4-1 Study Guide and Intervention 8 Chapter 4... · Study Guide and Intervention Ratios and Rates 4-1 ... Express this ratio in simplest form. 2. ... For each exercise below, rates

Chapter 4 10 Course 3

NAME ________________________________________ DATE ______________ PERIOD _____

Study Guide and Intervention

Ratios and Rates

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Express 35 wins to 42 losses in simplest form.

}3452} 5 }

56

} Divide the numerator and denominator by the greatest common factor, 7.

The ratio in simplest form is }56

} or 5:6.

Express 1 foot to 3 inches in simplest form.

To simplify a ratio involving measurements, both quantities must have the same unitof measure.

}3

1in

fcohotes

} 5 }132

iinncchheess

} Convert 1 foot to 12 inches.

5 }41ininchch

es} Divide the numerator and denominator by 3.

The ratio in simplest form is }41

} or 4:1.

Express 309 miles in 6 hours as a unit rate.

}3609

homuirless

} 5 }51

1.5

hmou

irles

} Divide the numerator and denominator by 6 to get a denominator of 1.

The unit rate is 51.5 miles per hour.

Express each ratio in simplest form.

1. 3 out of 9 students 2. 8 passengers:2 cars

3. 5 out of 10 dentists 4. 35 boys:60 girls

5. 18 red apples to 42 green apples 6. 50 millimeters to 1 meter

Express each rate as a unit rate.

7. 12 waves in 2 hours 8. 200 miles in 4 hours

9. 21 gallons in 2.4 minutes 10. $12 for 4.8 pounds

A rate is a ratio that compares two quanitities with different types of units. A unit rate is a rate with a

denominator of 1.

A ratio is a comparison of two numbers by quantities. Since a ratio can be written as a fraction, it

can be simplified.

Example 1

Example 2

Example 3

Exercises



1. 15 cats:50 dogs 2. 18 adults to 27 teens

3. 27 nurses to 9 doctors 4. 12 losses in 32 games

5. 50 centimeters:1 meter 6. 1 foot:1 yard

7. 22 players:2 teams 8. $28:8 pounds

9. 8 completions:12 passes 10. 21 hired out of 105 applicants

11. 18 hours out of 1 day 12. 64 boys to 66 girls

13. 66 miles on 4 gallons 14. 48 wins:18 losses

15. 112 peanuts:28 cashews 16. 273 miles in 6 hours


17. 96 students in 3 buses 18. $9,650 for 100 shares of stock

19. $21.45 for 13 gallons of gasoline 20. 125 meters in 10 seconds

21. 30.4 pounds of tofu in 8 weeks 22. 6.5 inches of rainfall in 13 days

23. 103.68 miles in 7.2 hours 24. $94.99 for 7 pizzas

Skills Practice

Ratios and Rates

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NAME ________________________________________ DATE ______________ PERIOD _____

Practice

Ratios and Rates

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1. 32 out of 200 adults like opera 2. 20 picked out of 65 who tried out

3. 48 robins to 21 blackbirds seen 4. 10 rock musicians to 22 classical

musicians in the concert

5. 2 feet long to 64 inches wide 6. 45 millimeters out of 10 centimeters

7. 10 ounces sugar for 1 pound apples 8. 2 quarts out of 4 gallons leaked out


9. 110 inches of snow in 8 days 10. 38 feet in 25 seconds

11. 594 cars crossing the bridge in 3 hours 12. 366 miles on 12 gallons

13. SHOPPING An 8-ounce box of Crispy Crackers costs $1.59 and a

2-pound box costs $6.79. Which box is the better buy? Explain your

reasoning.

14. ANIMALS Which animal listed in the

table consumes the least amount of food

compared to its body weight? Explain

your reasoning.

Animal Body Amount ofWeight Food per

(lb) Day (lb)

African Elephant 12,000 500

Blue Whale 286,000 8,000

Koala 22 2

Komodo Dragon 300 240

Source: Scholastic Book of World Records

4-1


Word Problem Practice

Ratios and Rates

NAME ________________________________________ DATE ______________ PERIOD _____

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1. COOKING In a bread dough recipe,

there are 3 eggs for every 9 cups of

flour. Express this ratio in simplest

form.

