Ratios and Rates

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Ratios

and Rates

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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.

© 2010 Encyclopædia Britannica, Inc. Britannica, Encyclopædia Britannica, the thistle logo, Mathematics in Context, and the Mathematics in Context logo are registered trademarks of Encyclopædia Britannica, Inc.

All rights reserved.

No part of this work may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage or retrieval system, without permission in writing from the publisher.

International Standard Book Number 978-1-59339-943-6 Printed in the United States of America

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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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

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

Rose Byrd Mary C. Shafer Marja van den Adri Treffers

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

Doug Clarke Marvin Smith Vincent Jonker

Beth R. Cole Stephanie Z. Smith Ronald Keijzer

Fae Dremock Mary S. Spence Martin Kindt

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 Project Staff

Sarah Ailts Margaret R. Meyer Arthur Bakker Nathalie Kuijpers

Beth R. Cole Anne Park Peter Boon Huub Nilwik

Erin Hazlett Bryna Rappaport Els Feijs Sonia Palha

Teri Hedges Kathleen A. Steele Dédé de Haan Nanda Querelle

Karen Hoiberg Ana C. Stephens Martin Kindt Martin van Reeuwijk

Carrie Johnson Candace Ulmer

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12, 14–16, 20 © Encyclopædia Britannica, Inc.; 19, 22, 23, 32 Holly Cooper-Olds;

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Getty Images; 30 Sam Dudgeon/HRW; 35 (top) Jim Vogel; (bottom) Kalmbach Publishing Co. collection; 39 © Corbis; 41 (left to right) © Digital Vision/

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Letter to the Student vi Section A Single Number Ratios

Car Pooling? 1

Miles per Gallon 4

Miles per Hour 6

Cruise Control 7

Summary 8

Check Your Work 9

Section B Comparisons

Telephones and Populations 11

Television Sets 15

Cell Phones 16

Summary 18

Check Your Work 19

Section C Different Kinds of Ratios

Too Fast 21

Percent 23

Part-Part and Part-Whole 25

Summary 28

Check Your Work 29

Section D Scale and Ratio

Scale Drawings 30

Scale Models 35

Maps 36

Summary 38

Check Your Work 39

Section E Scale Factor

Smaller or Larger 41

Enlarged or Reduced 43

Summary 48

Check Your Work 49

Additional Practice 50 Answers to Check Your Work 55

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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, T

Thhee MMaatthheemmaattiiccss iinn CCoonntteexxtt DDeevveellooppmmeenntt TTeeaamm

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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 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.

Single Number Ratios

Car Pooling?

They set up an experiment to count the number of cars and people on the East Side Highway adjacent to the school.

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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

Single Number Ratios

A

ver rs

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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 ratioof people to cars was 12 people to 10 cars or 12:10. Another way to describe this is it to use the average numberof 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.

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

Number of People 12 1.2 Number of Cars 10 1

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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.

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 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 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?

Single Number Ratios

A

Miles per Gallon

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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.

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.

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.

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

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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 is called the average speed. Average speed

is expressed in miles per hour (mi/h).

Average speed is expressed as a single number.

Single Number Ratios

A

Miles per Hour

Miles Hours

81.2 2

812 20

406 10

40.6 1

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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.)

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?

Cruise Control

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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?

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

A

Number of Miles Number of Gallons

75 2.5

750 25

30 1

 10  25

 10  25

Number of Miles Number of Gallons

90 2

45 1

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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

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Martha had her car repaired at a garage. Shown below is part of the bill 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 calculate gas mileage.

Village Automotive

Parts Labor Total

None 1.5 hr $90.00

Cost in Dollars Number of Hours

Single Number Ratios

A

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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?

Comparisons

Telephones and Populations

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)

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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?

Comparisons

B

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.

Number of People Number of Telephones

112,000 60,000

...

1

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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. ReflectWhich 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.

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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?

