Ratios
and Rates
Number
Mathematics in Context is a comprehensive curriculum for the middle grades.
It was developed in 1991 through 1997 in collaboration with the Wisconsin Center
for Education Research, School of Education, University of Wisconsin-Madison and
the Freudenthal Institute at the University of Utrecht, The Netherlands, with the
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., &
Pligge, M. A. (2010). Ratios and rates. In Wisconsin Center for Education
Research & Freudenthal Institute (Eds.), Mathematics in context. Chicago:
Encyclopædia Britannica, Inc.
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International Standard Book Number 978-1-59339-943-6
Printed in the United States of America
1 2 3 4 5 C 13 12 11 10 09
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
Director
Assistant to the Director
Director
Gail Burrill
Margaret R. Meyer
Els Feijs
Martin van Reeuwijk
Coordinator
Coordinator
Coordinator
Coordinator
Sherian Foster
James A, Middleton
Jasmina Milinkovic
Margaret A. Pligge
Mary C. Shafer
Julia A. Shew
Aaron N. Simon
Marvin Smith
Stephanie Z. Smith
Mary S. Spence
Mieke Abels
Nina Boswinkel
Frans van Galen
Koeno Gravemeijer
Marja van den
Heuvel-Panhuizen
Jan Auke de Jong
Vincent Jonker
Ronald Keijzer
Martin Kindt
Jansie Niehaus
Nanda Querelle
Anton Roodhardt
Leen Streefland
Adri Treffers
Monica Wijers
Astrid de Wild
Project Staff
Jonathan Brendefur
Laura Brinker
James Browne
Jack Burrill
Rose Byrd
Peter Christiansen
Barbara Clarke
Doug Clarke
Beth R. Cole
Fae Dremock
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.
Wisconsin Center for Education
Freudenthal Institute Staff
Research Staff
Thomas A. Romberg
David C. Webb
Jan de Lange
Truus Dekker
Director
Coordinator
Director
Coordinator
Gail Burrill
Margaret A. Pligge
Mieke Abels
Monica Wijers
Editorial Coordinator
Editorial Coordinator
Content Coordinator
Content Coordinator
Margaret R. Meyer
Anne Park
Bryna Rappaport
Kathleen A. Steele
Ana C. Stephens
Candace Ulmer
Jill Vettrus
Arthur Bakker
Peter Boon
Els Feijs
Dédé de Haan
Martin Kindt
Nathalie Kuijpers
Huub Nilwik
Sonia Palha
Nanda Querelle
Martin van Reeuwijk
Project Staff
Sarah Ailts
Beth R. Cole
Erin Hazlett
Teri Hedges
Karen Hoiberg
Carrie Johnson
Jean Krusi
Elaine McGrath
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Contents
Letter to the Student
Section A
Single Number Ratios
Car Pooling?
Miles per Gallon
Miles per Hour
Cruise Control
Summary
Check Your Work
Section B
21
23
25
28
29
Scale and Ratio
Scale Drawings
Scale Models
Maps
Summary
Check Your Work
Section E
11
15
16
18
19
Different Kinds of Ratios
Too Fast
Percent
Part-Part and Part-Whole
Summary
Check Your Work
Section D
1
4
6
7
8
9
Comparisons
Telephones and Populations
Television Sets
Cell Phones
Summary
Check Your Work
Section C
vi
30
35
36
38
39
Scale Factor
Smaller or Larger
Enlarged or Reduced
Summary
Check Your Work
41
43
48
49
Additional Practice
50
Answers to Check Your Work
55
Contents v
Dear Student,
Welcome to the unit Ratios and Rates. In this unit you will learn many
different ways to make comparisons.
Do you have more boys or girls in your class? If you count, you might
use a ratio to describe this situation. You can make comparisons
using 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 objects
they want to construct. Many people have hobbies creating miniature
worlds using trains, planes, ships, and automobiles. When you look
through a microscope, you see enlargements of small objects.
In all instances, ratios keep everything real. We hope you learn efficient
ways to work with ratios and rates.
Sincerely,
The Mathematics in Context Development Team
vi Ratios and Rates
A
Single Number Ratios
Car Pooling?
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 project
where they investigated the ways smog destroys plants, corrodes
buildings 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 so
many cars because most people do not carpool. They want to find
out if people carpool.
They set up an experiment to
count the number of cars and
people on the East Side Highway
adjacent to the school.
Section A: Single Number Ratios 1
A Single Number Ratios
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 group
counts cars and people for three minutes.
The second and third groups each calculate the average number of
people per car. They are surprised to find that both groups got an
average of 1.2 people per car.
2. How many cars and how many people might each group have
counted?
2 Ratios and Rates
Single Number Ratios A
A fourth group counts cars and people for one minute on the north
side 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 ratio of people to cars was
12 people to 10 cars or 12:10. Another way to describe this is it to
use the average number of people per car. The first three groups
calculated an average of 1.2 people per car. They might have found
this average by calculating the result of the division 12 10.
You can show both the ratio and the average in a ratio table.
Number of People
12
1.2
Number of Cars
10
1
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.
5. Explain why a higher average of people per car will result in
fewer cars. You may use examples in your explanation.
Section A: Single Number Ratios 3
A Single Number Ratios
Some students recommend that the average number of people per
car 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 of
people per car from 1.2 to 1.5 will lessen traffic congestion and
improve the quality of air.
Miles per Gallon
Another way to reduce air pollution is to encourage drivers to use
automobiles 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 mileage
pollute less than cars with low gas mileage.”
Gas mileage is the average number of miles (mi) a car can travel on
1 gallon (gal) of gasoline. It is represented by the ratio of miles per
gallon (mpg).
John says, “My car’s gas mileage is 25 mpg.”
7. How many miles can John travel on 12 gal of gas?
4 Ratios and Rates
Single Number Ratios A
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.
Miles
Gallons
50
2.5
100
5
20
1
Cindy says, “My gas mileage is 20 mpg.”
8. Explain Cindy’s calculation and answer.
The last time Arturo filled up his car, he
had driven 203 mi on 8.75 gal of gas.
9. Explain whether Arturo’s gas
mileage will be more or less than
Cindy’s gas mileage.
Arturo set up this ratio table to calculate his gas mileage.
Number of Miles
203
2,030
20,300
Number of Gallons
8.75
87.5
875
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.
Sheena traveled 81.2 mi on 3.75 gal of gas.
11. Compare the gas mileage of Sheena’s, Arturo ’s and Cindy’s cars.
What conclusions can you draw?
Section A: Single Number Ratios 5
A Single Number Ratios
Miles per Hour
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.
