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2013Edition

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MATHEMATICS

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Taken from:

Prentice Hall Mathematics, Course 2, All-in-One Workbook, Version A

ALL-IN-ONEStudent WorkbookVersion A

Common Core

Course 2

MATHEMATICS

Pre

ntic

e H

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441197_fm_i-ii.indd 1 10/10/12 2:45 PM

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Course 2Lesson4-1 DailyNotetakingGuide

Lesson 4-1 Ratios

Lesson Objective

To write ratios and use them to compare quantities

Vocabulary and Key Concepts

Ratio

A ratio is

You can write a ratio in three ways.

Arithmetic Algebra

5 to 7 a : b ,where b 0

Equivalent ratios are

Example

1 Writing Ratios There are 7 red stripes and 6 white stripes on the flag of the United States. Write the ratio of red stripes to white stripes in three ways.

red stripes S d white stripes

red stripes S d white stripes

d red stripes

d white stripes

Quick Check

1. Write each ratio in three ways. Use the pattern of piano keys shown at the right.a. white keys to all keys

b. white keys to black keys

441197_C2_Ch4_WKbk_Ver-1.indd 38 10/11/12 1:23 PM

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DailyNotetakingGuide Course 2Lesson4-1

Examples

2 Writing Equivalent Ratios Find a ratio equivalent to 14 4 .

14 4 5

4 4

3 Writing Equivalent Ratios Write the ratio 2 lb to 56 oz as a fraction in simplest form.2 lb

56 oz 52 3 16 oz56 oz

5 oz

56 oz

5 4 oz

56 4 oz

5

4 Comparing Ratios The ratio of girls to boys enrolled at King Middle School is 15 : 16. There are 195 girls and 208 boys in Grade 8. Is the ratio of girls to boys in Grade 8 equivalent to the ratio of girls to boys in the entire school?

EntireSchool Grade8

d girls S

5 d Write as a decimal. S 5

Since the two decimals are , the ratio of girls to boys in Grade 8 is to the ratio of girls to boys in the entire school.

Quick Check

d boys S

d Divide the numerator and denominator by 2.

d There are 16 oz in each pound.

d Multiply.

d Simplify.

d Divide by the GCF, oz.

2. Find a ratio equivalent to 7 9. 3. Write the ratio 3 gal to 10 qt as a fraction in

simplest form.

4. Tell whether the ratios below are equivalentornotequivalent.

a. 7 : 3, 128 : 54 b. 180 240, 25

34 c. 6.1 to 7, 30.5 to 35


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Vocabulary

A rate is

A unit rate is

A unit cost is

Examples

1 Finding a Unit Rate Using Whole Numbers You earn $33 for 4 hours of work. Find the unit rate of dollars per hour.

dollars Shours S

334

5 d Divide the first quantity by the second quantity.

The unit rate is , or per hour.

2 Finding a Unit Rate Using Fractions Ely walks 78 mile in 13 hour. What is his speed in miles per hour?

miles to hours 578

to 13 S Write the ratio.

miles hours 578 4 S

Divide the first quantity by the second quantity.

5 S Simplify.

5 S Write as a mixed number.

Ely walks miles .

Lesson Objective

To find unit rates and unit costs using proportional reasoning

Lesson 4-2 Unit Rates and Proportional Reasoning

Common Core Standard

Ratios and Proportional Relationships: 7.RP.1


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Quick Check

1. Find the unit rate for 210 heartbeats in 3 minutes.

2. Find the unit rate for 310

mile in 34 hour.

Example

3 Using Unit Cost to Compare Find each unit cost. Which is the better buy?3 lb of potatoes for $.895 lb of potatoes for $1.59

Divide to find the unit cost of each size.

cost Ssize S

$.893 lb

¯

cost Ssize S

$1.595 lb

¯

Since , , for

is the better buy.

Quick Check

3. Which bottle of apple juice is the better buy: 48 fl oz of fruit juice for $3.05 or 64 fl oz for $3.59?



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Lesson Objective

To test whether ratios form a proportion by using equivalent ratios and cross products

Lesson 4-3 Proportions

Common Core Standards

Ratios and Proportional Relationships: 7.RP.2, 7.RP.2.a

Key Concepts

Proportion

A proportion is

Arithmetic Algebra12 5 24 a

b 5 c

d, b0, d0

Cross Products Property

Cross products are

If two ratios form a proportion, the cross products are equal. If two ratios have equal cross products, they form a proportion.

Arithmetic Algebra68 5 9

12 ab 5 cd

6 ? 12 5 8 ? 9 ad 5bc,where b0, and d0

Example

1 Writing Ratios in Simplest Form Do the ratios 42 56 and 56

64 form a proportion?

42 56 5

42 4

56 4 5 d Divide the numerator and denominator by the GCF.

S 56 64 5

56 4

64 4 5

The ratios in simplest form are not equivalent. They form a proportion.

Quick Check

1. Do 10 12 and 40



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Example

2 Using Cross Products Do the ratios in each pair form a proportion?

a. 410, 615 b. 8

6 , 97

410 6

15 d Test each pair of ratios. S 86 9

7

4 10 d Write cross products. S 8 6

60 d Simplify. S 54

, 410 and 6

15 , 86 and 9

7

a proportion. a proportion.

Quick Check

2. Determine whether the ratios form a proportion.

a. 38, 6

16 b. 69 , 4

6 c. 48 , 5

9



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Lesson Objective

To solve proportions using unit rates, mental math, and cross products

Lesson 4-4 Solving Proportions

Examples

1 Using Unit Rates The cost of 4 lightbulbs is $3. Use the information to find the cost of 10 lightbulbs.

Step 1 Find the unit price.3 dollars

4 lightbulbs 5$3 4 4 lightbulbs d Divide to find the unit price.

lightbulb

Step 2 You know the cost of one lightbulb. Multiply to find the cost of 10 lightbulbs.

5 d Multiply the unit rate by the number of lightbulbs.

The cost of 10 lightbulbs is .

2 Solving Using Mental Math Solve each proportion using mental math.

a. 5c 5 3042

5c

3042 Since 5 �d � 30, the common multiplier is .

c Use mental math to find what number times equals 42.d

b. 94 5 72

t

94

72t Since 9 �d � t.

t Use mental math.d

� 72, 4 �



Ratios and Proportional Relationships: 7.RP.1, 7.RP.2


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3 Solving Using Cross Products Solve 68 5 9a using cross products.

68 5 9a

6a 5 8(9) d Write the cross products.

6a 5 d Simplify.

6a 5 d Divide each side by .

a 5 d Simplify.

Quick Check

1. a. Postcards cost $2.45 for 5 cards. How much will 13 cards cost?

b. Swimming goggles cost $84.36 for 12. At this rate, how much will new goggles for 17 members of a swim team cost?

2. Solve each proportion using mental math.

a. 38 5 b

24 b. m

5 5 16

40 c. 1530 5 5p

3. Solve each proportion using cross products.

a. 1215 5 x

21 b. 16

30 5 d51

c. 2035 5 110

m



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Lesson Objective

To use proportions to find missing lengths in similar figures

Lesson 4-5 Similar Figures


Ratios and Proportional Relationships: 7.RP.1, 7.RP.2, 7.G.1


Similar Polygons

Two polygons are similar if

• corresponding angles

• the lengths of corresponding sides

A polygon is

Indirect measurement is

Example

1 Finding a Missing Measure #ABC and #DEF are similar. Find the value of c.

Write in simplest form.d

Write a proportion.d

Substitute.d

ABDE

ACDF

c 6

c 2

2c

c

Find the common multiplier.d

Use mental math.d

6

�

�

�

�

69

c

14

18

AB

D

F

E

C 21


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2 Multiple Choice A 5-ft person standing near a tree has a shadow 12 ft long. At the same time, the tree has a shadow 42 ft long. What is the height of the tree?A. 17.5 ft B. 35 ft C. 49 ft D. 100.8 ft

42 ft 12 ft

5 ftx

Draw a picture and let x represent the height of the tree.x 5 42 d Write a proportion.

x 5 42 d Write the cross products.

12x12

5 5 4212 d Divide each side by 12.

x 5 d Simplify.

The height of the tree is ft. The correct answer is choice .

Quick Check

1. The trapezoids below are similar. Find x.

MN

L

37°

143°

x

6 6

E F10

310

O

D G

12

37°143° 5

2. A 6-ft person has a shadow 5 ft long. A nearby tree has a shadow 30 ft long. What is the height of the tree?


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Lesson Objective

To use proportions to solve problems involving scale


Ratios and Proportional Relationships: 7.G.1, 7.RP.1

Lesson 4-6 Maps and Scale Drawings

Course 2 Lesson 4-6 Daily Notetaking Guide

Vocabulary

A scale drawing is

A scale is

Example

1 Using a Scale Drawing The scale of a drawing is 1 in. : 6 ft. The length of awall is 4.5 in. on the drawing. Find the actual length of the wall.

You can write the scale of the drawing as 1 in.6 ft

. Then write a proportion.

Let n represent the actual length.

drawing (in.) →actual (ft) →

16

5 n d drawing (in.)

d actual (ft)

n 5 (4.5) d Write the cross products.

n 5 d Simplify.

