Ratios and rates grade 7

Section A Single Number Ratios Section C Different Kinds of Ratios Section D Scale and Ratio Section E Scale Factor... Sheena used a ratio table to calculate the average number of miles

Ratios

and Rates

Number

support of the National Science Foundation Grant No 9054928.

The revision of the curriculum was carried out in 2003 through 2005, with the support of the National Science Foundation Grant No ESI 0137414.

National Science Foundation

Opinions expressed are those of the authors and not necessarily those of the Foundation.

Keijzer, R., Abels, M., Wijers, M., Brinker, L J., Shew, J A., Cole, B R., and

Pligge, M A (2006) Ratios and rates In Wisconsin Center for Education

Research & Freudenthal Institute (Eds.), Mathematics in Context Chicago: Encyclopædia Britannica, Inc.

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ISBN 0-03-039629-8

The Mathematics in Context Development Team

Development 1991–1997

The initial version of Ratios and Rates was developed by Ronald Keijzer and Mieke Abels

It was adapted for use in American schools by Laura J Brinker, Julia A Shew, and Beth R Cole.

Wisconsin Center for Education Freudenthal Institute Staff

Research Staff

Thomas A Romberg Joan Daniels Pedro Jan de Lange

Project Staff

Jonathan Brendefur Sherian Foster Mieke Abels Jansie Niehaus Laura Brinker James A, Middleton Nina Boswinkel Nanda Querelle James Browne Jasmina Milinkovic Frans van Galen Anton Roodhardt Jack Burrill Margaret A Pligge Koeno Gravemeijer Leen Streefland

Peter Christiansen Julia A Shew Heuvel-Panhuizen Monica Wijers Barbara Clarke Aaron N Simon Jan Auke de Jong Astrid de Wild

Beth R Cole Stephanie Z Smith Ronald Keijzer

Mary Ann Fix

Revision 2003–2005

The revised version of Ratios and Rates was developed by Mieke Abels and Monica Wijers

It was adapted for use in American schools by Margaret A Pligge.

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Editorial Coordinator Editorial Coordinator Content Coordinator Content Coordinator

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Section A Single Number Ratios

Section C Different Kinds of Ratios

Section D Scale and Ratio

Section E Scale Factor

different ways to make comparisons

Do you have more boys or girls in your class? If you count, you mightuse a ratio to describe this situation You can make comparisonsusing different types of ratios

You might have noticed speed limit signs posted

along highways and streets The rate a car travels

on a highway is usually greater than the rate a

car travels on a street You can make comparisons

using rates

You use ratios to make scale drawings

Architects use scale drawings to design

and build buildings They create sets of

working documents, which contain a

floor plan, site plan, and elevation plan

Maps are also scale drawings

Have you ever looked at a cell through a microscope?

The magnification of the lens sets the ratio between

what you see and the actual size of the cell

Architects, engineers, and artists often make scale models of objectsthey want to construct Many people have hobbies creating miniatureworlds using trains, planes, ships, and automobiles When you lookthrough a microscope, you see enlargements of small objects

In all instances, ratios keep everything real We hope you learn efficientways to work with ratios and rates

Sincerely,

T

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The students in Ms Cole’s science class are concerned about the air quality around Brooks Middle School They noticed that smog frequently hangs over the area They just finished a science projectwhere they investigated the ways smog destroys plants, corrodesbuildings and statues, and causes respiratory problems.

The students hypothesize that the city has so much smog because

of the high number of cars on the roads Students think there are somany cars because most people do not carpool They want to find out if people carpool

One group spent exactly one minute and counted 10 cars and

12 people

1 a How many of these cars could have carried more than one

person? Give all possible answers

b Find the average number of people per car and explain how

you found your answer

At the same time, at a different point on the highway, a second group

of students counts cars and people for two minutes A third groupcounts cars and people for three minutes

The second and third groups each calculate the average number ofpeople per car They are surprised to find that both groups got anaverage of 1.2 people per car

2 How many cars and how many people might each group have

A fourth group counts cars and people for one minute on the northside of the school They count 18 cars and 21 people

3 Compare the results of the fourth group of students with those of

the other three groups What conclusions can you draw?

For the first group of students, the ratioof people to cars was

12 people to 10 cars or 12:10 Another way to describe this is it touse the average numberof people per car The first three groupscalculated an average of 1.2 people per car They might have foundthis average by calculating the result of the division 12
10

You can show both the ratio and the average in a ratio table

4 a How can you use the ratio table to find the average number of

people per car?

b You can also write the average number of people per car in a

ratio What ratio is this?

c Given this average, how many people would you expect to see

if you counted 15 cars?

d What can you say about the number of people in each of the

15 cars?