Miles
Hours
81.2
2
812
20
406
10
40.6
1
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 is
called the average speed. Average speed
is expressed in miles per hour (mi/h).
Average speed is expressed as a
single number.
6 Ratios and Rates
Single Number Ratios A
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 ratio
expressed 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.)
Cruise Control
Many modern cars are equipped with cruise control, which allows
the driver to set the car’s speed to be constant. This makes highway
driving easier and saves gas. Sheena used this feature to take two
trips.
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 the
cruise 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?
Section A: Single Number Ratios 7
A Single Number Ratios
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?
10
25
Number of Miles
75
750
30
Number of Gallons
2.5
25
1
10
25
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
8 Ratios and Rates
Number of Miles
90
45
Number of Gallons
2
1
Karla’s trip
Number of Miles
75
150
50
Number of Hours
1.5
3
1
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
Gallons of Gas
Section A: Single Number Ratios 9
A Single Number Ratios
Martha had her car repaired at a garage. Shown below is part of the
bill she received from her mechanic.
Village Automotive
Parts
Labor
Total
None
1.5 hr
$90.00
3. Use the following ratio table to find how much her mechanic
charged per hour.
Cost in Dollars
Number of Hours
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 calculate
gas mileage.
10 Ratios and Rates
B
Comparisons
Telephones and Populations
The table below shows the population and the total number of
telephones for 15 different countries.
Country
Population
Number of Telephones
Bolivia
8.4 million
1.26 million
Chad
9.0 million
44,000
China
1.3 billion
430.50 million
Cuba
11.2 million
580,700
Finland
5.2 million
6.3 million
France
59.8 million
73 million
Hungary
10.1 million
10.1 million
India
1.05 billion
54.6 million
Japan
127 million
150.82 million
Tonga
102,000
14,500
Micronesia
112,000
60,000
Solomon Islands
450,000
7,600
South Africa
45.3 million
17.17 million
Sudan
38.1 million
872,000
United States
292.6 million
331 million
Source: Encyclopaedia Britannica Almanac, 2005 (Chicago: Encyclopaedia Britannica, Inc. 2005)
1. According to the data, which countries in the table have more
telephones than people?
Section B: Comparisons 11
B Comparisons
Joan looks at the numbers in the table and says, “the United States
has the largest population because 292.6 is the highest number
before 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 60,000
telephones and a population of 112,000 people. The ratio of
people to telephones is 112,000:60,000.
12 Ratios and Rates
Number of People
112,000
.......
Number of Telephones
60,000
1
Comparisons B
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 share
one telephone.
It is also possible to look at the ratio of telephones to people. For
Micronesia, this ratio is 60,000:112,000.
5. a. Use this ratio to calculate the average number of telephones
per person.
b. Reflect Which number do you find the most useful to tell
something about the use of telephones in a country—the
number of people per telephone or the number of
telephones per person? Explain your choice.
If you compare countries with respect to the number of telephones
without 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 telephones
and consider the number of people living in these countries, the
comparison is a relative comparison, comparing telephones per
person.
Section B: Comparisons 13
B Comparisons
FINLAND
CHINA
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
Number of Telephones (in millions)
6.3
Population
5.2
(in millions)
1
China
Number of Telephones (in millions)
430.50
Population
1,300
(in millions)
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?
14 Ratios and Rates
Comparisons B
Television Sets
CANADA
FRANCE
BRAZIL
TV Sets
there are 317 TVs
Brazil has about 176 million people, and
e are about
for every 1,000 citizens. For Canada, ther
y 1,000 citizens.
ever
for
TVs
31.9 million people and 708
606 TVs for every
and
ple
France has about 59.7 million peo
1,000 citizens.
The paragraph on the left is
from a paper Brian wrote that
compares the numbers of
television sets in several of
the 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.
Section B: Comparisons 15
B Comparisons
Cell Phones
Since the 1990s, more and more people all over the world have cell
phones.
The table shows the number of cell phones per 1,000 people in the
year 2001 for some countries and for the world.
Denmark
United
States
Canada
Taiwan
Poland
World
740
440
320
970
260
160
Number of Cell
Phones per 1,000
People
CANADA
UNITED
STATES
16 Ratios and Rates
DENMARK
POLAND
TAIWAN
Comparisons B
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 countries
compare to the world average?
Math History
Ratios and Music
do
re
mi
la
sol
la
ti
do
C
D
E
F
G
A
B
C
Pythagoras (around 500 B.C.) was a Greek mathematician, teacher,
and philosopher. He found a relationship between ratios and the
musical scale as a result of his experiments with a monochord, a one
string musical instrument. He found that the shorter the string, the
higher 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).
(3)
do
(2)
Movable bridge
The other ratios are
C (do) 1:1
G (sol) 3:2
D (re) 9: 8
A (la) 5: 3
E (mi) 5 :4
B (te) 15 : 8
F (fa) 4 : 3
C (do) 2 :1
Section B: Comparisons 17
B Comparisons
You use numbers to make comparisons.
Absolute Comparisons
Comparisons can be absolute. When you make an absolute
comparison, you compare things without taking into consideration
anything 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 different
countries
comparing the number of telephones per thousand people in
different countries
When making a relative comparison, a ratio written as a single
number (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/h
versus 30 mi/h
Ratio tables are useful tools for making relative comparisons.
18 Ratios and Rates
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.
Population
(in millions)
Number of Cows
(in millions)
California
34.5
5.2
Colorado
4.4
3.1
12.5
1.4
Iowa
2.9
3.6
Kansas
2.7
6.6
Montana
0.9
2.5
Nebraska
1.7
6.4
South Dakota
0.8
4
21.3
13.6
5.5
3.3
State
Illinois
Texas
Wisconsin
U.S. Bureau of the Census. Statistical Abstract of the United State 1994 (114th edition) Washington, DC 1994.
Section B: Comparisons 19
B Comparisons
3. In your opinion, which of the countries below has the greatest
number of people per square mile? Show your work.
Country
Area (in sq mi)
Population
1.1 million
36.8 million
Japan
146,000
127 million
Brazil
3.3 million
176 million
Argentina
Source: Data from Encyclopaedia Britannica Almanac, 2005 (Chicago: Encyclopaedia Britannica, Inc., 2005)
JAPAN
BRAZIL
ARGENTINA
In your math class, determine the number of phones and people in
each household. Then find the number of phones per person.
20 Ratios and Rates
C
Different Kinds of Ratios
Too Fast
The citizens of Wrigley are concerned about
the number of people who speed through
town. The local police have identified the
four worst areas for speeding. The city
council has agreed to install traffic lights to
slow down the speeding cars.
