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# Comparison of Unit Rates

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In the Identify and Write Equivalent Rates Concept, Kiley finished figuring out the cost of the almonds. The customer had another question. Take a look at what happened.

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 Concept. ### Guidance Previously we worked on writing equivalent rates that were compared to one. These are unit rates . In this Concept, 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. $\frac{8\ apples}{1\ student}$ 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. 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. 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. $\frac{8\ apples}{1\ student} = \frac{?\ apples}{7\ students}$ 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 $\times$ 7 $=$ 56 There are 56 apples picked by the seven students. $\frac{8\ apples}{1\ student} = \frac{56\ apples}{7\ students}$ 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. Laquita picked 12 peaches in 6 minutes. Begin by writing a rate that compares peaches to minutes. $\frac{12\ peaches}{6\ minutes}$ 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. $\frac{12\ peaches}{6\ minutes} = \frac{?\ peaches}{1\ minutes}$ To change 6 minutes to one minute we divided by 6. We need to do the same thing to the numerator. $\frac{12\ peaches}{6\ minutes} = \frac{2\ peaches}{1\ minutes}$ Our unit rate is two peaches picked in one minute. Practice writing a few unit rates on your own. #### Example A $\frac{24\ buttons}{4\ shirts}$ Solution: 6 buttons per shirt #### Example B $\frac{4\ ice\ cream\ cones}{2\ people}$ Solution: 2 ice cream cones per person #### Example C $\frac{45\ miles}{3\ gallons}$ Solution: 15 miles per gallon Now let's go back to the supermarket. Kiley is trying to use unit rates to figure out the cost of the cashews. Let's take a look. Now, 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.

$\frac{3.29}{1} = \frac{?}{4}$

To go from one to four pounds in the denominator, we multiplied by four. We do the same thing to the numerator.

3.29 $\times$ 4 $=$ $13.16 Four pounds of cashews cost$13.16.

### 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.

Harold cuts seven lawns in 4 hours. What is the unit rate per hour?

To figure this out, first we have to write a ratio that compares the lawns to hours.

$\frac{7}{4}$

Now we can convert this to a rate. We want to know how many lawns in one hour.

$\frac{7}{4} = \frac{x}{1}$

Now if we solve this problem, we will divide 7 by 4.

$7 \div 4 = 1.75$

Harold cuts 1.75 lawns in one hour.

### Practice

Directions: Use each rate to write a unit rate for each. Remember a unit rate is compared to one.

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