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Unit Rates (1.4)

Remember the reading challenge?

Manuel loves to read. He found a series of mystery books that take place in the world of medieval knights, and he has been working his way through the series. There are twelve books in the series.

After the first 5 weeks of school, Manuel has finished 8 of the 12 books.

“How many have you finished?” his sister Sarina asked at breakfast.

“I have read 8 out of 12,” Manuel said. “I have four more to go to finish the series.”

“How long is that going to take you?” Sarina asked.

Manuel had to think for a few minutes about this. Then he took out his notebook to do some figuring.

Think about this with Manuel. At this rate, how many books is he reading per week? How many weeks will it take him to read all 12?

To figure out this problem, you will need to know about rates and unit rates. In this Concept, you will learn everything that you need to know to solve this problem. By the end, you will be able to help Manuel keep track of books.

Guidance

There is a special kind of rate which is called a unit rate.

A unit rate has a denominator of 1, meaning that it is the measure for one of whatever you are talking about: 1 mile, 1 pound, 1 foot, etc.

When we talk about unit rates, we are talking about singular measurement and single rates.

1 pound of apples is $1.79.

This is a unit rate. It tells us that for 1 pound of apples, we will pay $1.79. Based on this number, we can figure out the rate for 2 pounds, 3 pounds, 5 pounds, etc.

This problem told us what the unit rate was, but sometimes you will need to figure out the unit rate. How do we figure out the unit rate?

To figure out the unit rate, we have to write the ratio as a fraction with a denominator of one.

Rochelle ran 15 miles in 2 hours. Express her speed as a unit rate.

To express this as a unit rate, we need to figure out Rochelle’s speed for one mile. Begin by setting up the rate as a fraction.

15 \ miles \ \text{in}\ 2 \ hours = \frac{15mi}{2h} = \frac{15}{2}

The second number in a unit rate is 1. Since 2 \div 2 = 1 , you can divide both terms by 2 to write this as a unit rate.

\frac{15mi}{2h} = \frac{15}{2} = \frac{15 \div 2}{2 \div 2}

Use long division to divide 15 by 2.

& \overset{ \ \ 7.5}{2 \overline{ ) {\ 15.0 \;}}}\\& \quad \underline{-14}\\& \quad \ \ 10\\& \ \ \ \underline{-10}\\& \qquad \ 0

So, \frac{15 mi}{2h} = \frac{15 \div 2}{2 \div 2} = \frac{7.5}{1} = \frac{7.5mi}{1h} or 7.5 miles per hour.

The unit rate was 7.5 miles in 1 hour, or 7.5 miles per hour.

A quicker strategy for finding a unit rate is to simply divide the first term by the second term.

Notice that to find the unit rate in the first situation, you could have simply divided 15 miles by 2 hours to find the unit rate in miles per hour. Notice that that is exactly what we did. The only difference is that we showed why that strategy works by writing the original rate as a fraction and then showing how to simplify it to find an equivalent rate with a 1 in the denominator.

Try this out on this one.

Kyle ran 4 miles in 28 minutes. How fast did he run per mile?

“Per mile” let’s us know that we are looking for his rate for 1 mile. That is the unit rate.

We write a fraction ratio and divide.

& \frac{min}{miles} = \frac{28}{4}\\& 28 \div 4 = 7\\&\frac{7}{1}

Kevin ran 1 mile in 7 minutes. That is the unit rate.

Try a few on your own. Find each unit rate.

Example A

5 pounds of apples for $2.00

Solution: 1 pound for $.40

Example B

45 miles for 3 gallons of gasoline

Solution: 15 miles for 1 gallon

Example C

18 inches in 2 minutes

Solution: 9 inches in 1 minute

Here is the original problem once again.

Manuel loves to read. He found a series of mystery books that take place in the world of medieval knights, and he has been working his way through the series. There are twelve books in the series.

After the first 5 weeks of school, Manuel has finished 8 of the 12 books.

“How many have you finished?” his sister Sarina asked at breakfast.

“I have read 8 out of 12,” Manuel said. “I have four more to go to finish the series.”

“How long is that going to take you?” Sarina asked.

Manuel had to think for a few minutes about this. Then he took out his notebook to do some figuring.

Think about this with Manuel. At this rate, how many books is he reading per week? How many weeks will it take him to read all 12?

There are two questions here to figure out. Let’s tackle the first one. We need to figure out how many books Manuel is reading per week. The word “per” lets us know that we want to figure out the number of books for one week. Let’s write a ratio to compare books read to weeks.

\frac{8 \ books}{5 \ weeks}

Next, we need to figure that out for 1 week. To do this, we divide 8 by 5.

\overset{ \ \ 1.6}{5 \overline{ ) {8.0 \;}}}

Manuel reads 1.6 books per week. We could also say that he reads a little more than 1 \frac{1}{2} books per week.

Given this rate, how long will it take him to read 12 books? We can set up a pair of ratios to figure this out.

\frac{1.6 \ books}{1 \ week}=\frac{12 \ books}{x \ weeks}

If we divide 12 by 1.6, we will have the number of weeks.

\overset{ \ \qquad 7.5}{1.6 \overline{ ) {12.00 \;}}}

It will take Manuel 7 \frac{1}{2} weeks to finish the series. In 2 \frac{1}{2} more weeks he will finish the series .

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.

Maria ran 5 miles in 60 minutes. Her brother Mario ran the same distance in half the time. What is Mario’s rate per mile?

Answer

To figure this out, first notice that we are looking for Mario’s unit rate. To find this, we need to start with figuring out Maria’s. Then we know that Mario ran the same distance in half the time, so we can say that Mario’s unit rate is one-half of Maria’s. This means that the time that it takes Maria to run 1 mile, that Mario runs 1 mile in one-half the time.

\frac{1}{2} (Maria’s unit rate) = Mario’s unit rate

First, we need to figure out Maria’s unit rate.

\frac{60 \ min}{5 \ miles} = \frac{12 \ min}{1 \ mile}

Maria runs 1 mile in 12 minutes.

Mario runs the same distance in half the time. Mario runs one mile in 6 minutes.

Video Review

This is a James Sousa video on rates and unit rates.

Practice

Directions: Write a unit rate for each ratio.

1. 2 for $10.00

2. 3 for $15.00

3. 5 gallons for $12.50

4. 16 pounds for $40.00

5. 18 inches for $2.00

6. 5 pounds of blueberries for $20.00

7. 40 miles in 80 minutes

8. 20 miles in 4 hours

9. 10 feet in 2 minutes

10. 12 pounds in 6 weeks

11. 14 pounds for $7.00

12. 18 miles in 3 hours

13. 21 inches of cloth costs $7.00

14. 45 miles on 3 gallons of gasoline

15. 200 miles in four hours

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