At the present time, there is only enough
money in the budget to install one traffic
light. The council asks the police to decide
which area needs the traffic light the most.
The police make plans to study the situation
and give a report at the next council meeting.
In order to monitor the number of drivers
who speed through the four areas of town,
the police set up a device to count and
record the speed of passing cars.
Below is a chart showing the count at each area during a one-hour
period in the morning.
Speeders
Non-Speeders
Area 1
11
15
Area 2
42
20
Area 3
30
29
Area 4
4
0
1. a. Compare the results from these four areas of town.
b. What recommendation would you make to the city council?
Section C: Different Kinds of Ratios 21
C Different Kinds of Ratios
Suppose the police found another area of town where they suspect a
lot of speeding takes place. When they count the cars and figure out
how 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 sign
constantly shows the percent of cars that pass the sign and are within
the speed limit.
3. a. Why do you think the city put up this sign,
and why do you think the sign shows the
percent 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 the
sign 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 the
problem of speeding on the six
o’clock news. The report gave
some statistics to emphasize the
seriousness of the situation.
The police reported that
on Highway 19, two
cars were speeding for
every three that were
not speeding.
5. Can you conclude that over half
of the cars were speeding on
Highway 19? Why or why not?
Another TV station picked up the story. The newscaster from this
station wanted to describe the speeding situation on Highway
19 in terms of percents.
6. What percents could be used?
22 Ratios and Rates
Different Kinds of Ratios C
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.
Car
Time
Speed (in mi/h)
1
2:00 AM
53
2
2:02 AM
60
3
2:03 AM
55
4
2:05 AM
52
5
2:10 AM
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.
Percent
One way to find a percent is to use the relationship between fractions
and percents.
For example, if 12 of the cars were speeding, 50% were speeding.
8. Write all of the relationships between fractions and percents that
you know.
Section C: Different Kinds of Ratios 23
C Different Kinds of Ratios
Another way to find a percent is to rewrite each ratio as a number
compared to 100 (or per 100). A ratio table or a calculator may be
helpful with this strategy.
Number of Cars Not Speeding
Total Number of Cars
100
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 among
ratios, fractions, decimals, and percents. You already know many of
these relationships. Look at the table below.
10. a. Copy and fill in the table to show equivalent fractions, decimals,
and percents.
Ratio
Fraction
Decimal
Percent
1:2
1:3
1:4
1:5
1:15
0.3
1
20
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
percents. You may use examples to explain this relationship.
24 Ratios and Rates
Different Kinds of Ratios C
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 his
calculator, 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.
Part-Part and Part-Whole
These two photos show Ms. Humphrey as
a baby and as an adult.
When Ms. Humphrey was a baby, her
height 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-tobody ratio as a baby?
Ms. Humphrey, 28 days
c. What was her head-to-height ratio?
Ms. Humphrey, 28 years
Section C: Different Kinds of Ratios 25
C Different Kinds of Ratios
Now that she is an adult, Ms. Humphrey’s height is 155 cm, and her
head 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? Describe
your 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-whole
ratio fitting this situation.
Newborn
2 years
6 years
12 years
25 years
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.
26 Ratios and Rates
Different Kinds of Ratios C
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.
Head-to-Height Ratios
Person A
1 to 8
Person B
2 to 15
Person C
2 to 16
Person D
2 to 20
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?
Section C: Different Kinds of Ratios 27
C Different Kinds of Ratios
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 total
number 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?
Girls: 5 out of 20
1
This is 4 , which is 25%.
So, 25% of the 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 :150
Using a ratio table, it is 80:100,
so it is 80% for Joshua.
28 Ratios and Rates
Votes for Joshua
120
40
80
Total Votes
150
50
100
•
You can use the relationships among fractions, decimals, and
percents.
There were 48 out of 73 cars speeding. What percent of
the cars were speeding?
Speeding: 48:73.
Using a calculator, it is 0.6575…,
so 66% were speeding.
1. Make up your own example to explain the difference between a
part-part ratio and a part-whole ratio.
For every three people who take a certain medication without having
any side effects, five other people will experience side effects.
2. a. Write a part-part ratio that goes with this situation.
b. Write a part-whole ratio that represents this situation.
c. Which of the two ratios above, the one from a or from b,
can be written as a percent? Write this ratio as a percent.
3. Write a fraction and a percent for each of the ratios representing
the situation.
a. One out of every five drivers is a teenager.
b. Three out of four cars on the road are red.
c. Twenty-one out of 130 of the drivers surveyed said they had
gotten parking tickets.
You are given the following head-to-height ratios: 1: 8, 1: 5, 1: 7. Explain
which of these ratios is from the youngest person and tell why.
Section C: Different Kinds of Ratios 29
D
Scale and Ratio
Scale Drawings
Tim wants to rearrange the furniture in his room. He decides to make
a scale drawing of his room, called a floor plan. He can use the floor
plan to try out different room arrangements. This will save him the
work of moving the actual furniture. He can move the paper furniture
on his scale drawing.
Tim’s actual room dimensions are 2.6 m wide and 3 m long.
Tim decides to use graph paper. His first idea is to draw a floor plan
with dimensions 26 cm by 30 cm.
1. a. Explain why you think Tim decided on these floor plan
dimensions.
b. What are some advantages and disadvantages of making a
plan with dimensions 26 cm by 30 cm?
30 Ratios and Rates
Scale and Ratio D
Tim decides to use dimensions of 13 cm by 15 cm for his floor plan.
2. a. Why do you think Tim decided to use these dimensions?
b. Use Student Activity Sheet 1 to draw the same floor plan Tim
will draw of his room. Indicate the location for the door to his
room on the floor plan.
Section D: Scale and Ratio 31
D Scale and Ratio
A double number line is a useful tool to show the relationship between
the dimensions in a drawing and the actual room dimensions. Here is
a double number line that belongs to the scale drawing of Tim’s room.
0
1
5
0
10
cm in drawing
15
300
cm in the room
3. Copy this double number line under your own scale drawing on
Student Activity Sheet 1 and fill in the missing numbers on the
bottom of the line.
Here is the furniture for Tim’s room.
desk
w ⴝ 110 cm
d ⴝ 60 cm
h ⴝ 72 cm
dresser
w ⴝ 80 cm
d ⴝ 30 cm
h ⴝ 170 cm
chair
w ⴝ 50 cm
d ⴝ 50 cm
h ⴝ 100 cm
bed
w ⴝ 100 cm
d ⴝ 170 cm
h ⴝ 100 cm
On a separate piece of graph paper, draw each piece of furniture to
the same scale as the floor plan. Each miniature piece of furniture
should represent the space the actual furniture takes up on the floor
of Tim’s room. Cut out these pieces and move them around on your
floor plan until you have an arrangement you like.