The actual length is ft.

Quick Check

1. The chimney of a house is 4 cm tall on the drawing. How tall is the chimney of the actual house?

1 cm � 2.5 m


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Examples

2 Finding the Scale of a Model The actual length of the wheelbase of a mountain bike is 260 cm. The length of the wheelbase in a scale drawing is 4 cm. Find the scale of the drawing.

scale length →actual length →

4260

5 4 4

260 4 5 d Write the ratio in simplest form.

The scale is cm : cm.

3 Multiple Choice You want to make a scale model of a house that is 72 feet long and 24 feet tall. You plan to make the model 12 inches long. Which equation can you use to find x, the height of the model?

A. 2472

5 x12

B.1272

5 x24

C. 1224

5 x72

D. x24

5 7212

model (in.) →actual (ft) → 5

?

?

d model (in.)

d actual (ft) d Write a proportion.

5 d Fill in the information you know. Use x for the information you don’t know.

The correct answer is .

Quick Check

2. The length of a room in an architectural drawing is 10 in. Its actual length is 160 in.What is the scale of the drawing?

3. You want to make a scale model of a sailboat that is 51 ft long and 15 ft wide. You plan to make the sailboat 17 in. long. How wide should the model be?



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Vocabulary

A constant of proportionality is

Examples

1 Using a Table to Determine a Proportional Relationship The table below shows the number of times Linda skipped rope in minutes during a fundraiser. Is there a proportional relationship between time and skips?

Compare the ratios of time and rope skips.

rope skips →time → 150

5 5 12 5 450 5

The ratios are , so there is a relationship between time and .

2 Using a Graph to Find a Unit Rate The graph below displays the data given in Example 1. What is Linda’s speed in skips per minute?

500600

400300200100

00

0150

360450

510

5 10 15 20Skips

Min

utes

Linda’s speed is a unit rate. Find the value of r in the ordered pair (1, r).

The graph of this relationship passes through (0, 0) and 1 , 1502. So, it must also

pass through 11, 2. Since r 5 , the unit rate is per .

Linda’s speed is per .

Lesson Objective

To identify proportional relationships and find constants of proportionality


Ratios and Proportions: 7.RP.2.a, 7.RP.2.b, 7.RP.2.c, 7.RP.2.d

Lesson 4-7 Proportional Relationships

Minutes 0 5 12 15 17

Skips 0 150 360 450 510


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3 Using a Ratio to Identify a Unit Rate The table below shows a proportional relationship between the number of songs downloaded on a music site and the amount the customer pays. Identify the constant of proportionality.

Step 1 Use one data point to find the constant of proportionality c.

pricesongs 5 d Find the price per song by the

by the number of .

5 d Simplify.

Step 2 Check by multiplying c times the first quantity.20 3 5 403 5

1003 5 120 3 5

The constant of proportionality is . This unit rate represents a payment of per song.

Quick Check

Songs Downloaded, s

Price, p (dollars)

20 $10

40 $20

100 $50

120 $60

Dave

Hours 0 3 6 8 9

Miles 0 18.6 35.2 49.6 56.8

1. The table at the right shows the distances Dave rode in a bike-a-thon. Is there a proportional relationship? Explain.

2. Use the graph at the right. What is Damon’s reading speed in pages per day?

0

20

40

60

80

100

1 2 3 4 5 6Time (days)

(6, 90)

(5, 75)

Pag

es

3. Find the constant of proportionality for each table of values.a. yards of cloth per blanket b. pay per hour

Yards( y) 16 32 40

Blankets (b) 8 16 20

Hours( h) 2 10 16

Pay ( p) $11 $55 $88


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Practice Course 2 Lesson 4-1 161

Practice 4-1 Ratios

Write a ratio for each situation in three ways.

1. Ten years ago in Louisiana, schools averaged 182 pupils for every 10 teachers.

2. Between 1899 and 1900, 284 out of 1,000 people in the United States were 5–17 years old.

Use the chart below for Exercises 3–4.

Three seventh-grade classes were asked whether they wanted chicken or pasta served at their awards banquet.

Room Number Chicken Pasta

201 10 12

202 8 17

203 16 10

3. In room 201, what is the ratio of students who prefer chicken tostudents who prefer pasta?

4. Combine the totals for all three rooms. What is the ratio of thenumber of students who prefer pasta to the number of students who prefer chicken?

Write each ratio as a fraction in simplest form.

5. 12 to 18 6. 81 : 27 7. 628

Tell whether the ratios are equivalent or not equivalent.

8. 12 : 24, 50 : 100

9. 221 , 1

22

10. 2 to 3, 24 to 36

11. A bag contains green, yellow, and orange marbles. The ratio of green marbles to yellow marbles is 2 : 5. The ratio of yellow marbles to orange marbles is 3 : 4. What is the ratio of green marbles to orange marbles?


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162 Course 2 Lesson 4-1 Guided Problem Solving

4-1 • Guided Problem Solving

GPS Student Page 130, Exercise 27:

Cooking To make pancakes, you need 2 cups of water for every3 cups of flour. Write an equivalent ratio to find how much water you will need with 9 cups of flour.

Understand

1. Circle the information you will need to solve.

2. What are you being asked to do?

3. Why will a ratio help you to solve the problem?

Plan and Carry Out

4. What is the ratio of the cups of waterto the cups of flour?

5. How many cups of flour are you using?

6. Write an equivalent ratio to use 9 cups of flour.

7. How many cups of water areneeded for 9 cups of flour?

Check

8. Why is the number of cups of water triple the number of cupsneeded for 3 cups of flour?

Solve Another Problem

9. Rebecca is laying tile in her bathroom. She needs 4 black tiles forevery 16 white tiles. How many black tiles are needed if she uses 128 white tiles?


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Practice 4-2 Unit Rates and Proportional Reasoning

Write the unit rate for each situation.

1. travel 250 mi in 5 h 2. earn $75.20 in 8 h

3. read 80 pages in 2 h 4. type 8,580 words in 2 h 45 min

5. complete 34 of a puzzle in 7

8 h 6. drink 4

5 L in 1

4 h

Find each unit price. Then determine the better buy.

7. paper: 100 sheets for $.99 8. peanuts: 1 lb for $1.29500 sheets for $4.29 12 oz for $.95

9. crackers: 15 oz for $1.79 10. apples: 3 lb for $1.8912 oz for $1.49 5 lb for $2.49

11. mechanical pencils: 4 for $1.25 12. bagels: 4 for $.8925 for $5.69 6 for $1.39

13. a. Yolanda and Yoko ran in a 100-yd dash.When Yolandacrossed the finish line,Yoko was 10 yd behind her. The girls then repeated the race, with Yolanda starting 10 yd behind the starting line. If each girl ran at the same rate as before, who won the race? By how many yards?

b. Assuming the girls run at the same rate as before, how far behind the starting line should Yolanda be in order for the two to finish in a tie?

Practice Course 2 Lesson 4-2


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GPS Student Page 135, Exercise 27a:

Geography Population density is the number of people per unit ofarea. Alaska has the lowest population density of any state in the United States. It has 626,932 people in 570,374 mi2. What is itspopulation density? Round to the nearest person per square mile.

Understand

1. What is population density?


3. What does the phrase people per unit of area imply?

Plan and Carry Out

4. What is the population of Alaska?

5. What is the area of Alaska?

6. Write a division expression forthe population density.

7. What is its population density?

8. Round to the nearest personper square mile.

Check

9. Why is the population density only about 1 person/mi2?


10. Mr. Boyle is buying pizza for the percussion band. The bill is$56.82 for 5 pizzas. If there are 12 members of the band, how much does the pizza cost per member? Round to the nearest cent.



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165Practice Course 2 Lesson 4-3

Determine if the ratios in each pair are proportional.

1. 1216

, 3040

2. 812

, 1521

3. 2721, 81

56

4. 4524

, 7540

5. 59

, 80117

6. 1525, 75

125

7. 214

, 2035

8. 96

, 2114

9. 2415, 16

10

10. 34

, 810

11. 204

, 173

12. 256 , 98

Decide if each pair of ratios is proportional.

13. 1410

97

14. 188

3616

15. 610

1525

16. 716

49

17. 64

128

18. 193

1148

19. 514

615

20. 627

836

21. 2715

4525

22. 318

420

23. 52

156

24. 2015

43

Solve.

25. During the breaststroke competitions of the 1992 Olympics,Nelson Diebel swam 100 meters in 62 seconds, and Mike Bowerman swam 200 meters in 130 seconds. Are the rates proportional?

26. During a vacation, the Vasquez family traveled 174 miles in3 hours on Monday, and 290 miles in 5 hours on Tuesday. Are the rates proportional?

Practice 4-3 Proportions


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Decorating A certain shade of green paint requires 4 parts blue to5 parts yellow. If you mix 16 quarts of blue paint with 25 quarts of yellow paint, will you get the desired shade of green? Explain.

