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5.6: Comparison of Unit Rates

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Remember the book challenge from the Unit Rates Concept?

Kelly read 8 books in three weeks. Marc read 10 books in 4 weeks.

Who has the faster rate?

To figure this out, you will need to write and compare unit rates. This Concept will teach you just how to do that.

Guidance

Sometimes, it is helpful to compare unit rates. For example, a unit price is a type of unit rate. In real life, you may want to compare two unit prices to determine which one is the better buy.

Felicia needs to buy sugar. She could buy a 16-ounce box of sugar for $1.12, or she could buy a 24-ounce box of sugar for $1.44. Which is the better buy? How many cents per ounce cheaper is that buy?

Find the unit price per ounce for the 16-ounce box. Remember, you can find the unit price by dividing the first term by the second term.

\$1.12 \ \text{for}\ 16 \ oz = \frac{\$1.12}{16 oz} && & \overset{ \ \ \$0.07}{16 \overline{ ) {\$1.12 \;}}}\\&&& \quad \underline{-1.12}\\&&& \qquad \quad 0

Find the unit price per ounce for the 24-ounce box.

\$1.44 \ \text{for} \ 24 \ oz = \frac{\$1.44}{24oz} && & \overset{ \ \ \$0.06}{24 \overline{ ) {\$1.44 \;}}}\\&&& \quad \underline{-1.44}\\&&& \qquad \quad 0

Since $0.06 < $0.07, the 24-ounce box has a cheaper unit price, $0.06 per ounce.

Subtract to find how much cheaper per ounce the better buy is $0.07 - $0.06 = $0.01

So, the 24-ounce box of sugar is the better buy. Felicia will pay $0.01 less per ounce if she buys the 24-ounce box.

Once you can figure out the unit rate, you can easily compare which unit rate is greater and which is less.

Now it's time for you to compare a few unit rates on your own. Use <,>,or =.

Example A

3 to 1 ____ 6 to 2

Solution: =

Example B

4 to 5 ____ \frac{16}{20}

Solution: =

Example C

18 to 3 ____ 15 to 5

Solution: >

Here is the original dilemma once again.

Kelly read 8 books in three weeks. Marc read 10 books in 4 weeks.

Who has the faster rate?

To figure this out, we have to turn each rate into a unit rate. We do this by dividing.

Kelly \frac{8}{3} = \frac{2.6}{1}

Kelly read more than 2 1/2 books per week.

Marc \frac{10}{4} = \frac{2.5}{1}

Marc read 2 1/2 books per week.

Kelly has the faster reading rate.

Vocabulary

Rate
a special kind of ratio that compares two different quantities.
Per
a word that signals you that a rate is being used.
Unit Rate
a rate that is compared to 1. A rate can be for 1 pound, 1 mile, 1 second, 1 of any unit.

Guided Practice

Here is one for you to try on your own.

Marc put labels on 80 boxes in 6 minutes. Jeff put labels on 90 boxes in 8 minutes.

Who has the better rate?

Answer

To compare these two rates, we have to figure out the unit rates. This will tell us how many each man can do in one minutes.

To figure this out, we write a ratio and divide.

\frac{80}{6} = \frac{13.3}{1}

\frac{90}{8} = \frac{11.25}{1}

Marc has a better rate because he puts more labels on per minute.

Video Review

This is a James Sousa video on finding unit rates. It is a support video to this Concept.

Practice

Directions: Determine whether each is equivalent or not.

1. 60 units in 3 minutes and 80 units in 4 minutes

2. $16 for 8 pounds and $22 for 10 pounds

3. 50 kilometers in 2 hours and 75 kilometers in 3 hours

Directions: Solve each problem.

4. Max paid $45 for 15 gallons of gasoline. What was the cost per gallon of gasoline?

5. Mr. Brown paid $8.28 for 12 cans of green beans. Express this cost as a unit price.

6. A train travels 480 kilometers in 3 hours. Express this speed as a unit rate.

7. Mrs. Jenkins paid $50 for 40 square feet of carpeting. What was the cost per square foot for the carpeting?

8. A copy machine can produce 310 copies in 5 minute. How many copies can the machine produce per minute?

Nadia needs to buy some cheddar cheese. An 8-ounce package of cheddar cheese costs $2.40. A 12-ounce package of cheddar cheese costs $3.36.

9. Find the unit price for the 8-ounce package.

10. Find the unit price for the 12-ounce package.

11. Which is the better buy? How many cents per ounce cheaper is the better buy?

Joe drove 141 miles in 3 hours. His cousin Amy drove 102 miles in 2 hours. Assume both cousins were driving at constant speeds.

12. How fast was Joe driving, in miles per hour?

13. How fast was Amy driving, in miles per hour?

14. Who was driving at a faster rate of speed?

15. How much faster was the faster person traveling?

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Basic

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Date Created:

Dec 21, 2012

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Aug 18, 2014
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