4. Draw your favorite arrangement for Tim’s room on your floor
plan on Student Activity Sheet 1.
32 Ratios and Rates
Scale and Ratio D
The double number line used for Tim’s floor plan indicates a scale
ratio of 1:20.
5. Reflect Look back at the double number line for Tim’s floor plan.
Describe how you would explain to someone what it means
that Tim’s floor plan has a scale ratio of 1:20.
Tim’s older sister, Jenna, wants to rent an apartment. Below is a floor
plan of an apartment she likes a lot. She wants to use the floor plan to
find the dimensions of the living room.
Scale 1:75
6. a. Use this ratio table to help Jenna find the length of the living
room.
Length in Drawing (in cm)
1
Actual Length
75
(in cm)
b. What is the actual width of the living room? Show your
calculations.
Section D: Scale and Ratio 33
D Scale and Ratio
Tim and his friends want to build a sand volleyball court. They use the
scale drawing below to begin to figure out the actual dimensions. Tim
says, “One centimeter in the drawing is actually 3 meters.”
Volleyball court
Scale 1:300
7. a. Do you agree or disagree with Tim’s statement? Explain.
b. What are the actual dimensions of the court? Of the total
volleyball space (including the part around the actual court)?
A scale drawing represents objects that are too large or too small to
draw at actual size.
A scale ratio shows the relationship between the dimensions in the
drawing and the actual dimensions of the object. A scale ratio of
1:100 on a floor plan can mean:
1 centimeter represents 100 centimeters or
1 meter represents 100 meters or
1 millimeter represents 100 millimeters or
1 inch represents 100 inches
An architect makes a scale drawing. She uses 2 cm to represent 100 m.
8. a. What is the scale ratio for her drawing? Show your work.
b. What do you think she is drawing?
34 Ratios and Rates
Scale and Ratio D
Scale Models
Instead of a scale drawing on a piece of
paper, you can make a three-dimensional
scale model.
The photo on the left shows a plane with
a scale model of the plane on its wing.
The model is built with a scale of 1:6.
The actual length of the plane is 6.6 m
and its wingspan is 8 m.
9. a. What is the length of the scale
model airplane?
You may want to use a ratio table like the one below for your
calculations. (Note: Instead of using centimeters, you may
prefer to use meters.)
Length of Actual Plane (in cm)
Length of Scale Model (in cm)
b. How long is the wingspan of the scale model airplane?
The photo on the left shows six different model
trains. Each of them is built to a different scale.
The five scales below are commonly used.
Z scale: trains built to a ratio of 1:220
N scale: trains built to a ratio of 1:160
HO scale: trains built to a ratio of 1:87
S scale: trains built to a ratio of 1:64
Scale O: trains built to a ratio of 1:48
10. What scale was used to build the smallest train
shown? How do you know for sure?
Section D: Scale and Ratio 35
D Scale and Ratio
Maps
You may remember doing other work with scale lines on a map.
Scale lines are like a ruler. You can use scale lines to estimate or even
measure distances on a map. The map below shows the northern
part of San Francisco.
GO
LD
EN
Golden Gate Bridge
Fort Point National
Historic Site
E N AT
IO
G AT
NA
L REC
U.S. Coast
Guard Station
0
1
1
4
Yacht Harbor
DRIV
MARIN
E
1 km
Marina
Green
AT I O N AREA
DOYL
E
4
Ft. Mason
Palace of Fine Arts
(Exploratorium)
RE
3
2
.
A BLVD
Marina
Sarita walks from the Marina Green to Fort Point National Historic
Site. The black dotted line shows Sarita’s walking path.
11. Estimate the length of Sarita’s walking path.
If you want to find a distance on a map, you need to go from one
measurement unit to another. The following conversions are
common. Do you know them?
12. Check what you know by copying and filling in the following
measuring relationships. Add others that you might know.
1 meter
……. centimeters
1 kilometer
……. meters
You can transform a scale line on the map into a double number line.
Here is a double number line adapted from the scale line on the
San Francisco map.
0
0
36 Ratios and Rates
1
2
3
4
1,000
centimeters (on map)
meters (actual)
Scale and Ratio D
13. a. Describe the differences and similarities between the scale line
on the map and the double number line on the previous page.
b. Use the double number line to find the missing numbers in the
table below.
Distance on Map (in cm)
1
Actual Distance (in m)
Actual Distance (in cm)
c. What is the scale ratio of the map?
14. Suppose you have a map made on a scale of 1:50,000. You
measure 10 cm on the map. How many kilometers does
this distance represent?
Here are three different maps of three
different islands: Norfolk (Australia),
Iwo Jima (Japan), and Hierro (Spain).
Each map was made using a different
scale. The scale is indicated on each
map.
PA C I F I C
NORFOLK
ISLAND
OCEAN
Mt. Bates
318 m
29° S
Cascade
Burnt Pine
Middlegate
Kingston
Nepean I.
Philip I.
AUSTRALIA
NEW
ZEALAND
AT L A N T I C
168° E
HIERRO
ISLAND
OCEAN
Valverde
fo
1 : 500,000
27° 45' N
PA C I F I C
Sabinosa
El
Go
l
Isora
Taibique
IWO JIMA
Kitano Pt.
Kangoku Rock
Kama Rock
OCEAN
Restinga
ATLANTIC
OCEAN
EUROPE
SPAIN
Hanare Rock
Nishi
Motoyama
Air
Base
Hill
110 m
Minami
AFRICA
RUSSIA
Mt. Suribachi
170 m
29° 45' N
JAPAN
Tobiishi Pt.
CHINA
1:1,000,000
18° W
Source: Times Atlas of the World. plates 10, 20, and 96
PACIFIC
OCEAN
1 : 250,000
141° 20' E
If you compare the size of the islands visually, you might think the
three islands all look about the same size. In reality, this is not true!
15. Write the names of the islands in order from the largest to the
smallest island. Explain how you decided what the order was.
Section D: Scale and Ratio 37
D Scale and Ratio
You use a scale drawing to represent things that are too large or too
small to draw. A scale ratio indicates the relationship between the
dimensions on the scale drawing and the actual dimensions. You
use a scale ratio to create scale models.