Understand



3. Will a ratio help you to solve the problem? Explain.

Plan and Carry Out

4. What is the ratio of blue parts to yellow parts?

5. What is the ratio of blue quarts to yellow quarts?

6. Check to see if the cross products of the two ratios are equal.

7. Are the ratios the same?

8. Will you get the desired shade of green? Explain.

Check

9. How do you know that the ratios are not the same?


10. There are 15 boys and 12 girls in your math class. There are 5 boysand 3 girls in your study group. Determine if the boy to girl ratio is the same in study group as it is in your math class. Explain.


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Use mental math to solve for each value of n.

1. n14

5 2035

2. 96

5 21n

3. 24n

5 1610

4. 34

5 n10

Solve each proportion using cross products.

5. k8

5 144

6. u3

5 105

7. 146

5 d15

8. 51

5 m4

k 5 u 5 d 5 m 5

9. 3632

5 n8

10. 530

5 1x 11. t4

5 510

12. 92

5 v4

n 5 x 5 t 5 v 5

Solve.

13. A contractor estimates it will cost $2,400 to build a deckto a customer’s specifications. How much would it cost to build five similar decks?

14. A recipe requires 3 c of flour to make 27 dinner rolls. How much flour is needed to make 9 rolls?

Solve using a calculator, paper and pencil, or mental math.

15. Mandy runs 4 km in 18 min. She plans to run in a 15 km race.How long will it take her to complete the race?

16. Ken’s new car can go 26 miles per gallon of gasoline. The car’s gasolinetank holds 14 gal. How far will he be able to go on a full tank?

17. Eleanor can complete two skirts in 15 days. How long will it takeher to complete eight skirts?

18. Three eggs are required to make two dozen muffins. How manyeggs are needed to make 12 dozen muffins?

Practice 4-4 Solving Proportions


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Course 2 Lesson 4-4 Guided Problem Solving


There are 450 students and 15 teachers in a school. The school hires 2 new teachers. To keep the student-to-teacher ratio the same, how many students in all should attend the school?

Understand


2. Will a proportion help you to solve the problem? Explain.

Plan and Carry Out

3. Write a ratio for the current student-to-teacher ratio.

4. Write a ratio for the new student-to-teacher ratio.

5. Write a proportion using the ratios in Steps 3 and 4.

6. How many total students should attend the school?

Check

7. Are the two ratios equivalent? Explain.


8. There are 6 black marbles and 4 red marbles in a jar. If you add4 red marbles to the jar, how many black marbles do you need to add to keep the ratio of black marbles to red marbles the same?


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Practice 4-5 Similar Figures

k MNO , k JKL. Complete each statement.

32 20

24M

N

O

24

18

15

J

K

L

1. M corresponds to . 2. L corresponds to .

3. JL corresponds to . 4. MN corresponds to .

5. What is the ratio of the lengths of the corresponding sides?

The pairs of figures below are similar. Find the value of each variable.

6. 5 5

5 5

5

4 x 7.

24 18

16y

8.

x

206

14

9. 20

15

10 8

y

x

10.

8

4

5

x

11. 1596x

12. On a sunny day, if a 36-inch yardstick casts a 21-inch shadow,how tall is a building whose shadow is 168 ft?

13. Oregon is about 400 miles from west to east, and 300 miles fromnorth to south. If a map of Oregon is 15 inches tall (from north to south), about how wide is the map?


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Geometry A rectangle with an area of 32 in2 has one side measuring4 in. A similar rectangle has an area of 288 in2. How long is thelonger side in the larger rectangle?

Understand


2. Will a proportion that equates the ratio of the areas to the ratioof the shorter sides result in the desired answer? Explain.

3. What measure should you determine first?

Plan and Carry Out

4. What is the length of the longer side of the rectanglewhose area is 32 in.2 and whose shorter side is 4 in.?

5. What is the ratio of the longer side to the shorter side?

6. What pairs of factors multiply to equal 288?

7. Which pair of factors has a ratio of 21

?

8. What is the length of the longer side?

Check

9. Why must the ratio between the factors be 21

?


10. A triangle with perimeter 26 in. has two sides thatare 8 in. long. What is the length of the third side of a similar triangle which has two sides that are 12 in. long?


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The scale of a map is 2 cm : 21 km. Find the actual distances for the following map distances.

1. 9 cm 2. 12.5 cm 3. 14 mm

4. 3.6 m 5. 4.5 cm 6. 7.1 cm

A scale drawing has a scale of 14

in. : 12 ft. Find the length on thedrawing for each actual length.

7. 8 ft 8. 30 ft 9. 15 ft

10. 18 ft 11. 20 ft 12. 40 ft

Use a metric ruler to find the approximate distance between the towns.

13. Hickokburg to Kidville

14. Dodgetown to Earp City

15. Dodgetown to Kidville

16. Kidville to Earp City

17. Dodgetown to Hickokburg

18. Earp City to Hickokburg

Solve.

19. The scale drawing shows a two-bedroomapartment.The master bedroom is 9 ft 3 12 ft.Use an inch ruler to measure the drawing.

a. The scale is .b. Write the actual dimensions in place of the

scale dimensions.

Practice 4-6 Maps and Scale Drawings

47

180

47

Earp City

HickokburgKidville

DodgetownHoliday Lake

scale: 1 cm to 20 km

MasterBedroom Bedroom

LivingRoom

DiningRoom

Den Kitchen


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Writing in Math You are making a scale drawing with a scale of2 in. 5 17 ft. Explain how you find the length of the drawing of an object that has an actual length of 51 ft.

Understand


2. What points should you include in your explanation?

3. What is a scale?

Plan and Carry Out

4. What is the scale?

5. What is the actual length of the object?

6. Write a proportion using the scale, the actuallength, and the unknown length of the drawing.

7. What is the length of the object in a drawing?

Check

8. Use Steps 4–7 to explain how you decided how long to draw theobject.


9. The length of the wing of a model airplane is 3 in.If the scale of the model to the actual plane is 1 in. 5 25 ft, what is the length of the actual wing?



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Practice 4-7 Proportional Relationships

Determine whether each table or graph represents a proportional relationship. Explain your reasoning.

1. x 1 3 6 8 9

y 4.5 13.5 24 36 38

2. x 1 5 8 11 12

y 85

405

645

885

965

3. 1012

86420

0 2 4 6 8 10

4.

16

20

12

8

4

00 2 4 6 8 10

Find the constant of proportionality for each table of values.

5. Roses 6 12 24

Price $22.50 $45.00 $90.00

6. Tomatoes (lb) 3 7 9

Price $4.47 $10.43 $13.41

c 5 c 5

7. Gallons 5 10 15

Miles 120 240 360

8. Seconds 2 6 8

Feet 500 1,500 2,000

c 5 c 5

Write an equation using the constant of proportionality to describe the relationship.

9. A boat that has traveled 8 leagues from shore is 24 nauticalmiles out. Find the number of miles m in l leagues.

10. Four score years ago is 80 years past. Find the number of years y in s scores.


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Error Analysis A salesperson showed the table below while explaining that oranges are the same price per pound, no matter what size bag they come in. Why is the salesperson wrong?

$ $8 $10 $20

lbs 4 6 10

Understand


2. What will you use to find the answer?

Plan and Carry Out

3. What is the unit price for 4 pounds of oranges?



6. Why is the salesperson wrong?

Check

7. How can you check your answer?


8. A customer thinks pizzas cost the same per slice at a local restaurant. Why is the customer wrong?

$ $8 $9 $12

Slices 10 12 16


4A: Graphic Organizer For use before Lesson 4-1

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Vocabulary and Study Skills Course 2 Chapter 4

Study Skill As you read over the material in the chapter, keep apaper and pencil handy to write down notes and questions in your math notebook. Review notes taken in class as soon as possible.

Write your answers.

1. What is the chapter title?

2. How many lessons are there in this chapter?

3. What is the topic of the Test-Taking Strategies page?

4. Complete the graphic organizer below as you work through the chapter.

• In the center, write the title of the chapter.

• When you begin a lesson, write the lesson name in a rectangle.

• When you complete a lesson, write a skill or key concept in a circle linked to that lesson block.

• When you complete the chapter, use this graphic organizer to help you review.

441197_C2_Ch4_VocabPages.indd 175 10/10/12 1:35 PM

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Course 2 Chapter 4 Vocabulary and Study Skills176

Study Skill When you read mathematics, look for words like “morethan,” “less than,” “above,” “times as many,” “divided by.” These clues will help you decide what operation you need to solve a problem.

Read the paragraph and answer the questions that follow.

A tropical storm is classified as a hurricane when it has wind speeds in excess of 74 mi/h. The winds of Hurricane Gordon (1994) reached 12.4 mi/h above the minimum. How fast were the winds of Hurricane Gordon?

1. What numbers are in the paragraph?

2. What question are you asked to answer?

3. What units will you use in your answer?

4. Does a storm with winds of 74 mi/h qualify as a hurricane?Explain.

5. When did Hurricane Gordon occur?

6. How much above the minimum were Hurricane Gordon’s winds?

7. Let x represent Hurricane Gordon’s wind speed. Write anequation to help you solve the problem.

8. What is the answer to the question asked in the paragraph?

9. High-Use Academic Words In Exercise 7, what does it mean tosolve? a. to find an answer for b. to keep something going

4B: Reading Comprehension For use after Lesson 4-3


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Study Skill When you take notes in any subject, use abbreviationsand symbols whenever possible.