2. WILDLIFE Dena counted 14 robins out of

150 birds. Express this ratio in

simplest form.

3. INVESTMENTS Josh earned dividends of

$2.16 on 54 shares of stock. Find the

dividends per share.

4. TRANSPORTATION When Denise bought

gasoline, she paid $27.44 for 11.2

gallons. Find the price of gasoline per

gallon.

5. WATER FLOW Jacob filled his 60-gallon

bathtub in 5 minutes. How fast was the

water flowing?

6. TRAVEL On her vacation, Charmaine’s

flight lasted 4.5 hours. She traveled

954 miles. Find the average speed of

the plane.

7. HOUSING Mr. and Mrs. Romero bought

a 1,200 square-foot house for $111,600.

How much did they pay per square

foot?

8. SHOPPING A breakfast cereal comes in

two different sized packages. The

8-ounce box costs $2.88, while the

12-ounce box costs $3.60. Which box is

the better buy? Explain your reasoning.


NAME ________________________________________ DATE ______________ PERIOD _____

Enrichment4-1

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Bargain Hunting

Rates are useful and meaningful when expressed as a unit rate.For example, which is the better buy—one orange for $0.29 or12 oranges for $3.00?

To find the unit rate for 12 oranges, divide $3.00 by 12. Theresult is $0.25 per orange. If a shopper needs to buy at least12 oranges, then 12 oranges for $3.00 is the better buy.

For each exercise below, rates are given in Column A and Column B. In the blank next to each exercise number, write the letter of the column that contains the better buy.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

OR

AN

GE

S

Column A Column B

1 apple for $0.19 3 apples for $0.59

20 pounds of pet food for $14.99 50 pounds of pet food for $37.99

A car that travels 308 miles on11 gallons of gasoline

A car that travels 406 miles on14 gallons of gasoline

10 floppy discs for $8.99 25 floppy discs for $19.75

1-gallon can of paint for $13.995-gallon bucket of paint for$67.45

84 ounces of liquid detergentfor $10.64

48 ounces of liquid detergentfor $6.19

5,000 square feet of lawn foodfor $11.99

12,500 square feet of lawn foodfor $29.99

2 compact discs for $26.50 3 compact discs for $40.00

8 pencils for $0.99 12 pencils for $1.49

1,000 sheets of computer paperfor $8.95

5,000 sheets of computer paperfor $41.99

Exercises


TI-83/84 Plus Activity

Ratios and Rates

NAME ________________________________________ DATE ______________ PERIOD _____

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Example 2

You can simplify many ratios and rates mentally. A graphingcalculator can help you simplify more difficult ratios and rates byusing the FRAC mode under the Math menu.

Express the ratio 72 to 216 in simplest form.

Enter: 72 216 1 1/3

The ratio 72 to 216 in simplest form is }1

3}.

If the ratio or rate cannot be simplified, the input numbers will appear as theanswer in fraction form, or the decimal equivalent may be given as theanswer.

Express the ratio 25 inches to 4 yards in simplest form.(Remember: Both measurements must have the same unit.)

Enter: 25 144 1 25/144

The ratio of 25 inches to 4 yards does not simplify.

Express the ratio 50,000,000 people to 3,237 magazine subscribers in simplest form.

Enter: 50000000 3237 1 15446.40099

The ratio 50,000,000 people to 3,237 subscribers cannot

be simplified.

Express each ratio or rate in simplest form.

1. 88 to 104 2. 16 out of 228 3. 84:56

4. 35 inches to 2 yards 5. 115 miles per 5 gallons 6. 76.5 meters in 8.5 seconds

7. 3,000,000 seconds of television advertisements to 14,897 new customers

ENTERMATH

ENTERMATH

ENTERMATH

Example 1

Example 3

4-1

Get Ready for the Lesson

Read the introduction at the top of page 194 in your textbook.