To create a scale drawing or model, you need to know the relationship
between the scaled dimensions and the actual dimensions. This
relationship can be given with:
a scale line
0
10m
a scale ratio
1:1000
A scale ratio always begins with the number 1. Both numbers represent
identical units. The scale ratio 1:1,000 means that 1 cm on the drawing
represents 1,000 cm in reality.
a statement
On the map, a distance of 1 cm is actually 1,000 cm, which is 10 m.
A ratio table and a double number line can help you to organize your
work and make your calculations involving scale easy.
Ratio Table:
Distance on Map (in cm)
1
Actual Distance (in cm)
1000
Double Number Line
1
0
1000
1000 centimeters 10 meters
38 Ratios and Rates
cm on map
cm in reality
1. A room is 3 m wide and 4 m long.
Make a scale drawing of this room using a scale of 1:50
Here is a photo of a Swallowtail butterfly (Papilio
machaon). The wingspan of the actual butterfly
is 10 cm.
2. a. If you wanted to make a life-size drawing
of the butterfly, would it fit on a page in
this book?
b. What is the size of the wingspan in the
photo?
c. Use a double number line or a ratio table
to find the scale ratio of the photo.
d. What is the actual length of the body of the butterfly?
Show your calculations.
Here is a scale line from a map:
0
5
kilometers
3. a. What is the actual distance of 1 cm on this map?
b. What is the scale ratio of the map?
Section D: Scale and Ratio 39
D Scale and Ratio
The map below shows a part of downtown Philadelphia.
Scale 1:5,000
Apple
Tree St
Thomas Jefferson
Univ. Hospital
S 7th St
St
St
Mall
Independence
History
Museum of
Philadelphia
Balch
Institute
for Ethnic
Studies
Ranstead
N 6th St
N 7th St
N 8th St
N 9th St
S 9th St
St
Ionic St
St
Walnut
Walnut
Street
Theater
St
S 6th St
Sansom
National
Archives
Branch
St
S 8th St
Chestnut
N 10th St
Market
Reeding
Terminal
Market
St
S 10th St
N 11th St
Filbert
S 11th St
S 12th St
N 12th St
Arch
African-American
Historical and
Cultural Museum
4. a. Copy and complete the following ratio table for the map.
Distance on Map (in cm)
1
Actual Distance (in cm)
.............
b. How far is a walk from Sansom Street to Arch Street?
(Use meters or kilometers for your distance.)
Suppose a map has a scale ratio of 1:20,000.
5. a. Do you think this map was designed to be used by someone
who is walking or someone who is driving? Explain your answer.
b. Make a scale line for this map.
Write a paragraph describing the need for using scale lines and
scale ratios in designing toy cars. Be exact in your descriptions.
40 Ratios and Rates
E
Scale Factor
Smaller or Larger?
You can use ratios in many different ways. One of them is working
with scales. You can zoom in or out with a camera or microscope to
make the objects on photos or slides appear to be larger or smaller
than the actual object.
On the right is a photo of a ladybird beetle,
or ladybug.
1. a. Under the photo, notice the (4 )
next to the name, “Ladybird Beetle.”
What does this 4 mean?
b. What is the length of an actual
ladybird beetle?
Ladybird Beetle (4)
A ladybird beetle lays very
small eggs. They are about
1.5 mm long and 0.5 mm wide.
2. a. Try to make a life-size drawing of a ladybird beetle’s egg.
Above is an enlarged picture of these eggs.
b. Measure the length of the egg in the lower right corner of
the picture.
c. How many times has the egg been enlarged?
Section E: Scale Factor 41
E
Scale Factor
On the left is a picture of a frog.
3. a. What does 0.5 mean?
b. In reality, what is the actual length of this
frog? Show your work.
Frog (0.5x)
On the right is a picture of salamander larvae.
4. a. Are these larvae in the picture reduced
or enlarged from their actual size?
Salamander Larvae (4x)
b. Based on the larva shown above, what is the actual length of
the real salamander larva?
Here is a picture of a mature
salamander.
Mature Salamander (0.3x)
5. a. Which animal is longer, the frog (see problem 3) or the
salamander? Explain your reasoning.
b. Compare the salamander larva with the mature salamander.
How many times longer is the mature salamander than the
larva?
A scale factor indicates how many times a measurement of an object
has been enlarged or reduced. You can use the scale factor with
arrow language to describe the enlargement or reduction.
Here is an arrow string describing an enlargement with a scale factor
of 5.
scale factor
Measure of Original
5
Measure of Enlargement
6. What can you say about the numbers used as a scale factor for a
reduction? Write an arrow string example to describe a reduction.
42 Ratios and Rates
Scale Factor E
Enlarged or Reduced?
This is a picture of the smallest butterfly in the
world.
The scale factor of the picture is 4.
7. Make a life-size drawing of this butterfly.
The Western Pygmy blue
(Brephidium exilis)
This is a picture of the largest butterfly in the world: the female
Queen Alexandra birdwing butterfly (Ornithoptera alexandrae).
It lives in New Guinea.
The scale factor of the picture is 0.25.
wingspan
?
8. a. How many centimeters is the
wingspan of the actual butterfly?
b. Find the scale ratio of this picture.
You may want to look back over
to Section D, where you first worked
with a scale ratio.
c. Consider the scale ratio and the
scale factor. Explain how they
relate to each other.
9. a. Draw an enlargement of the
shape on the right using a
scale factor of 3. Use centimeter
graph paper for your drawing.
b. Are all sides tripled?
c. Reflect How has the area
changed? How do you know?
Section E: Scale Factor 43
E
Scale Factor
Four pictures are enlarged. Measurements from the original pictures
and the enlarged pictures are in the table below.
Length of Original
(in cm)
Length of Enlargement
(in cm)
Picture A
2
8
Picture B
6
18
Picture C
3.5
14
Picture D
7.5
32
10. a. For which two pictures are the scale factors the same?
b. Are the remaining two pictures enlarged more or less than the
two pictures with the same scale factor? How do you know?
To find a scale factor, you can
use arrow language.
Measure of Original
scale factor
?
Measure of Enlargement
scale factor
Fill in the measurements from
the problem.
?
6 cm
15 cm
11. a. What calculation can you make to find the scale factor for the
enlargement described above?
b. Find the scale factor.
Here is a ratio table for the enlargement above.
Length of Original Drawing (in cm)
6
1
Length of Enlarged Drawing (in cm)
15
....
12. a. Copy the ratio table and fill in the missing numbers, especially
the last entry.
b. Compare your answers to questions 11b and 12a. What do
you notice?