Write each statement or expression using the appropriate mathematical symbols.

1. the ratio of a to b

2. x to 4 is less than 5 to 2

3. 4 more than 5 times n

4. 5 : 24 is not equal to 1 : 5

Write each mathematical statement in words.

4C: Reading/Writing Math Symbols For use after Lesson 4-4

5. x 25

7. 1 oz < 28 g

6. )220) . )15|

8. 13

5 412

Match the symbolic statement or expression in Column A with its written form in Column B.

Column A Column B

9. k , 12 A. 12 times x

10. )25) B. negative 2 plus negative 4 is p

11. n 15 C. the ratio of 4 to 8

12. x 5 24 1 5 D. k is less than 12

13. 4 : 8 E. the quotient of x and 9

14. 12x F. x equals negative 4 plus 5

15. 22 1 (24) 5 p G. the absolute value of negative 5

16. x 4 9 H. n is greater than or equal to 15


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Study Skill When you come across something you don’t understand,view it as an opportunity to increase your brain power.

Concept List

cross products equivalent ratios indirect measurement

proportion rate scale

similar polygons unit cost unit rate

Write the concept that best describes each exercise. Choose from the concept list above.

1. 1816

and 4.5 : 4

2. A 6-ft-tall person standing

near a building has a shadow that is 60 ft long.This can be used to determine the height

of the building.

3. A bakery sells

a dozen donuts for $3.15. This can also

be represented as $3.1512 donuts

.

4. The expression “45 words

per minute” represents this.

5. 3075

5 25

6.

For the equation 1516

5 3z4

,these are represented by

15 3 4 and 3z 3 16.

7.

The equation 12

in. 5 50 mirepresents this on a map.

8. $4.255 lb

5 $0.85

lb

9.

4D: Visual Vocabulary Practice For use after Lesson 4-6

25.5

8.5

12

4


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Study Skill Strengthen your vocabulary. Use these pages and add cuesand summaries by applying the Cornell Notetaking style.

Write the definition for each word or term at the right. To check your work, fold the paper back along the dotted line to see the correct answers.

polygon

proportion

unit rate

ratio

scale drawing

4E: Vocabulary Check For use after Lesson 4-7


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4E: Vocabulary Check (continued) For use after Lesson 4-7

Write the vocabulary word or term for each definition. To check your work, fold the paper forward along the dotted line to see the correct answers.

a closed figure formed by three or more line segments that do not cross

an equation stating that two ratios are equal

the rate for one unit of a given quantity

a comparison of two quantities by division

an enlarged or reduced drawing of an object that is similar to the actual object


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Study Skill Use a special notebook or section of a loose-leaf binderfor math.

Complete the crossword puzzle. For help, use the Glossary in your textbook.

Here are the words you will use to complete this crossword puzzle:

equation factor figures fraction

inequality mixed number prime proportion

ratio scale drawing

1

2

3

5

6

7

10

9

8

4

ACROSS

5. enlarged or reduced drawing of an object

7. equation stating two ratios are equal

8. a statement of two equal expressions

10. a whole number that divides another wholenumber evenly

4F: Vocabulary Review Puzzle For use with the Chapter Review

DOWN

1. Similar ________________ have the same shape but not necessarily the same size.

2. a statement that two expressions are not equal

3. a number made up of a nonzero wholenumber and a fraction

4. a number with only two factors, one anditself

6. a number in the form ab

9. a comparison of two numbers by division


Taken from:

Prentice Hall Mathematics, Course 2, All-in-One Workbook, Teacher’s Guide, Regular Version A

TEACHER’S GUIDEALL-IN-ONEStudent WorkbookRegular Version A

Common Core

Course 2

MATHEMATICS

Pre

ntic

e H

all

2013Edition

448591_C2_TE.indd 1 10/23/12 12:17 PM

Course 2: All-In-One Answers Version A (continued)

Course 2 All-In-One Answers Version A10

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Quick Check

1. Find the unit rate for 210 heartbeats in 3 minutes.

70 heartbeats per min

2. Find the unit rate for 310

mile in 34 hour.

410

Example

3 Using Unit Cost to Compare Find each unit cost. Which is the better buy?3 lb of potatoes for $.895 lb of potatoes for $1.59

Divide to find the unit cost of each size.

cost Ssize S

$.893 lb

¯ $.30/lb

cost Ssize S

$1.595 lb

¯ $.32/lb

Since $.30/lb , $.32/lb , 3 lb of potatoes for $.89

is the better buy.

Quick Check

3. Which bottle of apple juice is the better buy: 48 fl oz of fruit juice for $3.05 or 64 fl oz for $3.59?

$.064/fl oz, $.056/fl oz; the 64-fl-oz choice is the better buy.

Daily Notetaking Guide Course 2 Lesson 4-2


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Vocabulary

A rate is

A unit rate is

A unit cost is

Examples

1 Finding a Unit Rate Using Whole Numbers You earn $33 for 4 hours of work. Find the unit rate of dollars per hour.

dollars Shours S

334

5 8.25 d Divide the first quantity by the second quantity.

The unit rate is $8.25

1 hour, or $8.25 per hour.

2 Finding a Unit Rate Using Fractions Ely walks 78 mile in 13 hour. What is his speed in miles per hour?

miles to hours 578

to 13 S Write the ratio.

miles 4 hours 578 4

1

3S

Divide the first quantity by the second quantity.

521

8S Simplify.

5 2 5

8S Write as a mixed number.

Ely walks 2 5

8 miles per hour .

Lesson Objective

To find unit rates and unit costs using proportional reasoning

Lesson 4-2 Unit Rates and Proportional Reasoning


Ratios and Proportional Relationships: 7.RP.1

the rate for one of a given quantity.

a unit rate that gives the cost per unit.

a ratio that compares two quantities measured in different units.


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Examples

2 Writing Equivalent Ratios Find a ratio equivalent to 14 4 .

14 4 2 5

4 4 2

3 Writing Equivalent Ratios Write the ratio 2 lb to 56 oz as a fraction in simplest form.2 lb

56 oz 5 2 3 16 oz56 oz

5 32 oz56 oz

5 32 4 8 oz56 4 8 oz

5 4

7

4 Comparing Ratios The ratio of girls to boys enrolled at King Middle School is 15 : 16. There are 195 girls and 208 boys in Grade 8. Is the ratio of girls to boys in Grade 8 equivalent to the ratio of girls to boys in the entire school?

Entire School Grade 8

15

16

d girls S

195

208

15

16 5 0.9375 d Write as a decimal. S 195

208 5 0.9375

Since the two decimals are equal , the ratio of girls to boys in Grade 8 is equivalent to the ratio of girls to boys in the entire school.

Quick Check

d boys S

7

2d Divide the numerator and denominator by 2.

d There are 16 oz in each pound.

d Multiply.

d Simplify.

d Divide by the GCF, 8 oz.

2. Find a ratio equivalent to 7 9.

Answers may vary. Sample answer: 1418

.

3. Write the ratio 3 gal to 10 qt as a fraction in simplest form.

65

4. Tell whether the ratios below are equivalent or not equivalent.

a. 7 : 3, 128 : 54 b. 180 240, 25

34 c. 6.1 to 7, 30.5 to 35

not equivalent not equivalent equivalent


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Lesson 4-1 Ratios

Lesson Objective

To write ratios and use them to compare quantities


Ratio

A ratio is

You can write a ratio in three ways.

Arithmetic Algebra

5 to 7 a : b , where b 0

Equivalent ratios are

Example

1 Writing Ratios There are 7 red stripes and 6 white stripes on the flag of the United States. Write the ratio of red stripes to white stripes in three ways.

red stripes S 7 to 6 d white stripes

red stripes S 7 : 6 d white stripes

7

6

d red stripes

d white stripes

Quick Check

1. Write each ratio in three ways. Use the pattern of piano keys shown at the right.a. white keys to all keys

7 to 12, 7 : 12, 712

b. white keys to black keys

7 to 5, 7 : 5, 75

5 : 7 a to b5

7

a

b

a comparison of two quantities by division.

two ratios that name the same number.


448591_C2_TE.indd 10 10/23/12 12:18 PM


All-In-One Answers Version A Course 2 11

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3 Solving Using Cross Products Solve 68 5 9a using cross products.

68 5 9a

6a 5 8(9) d Write the cross products.

6a 5 72 d Simplify.

6a

6 5

72

6 d Divide each side by 6 .

a 5 12 d Simplify.

Quick Check

1. a. Postcards cost $2.45 for 5 cards. How much will 13 cards cost?

$6.37

b. Swimming goggles cost $84.36 for 12. At this rate, how much will new goggles for 17 members of a swim team cost?

$119.51

2. Solve each proportion using mental math.

a. 38 5 b

24 b. m

5 5 16

40 c. 1530 5 5p

9 2 10

3. Solve each proportion using cross products.

a. 1215 5 x

21 b. 16

30 5 d51

c. 2035 5 110

m

16.8 27.2 192.5



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Lesson Objective

To solve proportions using unit rates, mental math, and cross products

Lesson 4-4 Solving Proportions

Examples

1 Using Unit Rates The cost of 4 lightbulbs is $3. Use the information to find the cost of 10 lightbulbs.

Step 1 Find the unit price.3 dollars

4 lightbulbs 5 $3 4 4 lightbulbs d Divide to find the unit price.

