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# Identification and Writing of Equivalent Rates

## Understand that equivalent rates are equivalent ratios.

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Identification and Writing of Equivalent Rates

Have you ever bought things in bulk at a supermarket? People do it all the time, and you need to know how to work with rates to be successful.

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? To complete this dilemma, you will need to understand rates and equivalent rates. Pay close attention and we'll revisit this problem at the end of the Concept. ### Guidance 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 Concept 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. 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. 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. Let's look at the dilemma above again. 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. 3 miles21 minutes\begin{align*}\frac{3\ miles}{21\ minutes}\end{align*} 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.

Price of applesnumber of pounds=.991\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.

Karen ran four miles in 20 minutes.

First, we write it as a ratio. We are comparing four miles to twenty minutes.

20 minutes4 miles\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.

20 minutes4 miles=5 minutes1 mile\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.

It’s time for you to apply these skills. Write an equivalent rate for each.

#### Example A

6 shirts2 boxes\begin{align*}\frac{6\ shirts}{2\ boxes}\end{align*}, how many shirts would there be in six boxes?

Solution: 18 shirts

#### Example B

How many shirts are there in one box?

Solution: Three shirts in one box

#### Example C

How many boxes would we need for 24 shirts?

Solution: 8 boxes

Now back to the dilemma with Kiley and the almonds.

The dilemma 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.

8.973 pounds\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.

8.973 pounds=?1 pound\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.

### Vocabulary

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.

### Guided Practice

Here is one for you to try on your own.

Ron ate five hot dogs in one minute. Now find a unit rate.

First, we write a ratio that compares hot dogs to minutes.

5 hot dogs1 minute\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.

5 hot dogs1 minutes=10 hot dogs2 minutes\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.

### 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

11. Six campers per tent

12. If there are six campers per tent, how many tents for 18 campers?

13. How many tents for 30 campers?

14. How many campers can you fit if you have 12 tents?

15. Sixty students on five teams

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

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 means equal in value or meaning.

Rate

A rate is a special kind of ratio that compares two quantities.

Unit Rate

A unit rate is a ratio that compares a quantity to one. The word “per” is a key word with unit rates.