44 Ratios and Rates
Scale Factor E
To find a scale factor, you can use a ratio table.
scale factor
Length of Original Drawing (in cm)
12
6
2
1
Length of Enlarged Drawing (in cm)
600
300
100
50
ⴛ 50
13. a. Describe the process of using a ratio table to find a scale factor.
b. The ratio between an original picture and its enlargement is
12:75. Use a ratio table to find the scale factor.
The scale factor for a reduction is a number between 0 and 1. You
can find the scale factor for a reduction in the same way as the scale
factor for an enlargement. You can use either arrow language or a
ratio table.
Anita found some large starfish; she measured them and made a
scale drawing of all three of them. Anita recorded the measurements
of the real starfish and the drawings in a table.
Length
Length of Original
Starfish (in cm)
Length of Starfish in
Drawing (in cm)
Gold-Colored Starfish
16
4
Red Starfish
25
5
Brownish Starfish
12
4
14. a. Did Anita use the same scale factor for the three drawings she
made? How do you know?
b. Use arrow language or a ratio table to find the scale factor of
each drawing.
Section E: Scale Factor 45
Scale Factor
E
The actual length of a mosquito from its head to the tips of its wings
is about 0.8 cm.
15. Find the scale factor for each picture of the mosquito shown
below.
a.
b.
c.
16. a. Describe why you might want to see an enlargement of an
object. Give an example.
b. What kind of numbers will describe the scale factor of an
enlargement?
c. Describe why you might want to see a reduction of an object.
d. What kind of numbers will describe the scale factor of an
reduction?
46 Ratios and Rates
Scale Factor E
Math History
The Golden Ratio
Let’s start with a small experiment. Here are 84 rectangles. They all
have the same height but the width varies. Select the one you prefer.
Have other people select the one they prefer as well.
Record the preferences!
It is very likely that a lot of people have chosen
the golden rectangle as their first choice. (This
is the fourth from the left in the third row.)
In the golden rectangle, the length (a) and the
width (b) relate to each other as a:b (a b):a.
Or in words: the ratio between the length (a) and
the width (b) of the golden rectangle is the same as
the ratio of the sum of the length and width (a + b)
to the length (a). This ratio is called the golden ratio
and is about 1.618 to 1.
The golden rectangle is used in art and architecture.
See if you can find the golden rectangle in these
buildings.
Section E: Scale Factor 47
E Scale Factor
A scale factor tells you how you enlarged or reduced every
measurement of the original picture or object. Note that a
scale factor is always expressed as a multiplier.
If a scale factor is greater than one, it is an enlargement.
If a scale factor is between zero and one, it is a reduction.
You can work with scale factors using arrow language or a ratio table
These tools help organize your work and make calculations easier.
For example, an enlargement with a scale factor of 2:
Using arrow language:
scale factor
Length of Original Drawing (in cm)
2
Length Enlargement (in cm)
Using a ratio table:
scale factor
Length of Original Drawing (in cm)
5
1
Length of Enlargement
10
2
(in cm)
2
For example, a reduction with a scale factor of 0.25:
Using arrow language:
scale factor
Length of Original Drawing (in cm)
0.25
Length Reduction (in cm)
Using a ratio table:
scale factor
Length of Original Drawing (in cm)
12
4
1
Length of Reduction
3
1
0.25
(in cm)
0.25
Note that the scale factor is always a multiplication factor. To make
the calculations you can use division as well.
48 Ratios and Rates
Four pictures are reduced. One measurement of the original
pictures and the reductions are listed in the table below.
Length of Original (in cm)
Length of Reduction (in cm)
Picture A
8
2
Picture B
24
8
Picture C
35
8.5
Picture D
30
7.5
1. a. For which two pictures are the scale factors the same?
b. Are the two remaining pictures reduced more or less than the
two pictures with the same scale factor? How do you know?
The actual length of an ant is about 4 mm.
2. What is the scale factor for the drawing
of the ant shown on the right?
3. Copy and complete the table below.
Scale Factor
Enlargement or Reduction?
Length
5
0.1
1
4
enlargement
1
Binoculars have a scale ratio. Describe in your own words what a
scale ratio of 1:35 would mean for binoculars.
Section E: Scale Factor 49
Additional Practice
Section A
Single Number Ratios
Mr. Adams asked his students to conduct a survey to find the average
number of children in a family.
Sarah surveyed 12 families in her neighborhood and counted a total
of 28 children.
1. a. In your opinion, how many of the families Sarah surveyed do
you think had exactly two children? Why do you think so?
b. What is the average number of children per family in Sarah’s
neighborhood survey?
c. What are some other numbers of families and children that
produce this same average?
Dennis surveyed his neighborhood. He found an average of
2.5 children per family.
2. List some different possibilities for the numbers of families and
children that Dennis could have counted.
Dave has a car. He made three different trips.
Trip A: 112 miles
Trip B: 70 miles
Trip C: 21 miles.
3. The gas mileage of Dave’s car is 28 mpg. For each of the trips
calculate how many gallons of gas Dave used.
Dave traveled non-stop for each trip and made a record of the times
listed below.
Trip A: 2 hours (112 miles)
Trip B: 1.5 hours (70 miles)
Trip C: 20 minutes (21 miles)
4. For each trip, calculate Dave’s average speed in miles per hour.
50 Ratios and Rates
Section B
Comparisons
The table below lists just a few of the countries you saw in the table
on page 11.
Country
Population
Number of
Telephones
Number of
Radios
Bolivia
8.4 million
1.26 million
5.5 million
Chad
9.0 million
44,000
1.9 million
Finland
5.2 million
6.3 million
8.4 million
Tonga
102,000
14,500
61,000
Solomon Islands
450,000
7,600
36,000
United States
292.6 million
317 million
598 million
Source: Encyclopaedia Britannica Almanac 2005 (Chicago: Encyclopaedia Britannica, Inc., 2005).
A fourth column with the number of radios has been added.
1. a. Which country has the fewest number of radios?
b. Do any countries have more telephones than radios? If so,
which ones?
c. Which country has the largest number of radios per person?
2. a. Choose two countries and compare the numbers of people per
telephone in an absolute and in a relative way.
b. For the same two countries you choose in part a, compare the
numbers of people per radio in these countries, both in an
absolute and in a relative way.
Additional Practice 51
Additional Practice
Section C
Different Kinds of Ratios
There are four city recreation centers where girls can participate
in after-school sports. Because funding is limited, only one
recreation center will have a girls’ basketball coach. The following
table shows the number of girls involved in basketball and the
number involved in sports other than basketball at each recreation
center.
Recreation
Center
Girls Involved
in Basketball
Girls Involved in
Other Sports
1
16
13
2
12
0
3
35
22
4
23
20
Compare the four centers.