$.75

lightbulb

Step 2 You know the cost of one lightbulb. Multiply to find the cost of 10 lightbulbs.

$.75 10 5 $7.50 d Multiply the unit rate by the number of lightbulbs.

The cost of 10 lightbulbs is $7.50 .

2 Solving Using Mental Math Solve each proportion using mental math.

a. 5c 5 3042

6

6

7

6

6

65c

3042 Since 5 �d � 30, the common multiplier is .

c Use mental math to find what number times equals 42.d

b. 94 5 72

t

8

8

88

32

94

72t Since 9 �d � t.

t Use mental math.d

� 72, 4 �



Ratios and Proportional Relationships: 7.RP.1, 7.RP.2


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Example

2 Using Cross Products Do the ratios in each pair form a proportion?

a. 410, 615 b. 8

6 , 97

410 6

15 d Test each pair of ratios. S 86 9

7

4 15 10 6 d Write cross products. S 8 7 6 9

60 5 60 d Simplify. S 56 fi 54

Yes , 410 and 6

15 No , 86 and 9

7

form a proportion. do not form a proportion.

Quick Check

2. Determine whether the ratios form a proportion.

a. 38, 6

16 b. 69 , 4

6 c. 48 , 5

9

yes; 48 5 48 yes; 36 5 36 no; 36 fi 40



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Lesson Objective

To test whether ratios form a proportion by using equivalent ratios and cross products

Lesson 4-3 Proportions


Ratios and Proportional Relationships: 7.RP.2, 7.RP.2.a

Key Concepts

Proportion

A proportion is

Arithmetic Algebra12 5 24 a

b 5 c

d, b 0, d 0

Cross Products Property

Cross products are

If two ratios form a proportion, the cross products are equal. If two ratios have equal cross products, they form a proportion.

Arithmetic Algebra68 5 9

12 ab 5 cd

6 ? 12 5 8 ? 9 ad 5 bc, where b 0, and d 0

Example

1 Writing Ratios in Simplest Form Do the ratios 42 56 and 56


42 56 5

42 4 14

56 4 14 5

3

4 d

Divide the numerator and denominator by the GCF.

S 56 64 5

56 4 8

64 4 8 5

7

8

The ratios in simplest form are not equivalent. They cannot form a proportion.

Quick Check

1. Do 10 12 and 40


No; 56 57.

an equation stating that two ratios are equal.

the two products found by multiplying the denominator of

each ratio by the numerator of the other ratio.


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Examples

2 Finding the Scale of a Model The actual length of the wheelbase of a mountain bike is 260 cm. The length of the wheelbase in a scale drawing is 4 cm. Find the scale of the drawing.

scale length →actual length →

4260

5 4 4 4

260 4 4 5

1

65 d Write the ratio in simplest form.

The scale is 1 cm : 65 cm.

3 Multiple Choice You want to make a scale model of a house that is 72 feet long and 24 feet tall. You plan to make the model 12 inches long. Which equation can you use to find x, the height of the model?

A. 2472

5 x12

B. 1272

5 x24

C. 1224

5 x72

D. x24

5 7212

model (in.) →actual (ft) →

12

72 5

?

?

d model (in.)

d actual (ft) d Write a proportion.

12

72 5

x

24 d Fill in the information you know. Use x

for the information you don’t know.

The correct answer is B .

Quick Check

2. The length of a room in an architectural drawing is 10 in. Its actual length is 160 in.What is the scale of the drawing?

1 in. : 16 in.

3. You want to make a scale model of a sailboat that is 51 ft long and 15 ft wide. You plan to make the sailboat 17 in. long. How wide should the model be?

5 in.



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Lesson Objective

To use proportions to solve problems involving scale


Ratios and Proportional Relationships: 7.G.1, 7.RP.1

Lesson 4-6 Maps and Scale Drawings


Vocabulary

A scale drawing is

A scale is

Example

1 Using a Scale Drawing The scale of a drawing is 1 in. : 6 ft. The length of awall is 4.5 in. on the drawing. Find the actual length of the wall.

You can write the scale of the drawing as 1 in.6 ft

. Then write a proportion.

Let n represent the actual length.

drawing (in.) →actual (ft) →

16

5 4.5

n d drawing (in.)

d actual (ft)

1 n 5 6 (4.5) d Write the cross products.

n 5 27 d Simplify.

The actual length is 27 ft.

Quick Check

1. The chimney of a house is 4 cm tall on the drawing. How tall is the chimney of the actual house?

10 m

1 cm � 2.5 m

an enlarged or reduced drawing of an object that is similar to

the actual object.

the ratio that compares a length in a drawing or model to the

corresponding length in the actual object.


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2 Multiple Choice A 5-ft person standing near a tree has a shadow 12 ft long. At the same time, the tree has a shadow 42 ft long. What is the height of the tree?A. 17.5 ft B. 35 ft C. 49 ft D. 100.8 ft

42 ft 12 ft

5 ftx

Draw a picture and let x represent the height of the tree.x

5 5 42

12 d Write a proportion.

12 x 5 5 42 d Write the cross products.

12x12

5 5 4212 d Divide each side by 12.

x 5 17.5 d Simplify.

The height of the tree is 17.5 ft. The correct answer is choice A .

Quick Check

1. The trapezoids below are similar. Find x.

MN

L

37°

143°

x

6 6

E F10

310

O

D G

12

37°143° 5

x 5 20

2. A 6-ft person has a shadow 5 ft long. A nearby tree has a shadow 30 ft long. What is the height of the tree?

36 ft


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Lesson Objective

To use proportions to find missing lengths in similar figures

Lesson 4-5 Similar Figures


Ratios and Proportional Relationships: 7.RP.1, 7.RP.2, 7.G.1


Similar Polygons

Two polygons are similar if

• corresponding angles

• the lengths of corresponding sides

A polygon is

Indirect measurement is

Example

1 Finding a Missing Measure #ABC and #DEF are similar. Find the value of c.

Write in simplest form.d

Write a proportion.d

Substitute.d

ABDE

ACDF

c 6

c 2

2

9

6

6

9

3

3

18

18

18

12

c

c

Find the common multiplier.d

Use mental math.d

6

�

�

�

�

69

c

14

18

AB

D

F

E

C 21

have the same measure.

form equivalent ratios.

a closed plane figure formed by three or more line segments that

do not cross.

measuring distances by using proportions and similar

figures.


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Examples

3 Writing Percents as Fractions Write each percent as a fraction in simplest form.

a. 12% b. 45%

12% 5 12

100 d

Write as a fraction with

a denominator of 100 . S 45% 5

45

100

5 12 4 4

100 4 4 d

Divide the numerator and the

denominator by the GCF. S 5

45 4 5

100 4 5

5 3

25 d Simplify the fraction. S 5

9

20

4 Ordering Rational Numbers Order 35,

210, 0.645, and 13% from least to greatest.

Write all numbers as decimals. Then graph each number on a number line.

35 5 0.6 d Divide the numerator by the denominator .

210 5 0.2 d Divide the numerator by the denominator .

0.645 d This number is already in decimal form.

13% 5 0.13 d Move the decimal point two places to the left .

0 0.1 0.2 0.3 0.4 0.5 0.8 0.9 10.6 0.7

35

21013% 0.645

Quick Check

3. An elephant eats about 6% of its body weight in vegetation every day. Write this as a fraction in simplest form.

350

4. Order from least to greatest.a. 3

10, 0.74, 29%, 11

25 b. 15%, 7

20, 0.08, 500%

29%, 310, 11

25, 0.74 0.08, 15%, 720, 500%


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Lesson 5-1 Percents, Fractions, and Decimals

Lesson Objective

To convert between fractions, decimals, and percents

Key Concepts

Fractions, Decimals, and PercentsYou can write 21 out of 100 as a fraction, a decimal, or a percent.

Fraction Decimal Percent

21

1000.21 21%

Examples

1 Writing Decimals as Percents Write 0.101, .008, and 2.012 as percents.

0.101 5 101

1000 0.008 5

8

1000 2.012 5

2,012

1000 d Write as a fraction.

5 10.1

100 5

0.8

100 5

201.2

100 d

Write an equivalent fraction

with 100 in the denominator.

5 10.1% 5 0.8% 5 201.2% d Write as a percent.

2 Writing Percents as Decimals Write 6.4%, .07%, and 3250% as decimals.

6.4% 5 6.4

100 0.07% 5

0.07

100 3250% 5

3250

100 d Write the percent as a fraction.

5 0.064 5 0.0007 5 32.5 d Divide.

Quick Check

1. Write 0.607, 0.005, and 9.283 as percents.

60.7% 0.5% 928.3%

2. Write each percent as a decimal.a. 3500% 35 b. 12.5% 0.125 c. 0.78% 0.0078


Expressions and Equations: 7.EE.3


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3 Using a Ratio to Identify a Unit Rate The table below shows a proportional relationship between the number of songs downloaded on a music site and the amount the customer pays. Identify the constant of proportionality.