Comparisons

B

FINLAND

CHINA

Number of Telephones (in millions) Population (in millions)

6.3

5.2 1

Number of Telephones (in millions) Population (in millions)

430.50 1,300

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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.

Television Sets

FRANCE

BRAZIL CANADA

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

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United

Denmark States Canada Taiwan Poland World Number of Cell

Phones per 1,000 740 440 320 970 260 160

People

Comparisons

B

Cell Phones

DENMARK POLAND

TAIWAN UNITED

STATES

CANADA

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.

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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 countries compare 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 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).

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

sol la ti do do re mi la

C D E F G A B C

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Movable bridge

do

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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.

B

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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.

State Population

(in millions)

Number of Cows (in millions) California

Colorado Illinois Iowa Kansas Montana Nebraska South Dakota Texas

34.5 4.4 12.5 2.9 2.7 0.9 1.7 0.8 21.3

5.2 3.1 1.4 3.6 6.6 2.5 6.4 4 13.6

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Country Area (in sq mi) Population

Argentina 1.1 million 36.8 million

Japan 146,000 127 million

Brazil 3.3 million 176 million

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

Comparisons

B

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 in each household. Then find the number of phones per person.

BRAZIL

ARGENTINA

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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.

Different Kinds of Ratios

Too Fast

Speeders Non-Speeders

Area 1 11 15

Area 2 42 20

Area 3 30 29

Area 4 4 0

Below is a chart showing the count at each area during a one-hour period in the morning.

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

b. What recommendation would you make to the city council?

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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.

Different Kinds of Ratios

C

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.

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?

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

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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.

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

Percent

One way to find a percent is to use the relationship between fractions and percents.

For example, if 1of the cars were speeding, 50% were speeding.

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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.

Different Kinds of Ratios

C

Ratio Fraction Decimal Percent

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 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.

Number of Cars Not Speeding

Total Number of Cars 100

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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-to- body ratio as a baby?

c. What was her head-to-height ratio?

Ms. Humphrey, 28 days

Ms. Humphrey, 28 years

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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.

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.

Different Kinds of Ratios

C

Newborn 2 years 6 years 12 years 25 years

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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.

Head-to-Height Ratios

Person A 1 to 8

Person B 2 to 15

Person C 2 to 16

Person D 2 to 20

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?

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Different Kinds of Ratios

C

Girls: 5 out of 20

This 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 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?

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.

Votes for Joshua 120 40 80

Total Votes 150 50 100

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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 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.

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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?

D Scale and Ratio

Scale Drawings

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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.

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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.

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.

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

Scale and Ratio

D

0 1 5 10

cm in drawing 15

300

cm in the room 0

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 desk

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Scale 1:75

Length in Drawing (in cm) 1 Actual Length (in cm) 75

The double number line used for Tim’s floor plan indicates a scale ratioof 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.

6. a. Use this ratio table to help Jenna find the length of the living room.

b. What is the actual width of the living room? Show your calculations.

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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 drawingrepresents objects that are too large or too small to draw at actual size.

A scale ratioshows 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?

Scale and Ratio

D

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.”

Scale 1:300 Volleyball court

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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?

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?

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.)

b. How long is the wingspan of the scale model airplane?

Length of Actual Plane (in cm) Length of Scale Model (in cm)

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Scale and Ratio

D

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.

Maps

0

0 1,000

1 2 3 4 centimeters (on map)

meters (actual)

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.

GOLDEN

GAT E NAT

I ON AL REC

R E ATION AREA

MARINA BLVD.

DOYLE DRIVE

Ft. Mason Golden Gate Bridge

Fort Point National Historic Site

U.S. Coast Guard Station

Yacht Harbor

Marina Marina

Green Palace of Fine Arts

(Exploratorium)

0 14 3 1 km

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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.

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?

29° S

168° E

NORFOLK ISLAND

Kingston Cascade Middlegate Burnt Pine

P A C I F I C O C E A N

1 : 500,000

AUSTRALIA

NEW ZEALAND Nepean I.