1. a. Which center has the highest number of girls participating in
after-school sports?
b. Which center has the highest number of girls involved in
basketball?
c. Which center has the highest percent of its participating girls
involved in basketball? Explain.
Write a percent for each of the following:
2. a. One out of every four girls in the seventh grade plays
volleyball.
b. Three out of every five boys in the sixth grade play soccer.
c. Eleven out of every 15 students like to participate in sports
events.
d. Thirty-two out of the 78 boys in our school are on the football
team.
52 Ratios and Rates
Additional Practice
Section D
Scale and Ratio
Members of the Lewis and Clark expedition (1804–1806) searched for
an overland route from the Mississippi River to the Pacific Ocean.
ws
Yello
nsi n
o
rad
lo
Co
atte
Pl
GE
RAN
ba
W isc o
te
i
L a ke
Michigan
St. Louis
A
sas
an
rk
1:25,000,000
s
l at
S
VE
DESERT
pp
ou r i
Miss
N.
P
ED
I NT T
PA ES E R
D
DEATEHY
VALL
Lewis’s
MOHA return route
M i s si
IN
RA
SIER DA
A
NEV
Clark’s return route
si
S
TA
Lewis and Clark
L ak e
S up e r i o r
I N
UN
LEWIS AND CLARK ROUTES
S
Fort Mandan
BLACK
H I LL S
ny
Al
Lake Winnipeg
Missouri
tone
E
Re d
B I G S N OW
MTS.
W
A
P L
ADE
wan
BEAR PAW
MTS.
RANGE
SC
N
MO
CA
S.
Y
A ST RANG E
Sna
CO
ke
S a s k a tc he
OT
BITTERRO
CK
RO
IC
OLYMP.
MTS
Fort Clatsop
skatc
hew
an
T
E A
G R
Columbia
P A C I F I C
O C E A N
N. S a
O ZA
P
RK
U
EA
LAT
O hio
The scale of this map is 1:25,000,000.
1. Estimate the distance Lewis and Clark covered when they
traveled from St. Louis to Fort Clatsop.
Here is an excerpt from a journal.
May 14, 1804
Expedition begins in St. Louis.
October 24, 1804
Expedition discovers earth lodge villages
of the Mandan and Hidatsas Indians. The
captains decide to build Fort Mandan across
the river from the main village.
2. a. Use the information in the journal to estimate the average
distance the expedition covered per month during this period.
Note that Fort Mandan is about halfway between St. Louis
and Fort Clatsop.
b. Also find the average distance per day.
Additional Practice 53
Additional Practice
Eiffel Tower Puzzle
The wrought-iron original has
attracted millions of visitors and
is the symbol of Paris. Now you
can construct your own Eiffel
Tower to a scale of 1:500.
The height of the actual tower is
312 meters.
3. What is the height of the
model?
Section E
Scale Factor
Some ponds become green in the summer because of the large
number of algae in the water. Algae are actually very small plants.
The simplified drawing on the left shows the
Scenedesmus alga. This is the size they would
appear to be if seen through a microscope that
enlarges 250x.
1. Tamar wants to make a life-sized drawing
of one alga. She begins by calculating the
actual length of this type of alga. She is
suddenly very surprised. Why?
Alga (250 )
2. Would you be able to see a Scenedesmus
alga with a microscope that enlarges 80x?
Why or why not?
Electron microscopes are more powerful than ordinary
microscopes. This picture from an electron microscope
shows a blood cell enlarged by a scale factor of 10,000.
3. a. Which are larger Scenedesmus algae or blood cells?
Explain your reasoning.
Blood Cell (10,000 )
b. Could you make a drawing of the blood cell if it was
enlarged by a factor of 250? Why or why not?
c. How would a magnification of 80x work for looking
at blood cells? Explain your answer.
54 Ratios and Rates
Section A
Single Number Ratios
1. a. There are 2.5 people per car.
There are different ways to find this answer.
Using a ratio table:
Number of People
40
20
10
5
2.5
Number of Cars
16
8
4
2
1
Using division:
40 16 2.5
b. About 35.6 students per class.
Using a calculator, 320 9 35.5555, and rounding to one
decimal is 35.6.
2. 18 mpg
Sample strategy:
Miles
Gallons of Gas
108
54
18
6
3
1
3. The mechanic charged $60 per hour.
Here are two ratio table strategies.
Charge
$90
$900
$180
$60
Hours
1.5
15
3
1
or
Charge
$90
$30
$60
Hours
1.5
0.5
1
4. 25:15 written as a single number is 1.7.
Here is one ratio table strategy.
Number of People
25
5
1.7
Number of Cars
15
3
1
Answers to Check Your Work 55
Answers to Check Your Work
5. Ask a classmate to do your problem. Check and discuss the
answer. Here is one possible problem.
Talia makes about 82% of her free throws. This season she shot
40 free throws. How many free throws did she make?
Solution: 82% is 82:100, so setting up a ratio table, she made
about 33 free throws.
Section B
Number of Free Throws Made
82
8.2
≈33
Number of Free Throws Shot
100
10
40
Comparisons
1. Two different answers are possible. Your explanation is critical.
Tom is correct: I start with the ratio 14.2 million: 43.6 million.
This is the same ratio as 14.2:43.6. I use a ratio table to get a
ratio of phones per person. Then I find the number of telephones
per 100 people.
Number of Telephones
14.2
≈0.33
33
Number of People
43.6
1
100
Tom is incorrect:
I start with Tom’s statement that the ratio is 33 telephones for
every 100 people. Then I use the ratio table to build up to
43.6 million people.
Number of Telephones (in millions)
33
3.3
0.33
14.4
Number of People
100
10
1
43.6
(in millions)
This doesn’t match the data since 14.4 phones is not the same as
14.2 phones.
2. a. Texas, with 13.6 million cows.
b. The comparison is absolute because the number of people is
not involved.
c. Kansas has 244 cows per 100 people while Montana has
278 cows for every 100 people, so Montana has more cows
for every 100 people.
56 Ratios and Rates
Answers to Check Your Work
Kansas: 6.6 million cows per 2.7 million people; I calculated
6.6 2.7 ➝ 2.44. This is 244 cows per 100 people.
For this calculation, you can also use a ratio table.
Number of Cows (in millions)
6.6
66
2.44
244
Number of People (in millions)
2.7
27
1
100
Montana: 2.5 million cows per 0.9 million people; I calculated
2.5 0.9 ➝ 2.78. This is 278 cows per 100 people
d. The comparison is relative, because the number of cows is in
relation to the number of people.
3. Different answers are possible depending on how you compared
the data.