Step 1 Use one data point to find the constant of proportionality c.

pricesongs 5

10

20 d Find the price per song by dividing the

price by the number of songs .

5 0.5 d Simplify.

Step 2 Check by multiplying c times the first quantity.20 3 0.5 5 10 40 3 0.5 5 20

100 3 0.5 5 50 120 3 0.5 5 60

The constant of proportionality is 0.5 . This unit rate represents a payment of $.50 per song.

Quick Check

Songs Downloaded, s

Price, p (dollars)

20 $10

40 $20

100 $50

120 $60

Dave

Hours 0 3 6 8 9

Miles 0 18.6 35.2 49.6 56.8

1. The table at the right shows the distances Dave rode in a bike-a-thon. Is there a proportional relationship? Explain.

2. Use the graph at the right. What is Damon’s reading speed in pages per day?

0

20

40

60

80

100

1 2 3 4 5 6Time (days)

(6, 90)

(5, 75)

Pag

es

3. Find the constant of proportionality for each table of values.a. yards of cloth per blanket b. pay per hour

Yards( y) 16 32 40

Blankets (b) 8 16 20

Hours( h) 2 10 16

Pay ( p) $11 $55 $88

No; all the ratios are not equivalent.

15 pages per day

c 5 2 c 5 $5.50


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Vocabulary

A constant of proportionality is

Examples

1 Using a Table to Determine a Proportional Relationship The table below shows the number of times Linda skipped rope in minutes during a fundraiser. Is there a proportional relationship between time and skips?

Compare the ratios of time and rope skips.

rope skips →time → 150

5 5

36012 5 450

15 5

510

17

The ratios are equivalent , so there is a proportional relationship between time and rope skips .

2 Using a Graph to Find a Unit Rate The graph below displays the data given in Example 1. What is Linda’s speed in skips per minute?

500600

400300200100

00

0150

360450

510

5 10 15 20Skips

Min

utes

Linda’s speed is a unit rate. Find the value of r in the ordered pair (1, r).

The graph of this relationship passes through (0, 0) and 1 5 , 1502. So, it must also

pass through 11, 30 2. Since r 5 30 , the unit rate is 30 skips per minute .

Linda’s speed is 30 skips per minute .

Lesson Objective

To identify proportional relationships and find constants of proportionality


Ratios and Proportions: 7.RP.2.a, 7.RP.2.b, 7.RP.2.c, 7.RP.2.d

Lesson 4-7 Proportional Relationships

Minutes 0 5 12 15 17

Skips 0 150 360 450 510

the value of the ratio of quantities in a proportional

relationship.


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Practice Course 2 Lesson 4-1 161

Practice 4-1 Ratios

Write a ratio for each situation in three ways.

1. Ten years ago in Louisiana, schools averaged 182 pupils for every 10 teachers.

2. Between 1899 and 1900, 284 out of 1,000 people in the United States were 5–17 years old.

Use the chart below for Exercises 3–4.

Three seventh-grade classes were asked whether they wanted chicken or pasta served at their awards banquet.

Room Number Chicken Pasta

201 10 12

202 8 17

203 16 10

3. In room 201, what is the ratio of students who prefer chicken tostudents who prefer pasta?

4. Combine the totals for all three rooms. What is the ratio of thenumber of students who prefer pasta to the number of students who prefer chicken?

Write each ratio as a fraction in simplest form.

5. 12 to 18 23 6. 81 : 27

31 7. 6

28 314

Tell whether the ratios are equivalent or not equivalent.

8. 12 : 24, 50 : 100 Yes, they are equivalent.

9. 221 , 1

22 No, they are not equivalent.

10. 2 to 3, 24 to 36 Yes, they are equivalent.

11. A bag contains green, yellow, and orange marbles. The ratio of green marbles to yellow marbles is 2 : 5. The ratio of yellow marbles to orange marbles is 3 : 4. What is the ratio of green marbles to orange marbles?

182 to 10; 182 : 10; 18210

10 : 12, or 5 : 6

39 : 34

3 : 10

284 to 1,000; 284 : 1,000; 2841,000


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Vocabulary and Study Skills Course 2 Chapter 3 159

Vo

cabu

lary and

Stud

y Skills

Study Skill Review notes that you have taken in class as soon as possible to clarify any points you missed and to refresh your memory.

Circle the word that best completes the sentence.

1. A mathematical statement that contains , or . is called an (equation, inequality).

2. You use the (Addition, Multiplication) Property of Inequality if you add the same value to each side of an inequality.

3. Two numbers are (inverses, reciprocals) if their product is 1.

4. The solution to a division sentence is a (quotient, divisor).

5. If you use the (Division, Subtraction) Property of Inequality with a negative number, the direction of the inequality symbol is reversed.

6. You can use the (Multiplication, Addition) Property of Inequality to solve the inequality m 4 6 , 29.

7. You use the (Subtraction, Division) Property of Inequality if you take away the same value from each side of an inequality.

8. To solve an inequality involving addition, you use (subtraction, reciprocals).

9. When solving a two-step inequality, you need to get the (variable, reciprocal) along on one side of the inequality.

10. A (variable, solution) of an inequality is any value that makes the inequality true.

11. A number sentence is a (compound, rational) inequality if it has more than one inequality symbol.

12. The solution to a multiplication sentence is a (product, difference).

3F: Vocabulary Review For use with the Chapter Review

inequality

Addition

reciprocals

quotient

Division

Multiplication

Subtraction

subtraction

reciprocal

solution

compound

product


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Write the definition for each word or term at the right. To check your work, fold the paper forward along the dotted line to see the correct answers.

the solution to a division sentence

a mathematical sentence that contains ,, ., , , or

two numbers whose product is 1

the solution to a multiplication sentence

a number sentence with more than one inequality symbol

Check students’ answers.


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Vocabulary and Study Skills Course 2 Chapter 3 157

Vo

cabu

lary and

Stud

y Skills

Study Skill Strengthen your vocabulary. Use these pages and add cues and summaries by applying the Cornell Notetaking style.


quotient

inequality

reciprocals

product

compound inequality




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Determine if the ratios in each pair are proportional.

1. 1216

, 3040

yes 2. 812

, 1521

no 3. 2721, 81

56 no

4. 4524

, 7540

yes 5. 59

, 80117

no 6. 1525, 75

125 yes

7. 214

, 2035

no 8. 96

, 2114

yes 9. 2415, 16

10 yes

10. 34

, 810

no 11. 204

, 173

no 12. 256 , 98 no

Decide if each pair of ratios is proportional.

13. 1410

97

14. 188

3616

not proportional

proportional

15. 610

1525

16. 716

49

proportional

not proportional

17. 64

128

18. 193

1148

proportional

not proportional

19. 514

615

20. 627

836

not proportional

proportional

21. 2715

4525

22. 318

420

proportional

not proportional

23. 52

156

24. 2015

43

proportional

proportional

Solve.

25. During the breaststroke competitions of the 1992 Olympics,Nelson Diebel swam 100 meters in 62 seconds, and Mike Bowerman swam 200 meters in 130 seconds. Are the rates proportional?

no

26. During a vacation, the Vasquez family traveled 174 miles in3 hours on Monday, and 290 miles in 5 hours on Tuesday. Are the rates proportional?

yes

Practice 4-3 Proportions


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GPS Student Page 135, Exercise 27a:

Geography Population density is the number of people per unit ofarea. Alaska has the lowest population density of any state in the United States. It has 626,932 people in 570,374 mi2. What is itspopulation density? Round to the nearest person per square mile.

Understand

1. What is population density?


3. What does the phrase people per unit of area imply?

Plan and Carry Out

4. What is the population of Alaska?

5. What is the area of Alaska?

6. Write a division expression forthe population density.

7. What is its population density?

8. Round to the nearest personper square mile.

Check

9. Why is the population density only about 1 person/mi2?


10. Mr. Boyle is buying pizza for the percussion band. The bill is$56.82 for 5 pizzas. If there are 12 members of the band, how much does the pizza cost per member? Round to the nearest cent.


Population density is the number of people per unit

of area.

Find the population density for Alaska.

that you are going to divide the number of people

Because the number of people in Alaska is very close to

by the area

626,932 people

570,374 mi2

1.099 people/mi2

1 person/mi2

626,932 people570,374 mi2

the number of square miles.

$ 4.74/member


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163

Practice 4-2 Unit Rates and Proportional Reasoning

Write the unit rate for each situation.

1. travel 250 mi in 5 h 2. earn $75.20 in 8 h

50 mi/h

$9.40/h

3. read 80 pages in 2 h 4. type 8,580 words in 2 h 45 min

40 pages/h

52 words/min or 3,120 words/h

5. complete 34

of a puzzle in 78

h 6. drink 45

L in 14

h

67 puzzles/h

165 L/h

Find each unit price. Then determine the better buy.