In order to lessen air pollution, the students investigate ways to

increase the average number of people per car

Single Number Ratios

Number of People 12 1.2

Number of Cars 10 1

Some students recommend that the average number of people percar should increase from 1.2 to 1.5 people per car.

6 a Find 5 different groups of cars and people that will give you an

average of 1.5 people per car Put your findings in a table

b Work with a group of your classmates to make a poster that

will show the city council how raising the average number ofpeople per car from 1.2 to 1.5 will lessen traffic congestion andimprove the quality of air

Another way to reduce air pollution is to encourage drivers to useautomobiles that are more efficient A local TV station decides to

do a special series on how to reduce air pollution

In one report, the newscaster mentions, “Cars with high gas mileagepollute less than cars with low gas mileage.”

Gas mileageis the average number of miles (mi) a car can travel on

1 gallon (gal) of gasoline It is represented by the ratio of miles pergallon (mpg)

John says, “My car’s gas mileage is 25 mpg.”

7 How many miles can John travel on 12 gal of gas?

Miles per Gallon

Cindy, Arturo, and Sheena see the report on TV They decide to

calculate their gas mileage to see whose car pollutes the least

Cindy remembers that she drove 50 mi on 2.5 gal of gasoline

She creates the following ratio table on a scrap paper

Cindy says, “My gas mileage is 20 mpg.”

8 Explain Cindy’s calculation and answer.

Single Number Ratios

The last time Arturo filled up his car, hehad driven 203 mi on 8.75 gal of gas

9 Explain whether Arturo’s gas

mileage will be more or less thanCindy’s gas mileage

Arturo set up this ratio table to calculate his gas mileage

10 a What did Arturo do in his ratio table to make the number of

gallons a whole number?

b Calculate Arturo ’s gas mileage.

Number of Miles 203 2,030 20,300

Number of Gallons 8.75 87.5 875

Miles Gallons

50 2.5

100 5

20 1

It took Sheena 2 hours to travel 81.2 mi Sheena used a ratio table

to calculate the average number of miles she drove per hour Here

is Sheena’s scrap of paper

12 a Explain Sheena’s calculation method.

b What is the average number of miles Sheena drove per hour?

c How would you calculate the average number of miles per

hour for Sheena?

The average number of miles per hour iscalled the average speed Average speed

is expressed in miles per hour (mi/h).Average speed is expressed as asingle number

Miles per Hour

Miles Hours

81.2 2

812 20

406 10

40.6 1

Consider for example that Cindy traveled at an average speed of

55 mi/h An average speed of 55 mi/h is the ratio 55:1, read as

“fifty-five to one.” This ratio can be written in a ratio table like the one for problem 12

13 Reflect. Describe another situation where the average is a ratioexpressed as a single number

Nick traveled 72 miles to Lincoln, Nebraska

He departed at 8:00 A.M and arrived at 9:30 A.M

Kendra traveled 140 mi to Louisville, Kentucky

She departed at 2:00 P.M and arrived at 5:20 P.M

14 Who traveled at a higher average speed, Nick or Kendra?

(Hint: Ratio tables can be very useful to solve this problem.)

Many modern cars are equipped with cruise control, which allows the driver to set the car’s speed to be constant This makes highwaydriving easier and saves gas Sheena used this feature to take twotrips

On Monday, Sheena drove from 1:00 P.M until 2:30 P.M with a constant average speed of 48 mi/h

15 How far did Sheena drive on Monday?

(Hint: Ratio tables can be very useful to solve this problem.)

On Tuesday, Sheena drove from 9:00 A.M until 9:45 A.M with thecruise control set at the same average speed of 48 mi/h

16 What is Sheena’s distance for Tuesday’s trip?

Sheena’s gas mileage was 24 mpg for both trips

17 How many gallons of gas did she use on these trips?

Single Number Ratios

Cruise Control

You can use ratios to express relationships.

The ratio of girls to boys in one class is 15:12

The ratio of people to cars at one corner is 14:10

You can write ratios as single numbers to express averages

On average, in one class there are 1.25 girls for every boy

On average, at one corner there were about 1.4 people per car

To write ratios as single numbers, you may use ratio tables

Average gas mileage

Karla drove 75 mi on 2.5 gal of gas What is her gas mileage?