Write your answers below.

1. Copy and complete the table to determine the cost for different numbers

of pizzas ordered.

2. For each number of pizzas, write the relationship of the cost and number

of pizzas as a ratio in simplest form. What do you notice?

Read the Lesson

3. At Better Shirts, an order of 10 printed T-shirts is $45 and an order of

250 printed T-shirts is $875. What must you do before you can compare

the ratios to see if they are proportional?

4. What must be true in order for the ratios to be proportional?

5. What are the simplified ratios for the T-shirt orders? Are the ratios

proportional or nonproportional?

Remember What You Learned

6. A delivery service charges $7 per package delivered locally. There is also a $2 service

charge for registering an order of packages for any number of packages. Create a table

to show what the costs of sending 1, 2, 3, and 4 packages are, using the service. Is the

relationship between total cost and number of packages proportional or

nonproportional? Explain your reasoning.


NAME ________________________________________ DATE ______________ PERIOD _____

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Lesson Reading Guide

Proportional and Nonproportional Relationships

Cost of Order ($) 8Pizzas Ordered 1 2 3 4

Example 1

Example 2


The cost of one CD at a record store is $12. Create a table to showthe total cost for different numbers of CDs. Is the total cost proportional to thenumber of CDs purchased?

Total Cost 5

125

245

365

485 $12 per CD Divide the total cost for each by the number

Number of CDs 1 2 3 4 of CDs to find a ratio. Compare the ratios.

Since the ratios are the same, the total cost is proportional to the number of CDs purchased.

The cost to rent a lane at a bowling alley is $9 per hour plus $4 forshoe rental. Create a table to show the total cost for each hour a bowling lane isrented if one person rents shoes. Is the total cost proportional to the number ofhours rented?

Total Cost →

13 or 13 22 or 11 31 or 10.34 40 or 10 Divide each cost by the

Number of Hours 1 2 3 4 number of hours.

Since the ratios are not the same, the total cost is nonproportional to the number of hoursrented with shoes.

Use a table of values to explain your reasoning.

1. PICTURES A photo developer charges $0.25 per photo developed. Is the total cost

proportional to the number of photos developed?

2. SOCCER A soccer club has 15 players for every team, with the exception of two teams

that have 16 players each. Is the number of players proportional to the number of

teams?

NAME ________________________________________ DATE ______________ PERIOD _____

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Number of CDs 1 2 3 4Total Cost $12 $24 $36 $48

Two related quantities are proportional if they have a constant ratio between them. If two related

quantities do not have a constant ratio, then they are nonproportional.

Exercises

Number of Hours 1 2 3 4Total Cost $13 $22 $31 $40




For Exercises 1–3, use the table of values. Write the ratios in the table to show therelationship between each set of values.

1.

2.

3.

For Exercises 4–8 use the table of values. Write proportional or nonproportional.

4.

5.

6.

7.

8. Jack is ordering pies for a family reunion. Each pie costs $4.50. For orders smaller than

a dozen pies, there is a $5 delivery charge. Is the cost proportional to the number of pies

ordered?

NAME ________________________________________ DATE ______________ PERIOD _____

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Number of Hours 1 2 3 4Total Amount Earned $17.25 $35.50 $50.75 $70

Number of Lunches 1 2 3 4Total Cost $2.75 $5.50 $8.25 $11

Number of Hours 1 2 3 4Total Amount Earned $0.99 $1.98 $2.97 $3.96

Number of Hours 1 2 3 4Number of Pages Read in Book 37 73 109 145

Number of Hours 1 2 3 4Total Amount Earned $15 $30 $45 $60

Ratios

Number of Packages 1 2 3 4Total Cost $11 $20 $29 $38

Ratios

Number of Classrooms 1 2 3 4Total Students 24 48 72 92

Ratios

Skills Practice



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For Exercises 1–3, use a table of values to explain your reasoning.

