Ron ate five hot dogs in one minute.
First, we write a ratio that compares hot dogs to minutes.
\begin{align*}\frac{5\ hot \ dogs}{1\ minute}\end{align*}
This is a unit rate because it is compared to one.
Next, we write an equivalent ratio to this one. We can do this by multiplying the numerator and the denominator by the same number.
\begin{align*}\frac{5\ hot\ dogs}{1\ minutes} = \frac{10\ hot\ dogs}{2\ minutes}\end{align*}
Yes it is! But, we have two equivalent rates here! We have a unit rate and a rate that shows more time and more hot dogs.
It’s time for you to apply these skills. Write an equivalent rate for each.
- \begin{align*}\frac{6\ shirts}{2\ boxes}\end{align*}, how many shirts would there be in six boxes?
- How many shirts are there in one box?
- How many boxes would we need for 24 shirts?
Take a few minutes to check your work. Did you remember how to create equivalent fractions?
II. Write and Compare Unit Rates
In our last section, we began writing unit rates. In this lesson, we are going to continue to work on writing unit rates given other rates.
Remember that a unit rate is a rate written that compares a quantity to one.
Example
\begin{align*}\frac{8\ apples}{1\ student}\end{align*}
Here the unit rate says that there were eight apples per student.
Let’s build a word problem around this. You can be very creative with this.
Example
Mrs. Harris’ class went apple picking. Each student picked eight apples.
This is a perfect word problem for our unit rate. Now let’s expand this problem a little further.
Example
Mrs. Harris’ class went apple picking. Each student picked eight apples. At this rate, how many apples were picked by seven students?
Here we are going to use our unit rate to solve a problem.
\begin{align*}\frac{8\ apples}{1\ student} = \frac{?\ apples}{7\ students}\end{align*}
Here we need to solve for how many apples were picked. We can do this by creating an equivalent rate. The denominator was multiplied by seven, one times seven is seven. We can do the same thing to the numerator.
8 \begin{align*}\times\end{align*} 7 \begin{align*}=\end{align*} 56
There are 56 apples picked by the seven students.
\begin{align*}\frac{8\ apples}{1\ student} = \frac{56\ apples}{7\ students}\end{align*}
How do we take a rate and write a unit rate?
We can also take a larger rate and figure out a unit rate. We do this by simplifying so that we are comparing the quantity with one.
Example
Laquita picked 12 peaches in 6 minutes.
Begin by writing a rate that compares peaches to minutes.
\begin{align*}\frac{12\ peaches}{6\ minutes}\end{align*}
Next, we look at the unit rate. The unit rate would compare peaches picked in one minute. We simplify the denominator to one and then simplify the numerator to create an equivalent rate.
\begin{align*}\frac{12\ peaches}{6\ minutes} = \frac{?\ peaches}{1\ minutes}\end{align*}
To change 6 minutes to one minute we divided by 6. We need to do the same thing to the numerator.
\begin{align*}\frac{12\ peaches}{6\ minutes} = \frac{2\ peaches}{1\ minutes}\end{align*}
Our unit rate is two peaches picked in one minute.
Practice writing a few unit rates on your own.
- \begin{align*}\frac{24\ buttons}{4\ shirts}\end{align*}
- \begin{align*}\frac{4\ ice\ cream\ cones}{2\ people}\end{align*}
- \begin{align*}\frac{45\ miles}{3\ gallons}\end{align*}
Take a few minutes to check your work with a partner.
Real Life Example Completed
The Measuring Dilemma
Did you pay close attention to calculating rates and unit rates? Well, it is time to apply what you have learned. Here is Kiley’s problem once again. Read it through and underline the important information.
Kiley is enjoying her work in the supermarket. Today, while she was working in the section of the supermarket that has nuts and other bulk items, a customer needed her help. This customer was trying to figure out a couple of different prices for almonds and cashews.
The customer had measured out three pounds of almonds. When she weighed the almonds and printed her price ticket, the price read “$8.97.”
“How much are these per pound?” the customer asked Kiley.
Kiley looked at the bin, but the label had become worn and she could not see the actual ticket. For Kiley to figure this out, she is going to have to use her arithmetic skills. How much are the almonds per pound?
The same customer weighed out four pounds of cashews. The cashews are $3.29 per pound. Given this information, how much did the four pounds of cashews cost?
There are two problems for Kiley to solve. The first has to do with the almonds. The customer wanted to know how much they were per pound. She is looking for the unit rate. Begin by writing a rate that compares three pounds of almonds to the price.
\begin{align*}\frac{8.97}{3\ pounds}\end{align*}
Next, we need to figure out the cost of one pound. We can create an equal fraction.
\begin{align*}\frac{8.97}{3\ pounds} = \frac{?}{1\ pound}\end{align*}
We divided by three to go from three pounds to one pound. We can divide 8.97 by three to get the unit price.
8.97 \begin{align*}\div\end{align*} 3 \begin{align*}=\end{align*} $2.99
The almonds cost $2.99 per pound.
Next, we need to figure out the cost of four pounds of cashews if the cashews cost $3.29 per pound. Here we have been given the unit rate and we are going to multiply to find the rate for four pounds.
Here is our unit rate.
\begin{align*}\frac{3.29}{1} = \frac{?}{4}\end{align*}
To go from one to four pounds in the denominator, we multiplied by four. We do the same thing to the numerator.
3.29 \begin{align*}\times\end{align*} 4 \begin{align*}=\end{align*} $13.16
Four pounds of cashews cost $13.16.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Rate
- a special ratio that compares two quantities. Often uses units such as miles, gallons or dollars to describe the rate.
- Unit Rate
- a unit rate compares a quantity to one. Rates can be simplified to be unit rates.
Technology Integration
Khan Academy Simplifying Rates and Ratios
James Sousa, Rates and Unit Rates
James Sousa, Example of Determining Unit Rate
James Sousa, Example of Determining Unit Rate in Gallons per Minute
James Sousa, Example of Determining the Best Buy Using Unit Rate
Time to Practice
Directions: Write a rate that compares the quantities described in each problem.
1. Fourteen apples in two barrels
2. Thirty-two crayons in two boxes
3. Eighteen bottles in three carriers
4. Twenty students on four teams
5. Twenty-five students on five teams
6. Fifty students in two classes
7. Ninety students on three buses
8. Thirteen students ate twenty-six cupcakes
9. Twelve campers in two tents
10. Twenty-four hikers per trail
Directions: Use the information in numbers 1 - 10 to write a unit rate for each.
11. How many apples were in one barrel?
12. How many crayons were in one box?
13. How many bottles were in one carrier?
14. How many students per team?
15. How many students per team?
16. How many students per class?
17. How many students per bus?
18. How many cupcakes per student?
19. How many campers per tent?
20. How many hikers per trail?
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