The ratio 75:2.5 is the same as 30:1 This ratio means that for this trip,Karla averaged 30 miles per gallon Her gas mileage was 30 mpg

• To write ratios as single numbers, you may also use division

75 mi
2.5 gal = 30 mpg

Using a ratio as a single number to express an average makes it easy

to compare different situations Here is an example

• Comparing average speed (mi/h)

It took Serena 2 hr to drive 90 mi Karla drove 75 mi in 1.5 hr.Compare their average speed

• Using ratio tables:

Serena’s trip

Number of Miles Number of Gallons

75 2.5

750 25

30 1

 10
25

 10
25

Number of Miles Number of Hours

90 2 45 1

Karla’s trip

Serena averaged 45 mi/h Serena’s average speed was 45 mi/h

Karla averaged 50 mi/h Karla’s average speed was 50 mi/h

So Karla drove faster

1 a Find the average number of people per car if you counted

16 cars and 40 people

b Find the average number of students per class if there are

320 students in 9 classes

2 Use a ratio table to calculate the gas mileage.

A car travels 108 mi on 6 gal of gas

Number of Miles Number of Hours

75 1.5

50 1

150 3

Number of Miles

Gallons of Gas

Martha had her car repaired at a garage Shown below is part of thebill she received from her mechanic.

3 Use the following ratio table to find how much her mechanic

charged per hour

David and his group counted cars and people

The ratio of people to cars is 25:15

4 Write this ratio as a single number to express the average

number of people per car

5 Make up your own problem about ratios and averages.

Of course, you will have to provide an answer to your problem as well

Describe how you would explain to a car owner the way to calculategas mileage

Village Automotive

Cost in Dollars Number of Hours

The table below shows the population and the total number of telephones for 15 different countries.

1 According to the data, which countries in the table have more

telephones than people?

B

Comparisons

Telephones and Populations

Country Population Number of Telephones

United States 292.6 million 331 million

Source: Encyclopaedia Britannica Almanac, 2005 (Chicago: Encyclopaedia Britannica, Inc 2005)

Joan looks at the numbers in the table and says, “the United Stateshas the largest population because 292.6 is the highest numberbefore the million.”

Brian disagrees; he says that the population of China is larger

2 Explain who is right.

3 a Based on the data in the table, in which countries do you

think people rely the most on the use of telephones for communication? Explain

b In which countries did people rely less on the use of

telephones for communication?

MICRONESIA

The data table on page 11 shows that Micronesia has 60,000telephones and a population of 112,000 people The ratio of people to telephones is 112,000:60,000

Number of People Number of Telephones

112,000 60,000

1

4 a Do you think it is true that in Micronesia everybody has a

phone? Explain your thinking

b Use the ratio table on page 12 to find the average number of

people per telephone in Micronesia

c Is the average number of people per telephone in Tonga

greater or smaller than in Micronesia? Explain how you

found your answer

In problem 4, you found the average number of people per telephone

in Micronesia This number tells you how many people would shareone telephone

It is also possible to look at the ratio of telephones to people ForMicronesia, this ratio is 60,000:112,000

5 a Use this ratio to calculate the average number of telephones

per person

b ReflectWhich number do you find the most useful to tell

something about the use of telephones in a country—thenumber of people per telephone or the number of

telephones per person? Explain your choice

If you compare countries with respect to the number of telephoneswithout considering the number of people living in these countries,the comparison is an absolute comparison.

If you compare countries with respect to the number of telephonesand consider the number of people living in these countries, the comparison is a relative comparison, comparing telephones per

person

B

Comparisons

Consider the data for China and Finland.

6 a Use an absolute comparison to answer.

Which of these countries had more telephones?

b Use a relative comparison to answer.

Which of these countries had more telephones per person?You may use the ratio tables set up below

Finland

China

7 Which of the comparisons between China and Finland, the

absolute comparison or the relative comparison, do you think gives a better picture of the number of telephones

in these countries? Why?

8 Reflect.When would an absolute comparison be most useful?When would a relative comparison be a better choice?

The paragraph on the left isfrom a paper Brian wrote thatcompares the numbers oftelevision sets in several ofthe world’s countries.

9 a What information could Brian have used to calculate that there

were 708 TV sets per 1,000 Canadians?

b Can you determine the number of TV sets for each Canadian?

Explain your answer

c What is the total number of TVs in Canada? Explain how you

found your answer

10 a Find the total number of TVs for Brazil.

b Find the total number of TVs in France.