7. paper: 100 sheets for $.99 8. peanuts: 1 lb for $1.29500 sheets for $4.29 12 oz for $.95

$.0099/sheet; $.00858/sheet;

$1.29/lb; $1.267/lb; 12 oz

500 sheets

9. crackers: 15 oz for $1.79 10. apples: 3 lb for $1.8912 oz for $1.49 5 lb for $2.49

$.1193/oz; $.1242/oz; 15 oz

$.63/lb; $.498/lb; 5 lb

11. mechanical pencils: 4 for $1.25 12. bagels: 4 for $.8925 for $5.69 6 for $1.39

$.3125/pencil; $.2276/pencil;

$.2225/bagel; $.2317/bagel;

25 pencils

4 bagels

13. a. Yolanda and Yoko ran in a 100-yd dash.When Yolandacrossed the finish line,Yoko was 10 yd behind her. The girls then repeated the race, with Yolanda starting 10 yd behind the starting line. If each girl ran at the same rate as before, who won the race? By how many yards?

b. Assuming the girls run at the same rate as before, how far behind the starting line should Yolanda be in order for the two to finish in a tie?

Practice Course 2 Lesson 4-2

Yolanda; 1 yd

11 19 yd, or 11 yd 4 in.


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Cooking To make pancakes, you need 2 cups of water for every3 cups of flour. Write an equivalent ratio to find how much water you will need with 9 cups of flour.

Understand



3. Why will a ratio help you to solve the problem?

Plan and Carry Out

4. What is the ratio of the cups of waterto the cups of flour?

5. How many cups of flour are you using?

6. Write an equivalent ratio to use 9 cups of flour.

7. How many cups of water areneeded for 9 cups of flour?

Check

8. Why is the number of cups of water triple the number of cupsneeded for 3 cups of flour?


9. Rebecca is laying tile in her bathroom. She needs 4 black tiles forevery 16 white tiles. How many black tiles are needed if she uses 128 white tiles?

Find the number of cups of water you will need with

You can use multiplication to find new numbers that

Since 9 cups is three times 3 cups, the number of cups of

32 black tiles

23 or 2 : 3

23 5 6

9

9 cups

6 cups

9 cups of flour.

share the same proportional relationship as the

numbers in the original recipe.

water is also tripled.


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Practice 4-5 Similar Figures

k MNO , k JKL. Complete each statement.

32 20

24M

N

O

24

18

15

J

K

L

1. M corresponds to /J . 2. L corresponds to /O .

3. JL corresponds to MO . 4. MN corresponds to JK .

5. What is the ratio of the lengths of the corresponding sides? 4 : 3 or 3 : 4

The pairs of figures below are similar. Find the value of each variable.

6. 5 5

5 5

5

4 x

4

7.

24 18

16y 12

8.

x

206

148 47

9. 20

15

10 8

y

xx 5 12; y 5 13 13

10.

8

4

5

x

2.5

11. 1596x

10

12. On a sunny day, if a 36-inch yardstick casts a 21-inch shadow,how tall is a building whose shadow is 168 ft?

13. Oregon is about 400 miles from west to east, and 300 miles fromnorth to south. If a map of Oregon is 15 inches tall (from north to south), about how wide is the map?

288 ft

20 in.


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168




There are 450 students and 15 teachers in a school. The school hires 2 new teachers. To keep the student-to-teacher ratio the same, how many students in all should attend the school?

Understand


2. Will a proportion help you to solve the problem? Explain.

Plan and Carry Out

3. Write a ratio for the current student-to-teacher ratio. 45015

4. Write a ratio for the new student-to-teacher ratio. x

17

5. Write a proportion using the ratios in Steps 3 and 4. 45015 5 x

17

6. How many total students should attend the school?

Check

7. Are the two ratios equivalent? Explain.


8. There are 6 black marbles and 4 red marbles in a jar. If you add4 red marbles to the jar, how many black marbles do you need to add to keep the ratio of black marbles to red marbles the same?

510 students

6 black marbles

yes, 45015 5 30 and 510

17 5 30

Find how many students should attend school to keep

the same student-to-teacher ratio.

Yes, because you have two ratios that need to be equal.


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Use mental math to solve for each value of n.

1. n14

5 2035

8 2. 96

5 21n

14

3. 24n

5 1610

15 4. 34

5 n10

7.5

Solve each proportion using cross products.

5. k8

5 144

6. u3

5 105

7. 146

5 d15

8. 51

5 m4

k 5 28 u 5 6 d 5 35 m 5 20

9. 3632

5 n8

10. 530

5 1x 11. t4

5 510

12. 92

5 v4

n 5 9 x 5 6 t 5 2 v 5 18

Solve.

13. A contractor estimates it will cost $2,400 to build a deckto a customer’s specifications. How much would it cost to build five similar decks?

14. A recipe requires 3 c of flour to make 27 dinner rolls. How much flour is needed to make 9 rolls?

Solve using a calculator, paper and pencil, or mental math.

15. Mandy runs 4 km in 18 min. She plans to run in a 15 km race.How long will it take her to complete the race?

16. Ken’s new car can go 26 miles per gallon of gasoline. The car’s gasolinetank holds 14 gal. How far will he be able to go on a full tank?

17. Eleanor can complete two skirts in 15 days. How long will it takeher to complete eight skirts?

18. Three eggs are required to make two dozen muffins. How manyeggs are needed to make 12 dozen muffins?

Practice 4-4 Solving Proportions

$12,000

1 c

67.5 min

364 mi

60 days

18 eggs


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Decorating A certain shade of green paint requires 4 parts blue to5 parts yellow. If you mix 16 quarts of blue paint with 25 quarts of yellow paint, will you get the desired shade of green? Explain.

Understand



3. Will a ratio help you to solve the problem? Explain.

Plan and Carry Out

4. What is the ratio of blue parts to yellow parts?

5. What is the ratio of blue quarts to yellow quarts?

6. Check to see if the cross products of the two ratios are equal.

7. Are the ratios the same?

8. Will you get the desired shade of green? Explain.

Check

9. How do you know that the ratios are not the same?


10. There are 15 boys and 12 girls in your math class. There are 5 boysand 3 girls in your study group. Determine if the boy to girl ratio is the same in study group as it is in your math class. Explain.

Determine whether you will get the desired shade of green

Yes; if the ratio of 16 to 25 is the same as the ratio of 4 to 5,

with 16 quarts of blue paint and 25 quarts of yellow paint.

you will get the desired shade of green.

no

No, the ratios are not the same.

The cross products are not equal.

No, it is not. The boy-to-girl ratio in your math class is 54;

451625

the boy-to-girl ratio in your study group is 53.

4 ∙ 25 5 5 ∙ 16; 100 fi 80?


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Practice 4-7 Proportional Relationships

Determine whether each table or graph represents a proportional relationship. Explain your reasoning.

1. x 1 3 6 8 9

y 4.5 13.5 24 36 38

2. x 1 5 8 11 12

y 85

405

645

885

965

Not proportional; the ratios for Proportional; the ratios for

each pair are not equivalent. each pair are equivalent.

3. 1012

86420

0 2 4 6 8 10

4.

16

20

12

8

4

00 2 4 6 8 10

Not proportional; the line does Proportional; the graph shows a

not pass through (0, 0). straight line that passes through (0, 0).

Find the constant of proportionality for each table of values.

5. Roses 6 12 24

Price $22.50 $45.00 $90.00

6. Tomatoes (lb) 3 7 9

Price $4.47 $10.43 $13.41

c 5 $3.75 c 5 $1.49

7. Gallons 5 10 15

Miles 120 240 360

8. Seconds 2 6 8

Feet 500 1,500 2,000

c 5 24 c 5 250

Write an equation using the constant of proportionality to describe the relationship.

9. A boat that has traveled 8 leagues from shore is 24 nauticalmiles out. Find the number of miles m in l leagues. m 5 3l

10. Four score years ago is 80 years past. Find the number of years y in s scores. y 5 20s


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172



Writing in Math You are making a scale drawing with a scale of2 in. 5 17 ft. Explain how you find the length of the drawing of an object that has an actual length of 51 ft.

Understand


2. What points should you include in your explanation?

3. What is a scale?

Plan and Carry Out

4. What is the scale?

5. What is the actual length of the object?

6. Write a proportion using the scale, the actuallength, and the unknown length of the drawing.

7. What is the length of the object in a drawing?

Check

8. Use Steps 4–7 to explain how you decided how long to draw theobject.


9. The length of the wing of a model airplane is 3 in.If the scale of the model to the actual plane is 1 in. 5 25 ft, what is the length of the actual wing?


Explain how you find the length of the drawing of an

the scale, the actual length, a ratio or proportion, and

the ratio that compares a length in a drawing to the

object with an actual length of 51 ft.

the answer

corresponding length in the actual object

Every 2 in. represents 17 ft. Fifty-one feet is 3 times

2 in. 5 17 ft

51 ft

6 in.

75 ft

217

5 x51

17 ft. Three times 2 in. is 6 in. Therefore, the object

should be 6 in. long in a drawing.


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The scale of a map is 2 cm : 21 km. Find the actual distances for the following map distances.

1. 9 cm 94.5 km 2. 12.5 cm 131.25 km 3. 14 mm 14.7 km

4. 3.6 m 3,780 km 5. 4.5 cm 47.25 km 6. 7.1 cm 74.55 km

A scale drawing has a scale of 14

in. : 12 ft. Find the length on thedrawing for each actual length.

7. 8 ft 16 in.

8. 30 ft 58 in.

9. 15 ft 516

in.