Philip I.

Mt. Bates 318 m

27° 45' N

18° W

HIERRO ISLAND

Valverde Sabinosa

Restinga Taibique

Isora

A T L A N T I C O C E A N

ElGolfo

1:1,000,000

ATLANTIC OCEAN

A F R I C A

SPAIN EUROPE

29° 45' N

141° 20' E

IWO JIMA

Motoyama Nishi

Minami

P A C I F I C O C E A N

1 : 250,000

RUSSIA

PACIFIC OCEAN CHINA

JAPAN Kangoku Rock

Kama Rock

Hanare Rock

Mt. Suribachi 170 m

Hill 110 m Air Base

Kitano Pt.

Tobiishi Pt.

Distance on Map (in cm) Actual Distance (in m) Actual Distance (in cm)

1

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.

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!

Source: Times Atlas of the World. plates 10, 20, and 96

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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

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:

Double Number Line

D

0 10m

1

0 1000

1000 centimeters  10 meters

cm on map

cm in reality Distance on Map (in cm) 1

Actual Distance (in cm) 1000

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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:

3. a. What is the actual distance of 1 cm on this map?

b. What is the scale ratio of the map?

5 kilometers

0

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Scale and Ratio

The map below shows a part of downtown Philadelphia.

D

IndependenceMall

Walnut St Market St

Ranstead St

Chestnut St

Sansom St

Arch St

Filbert St

Ionic St Apple Tree St

S 8th St S 6th St

S 7th St

S 9th St

S 10th St

S 11th St

S 12th St N 8th St N 6th St

N 7th St

N 9th St

N 10th St

N 11th St

N 12th St

Walnut Street Theater Thomas Jefferson

Univ. Hospital National Archives Branch

Balch Institute for Ethnic Studies Reeding

Terminal Market

History Museum of Philadelphia African-American Historical and Cultural Museum Scale 1:5,000

4. a. Copy and complete the following ratio table for the map.

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.

Distance on Map (in cm) 1 Actual Distance (in cm) ...

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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.

Scale Factor

Smaller or Larger?

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.

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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.

Scale Factor

E

On the right is a picture of salamander larvae.

4. a. Are these larvae in the picture reduced or enlarged from their actual size?

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.

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 factorindicates 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 enlargementor reduction.

Here is an arrow string describing an enlargement with a scale factor of 5.

Measure of Original  5 Measure of Enlargement

scale factor

Salamander Larvae (4x) Frog (0.5x)

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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.

Enlarged or Reduced?

The Western Pygmy blue (Brephidium exilis)

wingspan

?

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.

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?

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Four pictures are enlarged. Measurements from the original pictures and the enlarged pictures are in the table below.

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?

Scale Factor

E

Length of Original Length of Enlargement

(in cm) (in cm)

Picture A 2 8

Picture B 6 18

Picture C 3.5 14

Picture D 7.5 32

Measure of Original  ? Measure of Enlargement

scale factor

6 cm  ? 15 cm

scale factor

To find a scale factor, you can use arrow language.

Fill in the measurements from the problem.

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.

12. a. Copy the ratio table and fill in the missing numbers, especially the last entry.

Length of Original Drawing (in cm) 6 1

Length of Enlarged Drawing (in cm) 15 . . . .

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To find a scale factor, you can use a ratio table.

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.

14. a. Did Anita use the same scale factor for the three drawings she made? How do you know?

Length of Original Length of Starfish in Starfish (in cm) Drawing (in cm)

Gold-Colored Starfish 16 4

Red Starfish 25 5

Brownish Starfish 12 4

ⴛ 50 scale factor Length of Original Drawing (in cm) 12 6 2 1

Length of Enlarged Drawing (in cm) 600 300 100 50

Length

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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.

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?

Scale Factor

E

a.

b. c.

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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.

Afbeelding

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