Brazil has about 176 million people, and there are 317 TVs

for every 1,000 citizens For Canada, there are about

31.9 million people and 708 TVs for every 1,000 citizens.

France has about 59.7 million people and 606 TVs for every

1,000 citizens.

TV Sets

United Denmark States Canada Taiwan Poland World Number of Cell

Comparisons

Math History

11 a Can you conclude from the table that there are more cell

phones in Denmark than in Canada? Explain your answer

b What information do you need to be able to calculate the

number of cell phones in the U.S.?

c In Taiwan, the number of cell phones per person is

approximately 1.0 Explain how this number is calculated

d Select two other countries in the table and find the average

number of cell phones per person How do these countriescompare to the world average?

Ratios and Music

Pythagoras (around 500 B.C.) was a Greek mathematician, teacher,and philosopher He found a relationship between ratios and themusical scale as a result of his experiments with a monochord, a onestring musical instrument He found that the shorter the string, thehigher the pitch A movable bridge could make the string shorter.Here you see the ratio 3:2 between the lower C (do) and the G (sol)

The other ratios are

You use numbers to make comparisons.

Absolute Comparisons

Comparisons can be absolute When you make an absolute comparison, you compare things without taking into considerationanything else You compare numbers from only one category

Examples of absolute comparisons:

● comparing the number of people in different countries

● comparing the number of telephones in different countries

● comparing the number of TV sets in different countries

● comparing the amount of snowfall in different states

Relative Comparisons

Comparisons can also be relative When you make a relative comparison, you compare things related to something else

The comparison is in relation to a common base

Examples of relative comparisons:

● comparing the number of telephones per person in differentcountries

● comparing the number of telephones per thousand people indifferent countries

When making a relative comparison, a ratio written as a singlenumber (an average) is commonly used For example:

● comparing the number of telephones per person, 0.7 versus 0.2

● comparing the speed of two cars in miles per hour, 55 mi/hversus 30 mi/h

Ratio tables are useful tools for making relative comparisons

In 2002, the population of South Africa was about 43.6 million and the

number of telephones was about 14.2 million

Tom says that South Africa had about 33 telephones for every 100 people

1 Is Tom correct? Explain your answer.

The table below shows the population and number of cows for several

states in 1993

2 a Which state has the most cows?

b Is the comparison you made in problem 2a

absolute or relative? Explain why.

c Make a comparison of the number of cows

per 100 people for Kansas and Montana

d Is the comparison you made in problem 2c

absolute or relative? Explain why.

(in millions)

Number of Cows (in millions)

5.2 3.1 1.4 3.6 6.6 2.5 6.4

4

Country Area (in sq mi) Population

Source: Data from Encyclopaedia Britannica Almanac, 2005 (Chicago: Encyclopaedia Britannica, Inc., 2005)

3 In your opinion, which of the countries below has the greatest

number of people per square mile? Show your work

In your math class, determine the number of phones and people ineach household Then find the number of phones per person

BRAZIL

ARGENTINA

JAPAN

The citizens of Wrigley are concerned aboutthe number of people who speed throughtown The local police have identified thefour worst areas for speeding The citycouncil has agreed to install traffic lights toslow down the speeding cars.

At the present time, there is only enoughmoney in the budget to install one trafficlight The council asks the police to decidewhich area needs the traffic light the most.The police make plans to study the situationand give a report at the next council meeting

In order to monitor the number of driverswho speed through the four areas of town,the police set up a device to count andrecord the speed of passing cars

1 a Compare the results from these four areas of town.

b What recommendation would you make to the city council?

Suppose the police found another area of town where they suspect alot of speeding takes place When they count the cars and figure outhow many people speed in this area, they find that the ratio of speeders to non-speeders is one to three, or 1:3.

2 Will this change the recommendation you made in problem 1b?

Why or why not?

A neighboring town, Brighton, uses a sign on the highway The signconstantly shows the percent of cars that pass the sign that are withinthe speed limit

3 a Why do you think the city put up this sign,

and why do you think the sign shows thepercent of drivers who are not speeding?

b How is percent related to ratio?

c Suppose the next car that passes the sign

is speeding How will the percent on thesign change? Explain your answer

4 a According to the sign, what part of the total number of cars

was speeding?

b Suppose 269 cars have passed the sign shown Estimate the

number of cars that were speeding

One local TV station covered theproblem of speeding on the sixo’clock news The report gave some statistics to emphasize the seriousness of the situation

5 Can you conclude that over half

of the cars were speeding onHighway 19? Why or why not?