10. 18 ft 38 in.

11. 20 ft 512

in. 12. 40 ft

56 in.

Use a metric ruler to find the approximate distance between the towns.

13. Hickokburg to Kidville 80 km

14. Dodgetown to Earp City 50 km

15. Dodgetown to Kidville 55 km

16. Kidville to Earp City 95 km

17. Dodgetown to Hickokburg 50 km

18. Earp City to Hickokburg 20 km

Solve.

19. The scale drawing shows a two-bedroomapartment.The master bedroom is 9 ft 3 12 ft.Use an inch ruler to measure the drawing.

a. The scale is 1 in. : 12 ft .b. Write the actual dimensions in place of the

scale dimensions.

Practice 4-6 Maps and Scale Drawings

47

180

47

Earp City

HickokburgKidville

DodgetownHoliday Lake

scale: 1 cm to 20 km

MasterBedroom Bedroom

LivingRoom

DiningRoom

Den Kitchen

12 ft 9 ft

9 ft

12 ft

6 ft

12 ft 9 ft

9 ft

12 ft

6 ft


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Geometry A rectangle with an area of 32 in2 has one side measuring4 in. A similar rectangle has an area of 288 in2. How long is thelonger side in the larger rectangle?

Understand


2. Will a proportion that equates the ratio of the areas to the ratioof the shorter sides result in the desired answer? Explain.

3. What measure should you determine first?

Plan and Carry Out

4. What is the length of the longer side of the rectanglewhose area is 32 in.2 and whose shorter side is 4 in.?

5. What is the ratio of the longer side to the shorter side?

6. What pairs of factors multiply to equal 288?

7. Which pair of factors has a ratio of 21

?

8. What is the length of the longer side?

Check

9. Why must the ratio between the factors be 21

?


10. A triangle with perimeter 26 in. has two sides thatare 8 in. long. What is the length of the third side of a similar triangle which has two sides that are 12 in. long?

Find the longer side of the larger rectangle.

No, the ratio of the areas cannot be set equal to the

The length of the longer side of the rectangle whose

15 in.

8 in.84 5

21

Since the rectangles are similar, the lengths of the

1 3 288, 2 3 144, 3 3 96, 4 3 72, 6 3 48, 8 3 36, 9 3 32, 12 3 24, 16 3 18

12 3 24

24 in.

ratio of the shorter sides because the area is in square

units and the length of the shorter side is not.

area is 32 in2.

corresponding sides must be in proportion.


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Study Skill When you take notes in any subject, use abbreviationsand symbols whenever possible.

Write each statement or expression using the appropriate mathematical symbols.

1. the ratio of a to b ab, or a : b

2. x to 4 is less than 5 to 2 x4, , 52

3. 4 more than 5 times n 5n 1 4

4. 5 : 24 is not equal to 1 : 5 524

15

Write each mathematical statement in words.

4C: Reading/Writing Math Symbols For use after Lesson 4-4

5. x 25

7. 1 oz < 28 g

6. )220) . )15|

8. 13

5 412

Match the symbolic statement or expression in Column A with its written form in Column B.

Column A Column B

9. k , 12 A. 12 times x

10. )25) B. negative 2 plus negative 4 is p

11. n 15 C. the ratio of 4 to 8

12. x 5 24 1 5 D. k is less than 12

13. 4 : 8 E. the quotient of x and 9

14. 12x F. x equals negative 4 plus 5

15. 22 1 (24) 5 p G. the absolute value of negative 5

16. x 4 9 H. n is greater than or equal to 15

x is less than or equal to 25.

One third is equal to four twelfths.One ounce is approximately equal

to twenty-eight grams.

The absolute value of negative 20 is

greater than the absolute value of 15.

D

G

H

F

A

C

B

E


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Study Skill When you read mathematics, look for words like “morethan,” “less than,” “above,” “times as many,” “divided by.” These clues will help you decide what operation you need to solve a problem.

Read the paragraph and answer the questions that follow.

A tropical storm is classified as a hurricane when it has wind speeds in excess of 74 mi/h. The winds of Hurricane Gordon (1994) reached 12.4 mi/h above the minimum. How fast were the winds of Hurricane Gordon?

1. What numbers are in the paragraph?

2. What question are you asked to answer?

3. What units will you use in your answer?

4. Does a storm with winds of 74 mi/h qualify as a hurricane?Explain.

5. When did Hurricane Gordon occur?

6. How much above the minimum were Hurricane Gordon’s winds?

7. Let x represent Hurricane Gordon’s wind speed. Write anequation to help you solve the problem.

8. What is the answer to the question asked in the paragraph?

9. High-Use Academic Words In Exercise 7, what does it mean tosolve? a. to find an answer for b. to keep something going

4B: Reading Comprehension For use after Lesson 4-3

74, 1994, 12.4

How fast were the winds of Hurricane Gordon?

mi/h

No, the winds need to be in excess of, or more than, 74 mi/h

in order to be classified as a hurricane. Winds of 74 mi/h would

not qualify as a hurricane.

1994

12.4 mi/h

x 2 12.4 5 74

86.4 mi/h

a


4A: Graphic Organizer For use before Lesson 4-1

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Study Skill As you read over the material in the chapter, keep apaper and pencil handy to write down notes and questions in your math notebook. Review notes taken in class as soon as possible.

Write your answers.

1. What is the chapter title?

2. How many lessons are there in this chapter?

3. What is the topic of the Test-Taking Strategies page?

4. Complete the graphic organizer below as you work through the chapter.

• In the center, write the title of the chapter.

• When you begin a lesson, write the lesson name in a rectangle.

• When you complete a lesson, write a skill or key concept in a circle linked to that lesson block.

• When you complete the chapter, use this graphic organizer to help you review.

Ratios, Rates, and Proportions

7

Using a Variable

Check students’ diagrams.


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Error Analysis A salesperson showed the table below while explaining that oranges are the same price per pound, no matter what size bag they come in. Why is the salesperson wrong?

$ $8 $10 $20

lbs 4 6 10

Understand


2. What will you use to find the answer?

Plan and Carry Out




6. Why is the salesperson wrong?

Check

7. How can you check your answer?


8. A customer thinks pizzas cost the same per slice at a local restaurant. Why is the customer wrong?

$ $8 $9 $12

Slices 10 12 16

Determine if oranges and pounds form

a proportional relationship.

$2 per lb

$1.67 per lb

$2 per lb

The price per pound of oranges is not the same for

You can multiply the unit price by the number of pounds

The cost per slice is $.80 for 10 slices but only $.75 per

Use the prices given in the table to find the unit price

for different amounts of oranges.

every amount.

for each amount to be sure the unit price is correct.

slice for 12 or 16 slices.


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Study Skill Use a special notebook or section of a loose-leaf binderfor math.

Complete the crossword puzzle. For help, use the Glossary in your textbook.

Here are the words you will use to complete this crossword puzzle:

equation factor figures fraction

inequality mixed number prime proportion

ratio scale drawing

FIGURE

INEQU

S C ALIT OROPORPT

X

DNU

Y

MI

P

IME

MBEROTCF

F

ACTION

RATIO

L E D R A W I N G

Q U A T I O N

1

2

3

5

6

7

10

9

8

4

ACROSS

5. enlarged or reduced drawing of an object

7. equation stating two ratios are equal

8. a statement of two equal expressions

10. a whole number that divides another wholenumber evenly

4F: Vocabulary Review Puzzle For use with the Chapter Review

DOWN

1. Similar ________________ have the same shape but not necessarily the same size.

2. a statement that two expressions are not equal

3. a number made up of a nonzero wholenumber and a fraction

4. a number with only two factors, one anditself

6. a number in the form ab

9. a comparison of two numbers by division


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Write the vocabulary word or term for each definition. To check your work, fold the paper forward along the dotted line to see the correct answers.

a closed figure formed by three or more line segments that do not cross

an equation stating that two ratios are equal

the rate for one unit of a given quantity

a comparison of two quantities by division

an enlarged or reduced drawing of an object that is similar to the actual object



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Study Skill Strengthen your vocabulary. Use these pages and add cuesand summaries by applying the Cornell Notetaking style.


polygon

proportion

unit rate

ratio

scale drawing




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Study Skill When you come across something you don’t understand,view it as an opportunity to increase your brain power.

Concept List

cross products equivalent ratios indirect measurement

proportion rate scale

similar polygons unit cost unit rate

Write the concept that best describes each exercise. Choose from the concept list above.

1. 1816

and 4.5 : 4

2. A 6-ft-tall person standing

near a building has a shadow that is 60 ft long.This can be used to determine the height

of the building.

3. A bakery sells

a dozen donuts for $3.15. This can also

be represented as $3.1512 donuts

.

4. The expression “45 words

per minute” represents this.

5. 3075

5 25

6.

For the equation 1516

5 3z4

,these are represented by

15 3 4 and 3z 3 16.

7.

The equation 12

in. 5 50 mirepresents this on a map.

8. $4.255 lb

5 $0.85

lb

9.

4D: Visual Vocabulary Practice For use after Lesson 4-6

equivalent ratios

unit rate

scale

indirect measurement

proportion

unit cost

rate

cross products

similar polygons

25.5

8.5

12

4


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