If you made a relative comparison your can compare the area
per person, or you can compare the number of people per square
mile. For either strategy, the conclusion is that Japan is the most
populated. Japan, compared to Argentina and Brazil, has more
people per square mile and each person has the least area
available to them.
Here are the results for both strategies.
Country
Area
(sq mi)
Population
Square Miles Population per
per Person
Square Mile
Argentina
1.1 million
36.8 million
0.030
33.5
Japan
146,000
127 million
0.001
869.9
Brazil
3.3 million
176 million
0.019
53.3
If you made an absolute comparison, Brazil has the largest
population with 176 million people.
Answers to Check Your Work 57
Answers to Check Your Work
Section C
Different Kinds of Ratios
1. Discuss the examples you found with a classmate and check
whether they are right. Here is one example.
This season, our baseball team won 23 games and lost 20 games.
The win: loss ratio is 23:20, which is a part-part ratio.
The win: total ratio is 23:43, which is a part-whole ratio.
With the part-part ratio, you can see they won 3 games more than
they lost. Using the part-whole ratio, the team win average is
over 500 (0.535). You can use this average to compare our school
to other schools across the country that might play more or
fewer games than we do.
2. a. 3:5
b. 3:8 (people with side effects) or 5:8 (people without side effects)
c. Only a part-whole ratio can be written as a percent. This is the
1 is 0.125; this helps me with
ratio in 2b. I always remember 8
other eighths.
3 , which is 0.375 or 37.5%.
3:8 written as a fraction is 5:8 written as a fraction is
8
5 , which is 0.625 or 62.5%.
8
1 or 20%.
3. a. 1:5 written as a fraction is 5
3 or 75%.
b. 3:4 written as a fraction is 4
21 .
c. 21:130 written as a fraction is 130
Using a calculator 21 130 ≈ 0.16153 or about 16% (or 16.2%).
58 Ratios and Rates
Answers to Check Your Work
Scale and Ratio
1. Here is one sample drawing. Your room dimensions should be
8 cm by 6 cm.
Door
Bed
Window
Section D
6 cm
Desk
Chair
Dresser
8 cm
The scale ratio of 1:50, means 1 cm on the map is 50 cm in reality.
Working up to 300 cm (3 m) and 400 cm (4 m), you can get the
drawing dimensions needed.
1 ➝ 50, 2 cm ➝ 100 cm, 8 cm ➝ 400 cm and 6 cm ➝ 300 cm.
2. a. Yes, a life-size drawing of the butterfly would fit on a page in
this book because it is 10 cm. Here is how 10 cm looks.
10 cm
b. About 2.5 cm.
c. The scale ratio is 1:4.
Sample strategy using a double number line:
0
1
2.5
5
In Drawing (in cm)
0
4
10
20
Actual
(in cm)
Sample strategy using a ratio table:
In Drawing (in cm)
2.5
25
1
Actual
10
100
4
(in cm)
Answers to Check Your Work 59
Answers to Check Your Work
d. The actual length of the body of the butterfly is 3.6 cm.
Here is one strategy.
The body length in the reduction is about 0.9 cm. Since this
1 of the length, the body length is about 3.6 cm.
represents 4
(0.9 4 ➝ 3.6).
3. a. 1 cm represents 1 kilometer, which is 1,000 meters.
b. The scale ratio is 1:100,000
On the map, 1 cm represents 1,000 m.
In reality, 1,000 m is 100,000 cm. So 1 cm on the map represents
100,000 cm in reality. The scale ratio is 1:100,000.
4. a.
On Map (in cm)
1
In Reality (in cm)
1,000
1,000
In Reality (in cm)
100,000
Distance on a Map (in cm)
1
Actual Distance
(in cm)
5,000
b. About 385 m or 0.385 km.
The distance on the map is about 7.7 cm. Since 1 cm represents
5,000 cm, you can calculate 7.7 5,000 ➝ 38,500 cm or 385 m.
If you measured a distance between 7.3 cm and 7.8 cm on the
map, your answer must be between 365 m and 390 m.
5. a. The map is designed to be used by someone who is walking.
Here is one way of reasoning.
1 cm on the map represents 20,000 cm in reality. This is about
200 m. If 1 cm represents 200 m then 10 cm represents 2,000 m,
which is 2 km. I chose 10 cm, because that fits nicely on a page.
The map is not for driving because you would be off the map
before you knew it. 2 km is a short distance.
60 Ratios and Rates
Answers to Check Your Work
b. Here is one possible scale line.
0
meters
1,000
200
Your scale line might look different. You might have other
distances indicated like 400 m (at 2 cm); 600 m (at 3 cm); etc.
Instead of meters, it may show kilometers, and every 5 cm
is 1 km. Note that for a scale line to be correct, 1 cm must
represent 200 m.
Section E
Scale Factor
1. a. Pictures A and D have the same scale factor, 0.25.
Here are two strategies.
Calculating the scale factor of each:
Scale Ratio
Scale Factor
Picture A
2: 8
0.25
Picture B
8 : 24
about 0.33
Picture C
8.5 : 35
about 0.24
Picture D
7.5 : 30
0.25
Finding the scale factor using ratio tables:
Picture A
Original
(cm)
8
4
1
Reduction (cm)
2
1
0.25
(cm)
24
3
1
Reduction (cm)
8
1
0.33-
(cm)
35
70
1
Reduction (cm)
8.5
17
0.24-
(in cm)
30
60
20
4
1
Reduction (in cm)
7.5
15
5
1
0.25
0.25
Picture B
Original
⯝ 0.33
Picture C
Original
⯝ 0.24
Picture D
Original
0.25
Answers to Check Your Work 61
Answers to Check Your Work
b. Picture B is reduced less, and picture C is reduced a little more.
Sample Explanation:
Pictures A and D are one-fourth of their original size. Picture B
is one-third its original size. So picture B is not reduced as
much as pictures A and D. Similarly picture C is reduced a little
more than the other two because 0.24 is less than 0.25.
Note: When a reduction is minor (reduced less), the scale
factor is closer to 1. When a reduction is extensive (reduced
more), the scale factor is closer to 0.
2. The scale factor is 10.
Here is one sample strategy.
The measured length in the picture is about 4 cm or 40 mm.
I set up an arrow string to find the scale factor.
⎯⎯→
?
40 mm
4 mm ⎯⎯⎯
I found 4 10 40, so the scale factor is 10.
3.
62 Ratios and Rates
Scale Factor
Enlargement or Reduction?
5
enlargement
0.1
reduction
1
4
reduction
Any number that
is greater than 1
enlargement
1
Neither; exact replica