Another TV station picked up the story The newscaster from thisstation wanted to describe the speeding situation on Highway

19 in terms of percents

6 What percents could be used?

The police reported that

on Highway 19, two cars were speeding for every three that were not speeding.

The speed limit on Highway 19 where the sign is located is 55 mi/h.The sign is reset to zero at two o’clock every morning The table below shows the speed of the first four cars that pass the sign after it was reset.

7 a What percent did the sign display after the first car passed

the sign?

b What percent did the sign display after the fourth car passed?

c After the fifth car passes, the sign can display two possible

percents Explain why this is the case and calculate these percents

C

Different Kinds of Ratios

Another way to find a percent is to rewrite each ratio as a numbercompared to 100 (or per 100) A ratio table or a calculator may behelpful with this strategy.

9 a Why would it be helpful to rewrite the ratio as a number

compared to 100?

b Suppose 15 out of 25 cars were not speeding Show how to

write this ratio as a percent using the ratio table

c Do the same if 10 out of 24 cars were not speeding.

d Suppose 55 out of 76 cars were not speeding Show how to

write this ratio as a percent

Another way to find percents is by using the relationships amongratios, fractions, decimals, and percents You already know many ofthese relationships Look at the table below

10 a Copy and fill in the table to show equivalent fractions, decimals,

and percents

Number of Cars Not Speeding

1:2 1:3 1:4 1:5 1:15

1 20

0.3

10%

b Fill in three additional rows at the bottom of your table to show

other equivalent relationships that you know

c Explain the relationship between the equivalent decimals and

Joshua has to calculate the percentage of cars not speeding 55 out

of 76 cars were not speeding as they drove past the sign Using hiscalculator, he got the decimal 0.7236842 as a result

11 a What did Joshua enter in his calculator to get this result?

b What does the number Joshua got as a result mean?

c Explain how Joshua can use the decimal to determine the

percent of cars not speeding

C

Different Kinds of Ratios

Part-Part and Part-Whole

These two photos show Ms Humphrey as

a baby and as an adult

When Ms Humphrey was a baby, herheight was 60 cm and her head was

15 cm long

12 a As a baby, how long was her body

(not including the head)?

b What was Ms Humphrey’s

head-to-body ratio as a baby?

c What was her head-to-height ratio?

Ms Humphrey, 28 days

Ms Humphrey, 28 years

Now that she is an adult, Ms Humphrey’s height is 155 cm, and herhead is 27 cm long.

13 a As Ms Humphrey grew up, what happened to the size of her

head in relation to her height?

b Compare Ms Humphrey’s head-to-body and head-to-height

ratio as a baby and as an adult What do you notice? Describeyour findings

The head to body ratio is a part-part ratio.The head to height ratio is a part-whole ratio

14 a Explain what is meant by part-part ratio and part-whole ratio.

b Look back at the problems in this section about cars speeding

and not-speeding Describe a part-part ratio and a part-wholeratio fitting this situation

The head-to-height ratio changes over a person’s lifetime

15 a Use the chart above to estimate the head-to-height ratio of a

newborn baby

b What happens to the ratio as a person gets older? Explain.

Newborn 2 years 6 years 12 years 25 years

Jake’s head-to-height ratio is 1 to 8.

16 a How tall is Jake if his head is

20 cm long?

b How long is Jake’s head if he

is 168 cm tall?

c Find three other possible head

lengths and heights for Jake

Here are some head-to-height ratios for four different people

17 a Is it possible to determine which person has the longest head?

Explain your answer

b Which two people have the same head-to-height ratio?

How do you know?

Girls: 5 out of 20This is


14 , which is 25%.

So, 25% of the class is girls

In this section, you used two different kinds of ratios

You used the ratio of the number of cars speeding to the number not speeding

This is a part-part ratio

You used the ratio of the number of cars not speeding to the totalnumber of cars

This is a part-whole ratio

Sometimes this difference is hard to see, but it is important

A part-whole ratio can be written as a percent

A part-part ratio cannot be written as a percent

There are different strategies you can use to write a ratio as a percent.Here are some examples

• You can use the relationship between fractions and percents

In Ms William’s class, there are 20 students Five of these are girls What percent of this class is girls?

• You can rewrite the ratio as a comparison to 100

In one election, 120 out of 150 students voted for Joshua What percent of the students voted for Joshua?

Votes for Joshua: 120:150Using a ratio table, it is 80:100,

so it is 80% for Joshua

Votes for Joshua 120 40 80

Total Votes 150 50 100