1. ANIMALS The world’s fastest fish, a sailfish, swims at a rate of 69 miles

per hour. Is the distance a sailfish swims proportional to the number of

hours it swims?

FOSSILS For Exercises 2 and 3, use the following information.

In July, a paleontologist found 368 fossils at a dig. In August, she found about 14 fossils per day.

2. Is the number of fossils the paleontologist found in August proportional to

the number of days she spent looking for fossils that month?

3. Is the total number of fossils found during July and August proportional

to the number of days the paleontologist spent looking for fossils in

August?

Practice



NAME ________________________________________ DATE ______________ PERIOD _____

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For Exercises 1–8, use a table of values when appropriate to explain your reasoning.

1. SPORTS A touchdown is worth 6 points.

Additionally you score an extra point if

you can kick a field goal. Is the total

number of points scored equal to the

number of touchdowns?

2. DRIVING Gasoline costs $2.79 per

gallon. Is the number of gallons

proportional to the total cost?

3. JOBS Michael earns $3.90 per hour as a

server at a restaurant. In addition, he

earns an average of 18% tips on his

food sales. Is the amount of money that

he earns proportional to the number of

hours that he works?

4. RECREATION A outdoor swimming pool

costs $8 per day to visit during the

summer. There is also a $25 yearly

registration fee. Is the total cost

proportional to the total number of

days visited?

5. SCHOOL At a certain middle school,

there are 26 students per teacher in

every homeroom. Is the total number of

students proportional to the number of

teachers?

6. TEAMS A baseball club has 18 players

for every team, with the exception of

four teams that have 19 players each.

Is the number of players proportional

to the number of teams?

7. MONEY At the beginning of the

summer, Roger had $180 in the bank.

Each week he deposits another $64

that he earns mowing lawns. Is his

account balance proportional to the

number of weeks since he started

mowing lawns?

8. SHELVES A bookshelf holds 43 books on

each shelf. Is the total number of books

proportional to the number of shelves

in the bookshelf?




Enrichment

NAME ________________________________________ DATE ______________ PERIOD _____

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Saving with Larger Orders

Stores are able to make a profit by buying in bulk. They are charged less per item becausethey agree to buy a large number of items. When the unit price of bulk items is less thanthe unit price for a small order, the rates are nonproportional.

For each exercise below, the number of units and the total cost of the units isgiven in Column A and Column B. In the blank next to each exercise number,write whether the columns are proportional or nonproportional.

Column A Column B

1. ________ 100 T-shirts printed for $428.00 3,000 T-shirts printed for $12,180.00

2. ________ 3 compact discs for $23.34 10,000 compact discs for $32,900.00

3. ________ 8 books for $43.20 250 books for $1,350.00

4. ________ $546.00 earned in 42 hours $31,200.00 earned in 2,400 hours

5. ________ 36 photos printed for $7.20 800 photos printed for $96.00

6. ________ 1 bus token for $1.75 20 bus tokens for $34.00

7. ________ $3,840.00 earned in 160 hours $52,000 earned in 2,000 hours

8. ________ $730.00 earned in 40 hours $9,125.00 earned in 500 hours

9. ________ 1 bagel for $0.65 13 bagels for $7.80

10. _______ a series of 5 concert tickets for a series of 20 concert tickets for $187.50 $685.00


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NAME ________________________________________ DATE ______________ PERIOD _____


Rate of Change

4-3




1. What is the change in the number of entries from 2004 to 2006?

2. Over what number of years did this change take place?

3. Write a rate that compares the change in the number of entries to the

change in the number of years. Express your answer as a unit rate and

explain its meaning.

Read the Lesson

4. What does a rate of change measure on a graph?

5. On a graph, what does it mean when a rate of change is negative?

6. Complete the sentence: When a quantity does not change over a period of

time, it is said to have a __________ rate of change.


7. Write out in words the formula for finding a rate of change between two

data points (x1, y1) and (x2, y2).


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NAME ________________________________________ DATE ______________ PERIOD _____


Rate of Change

4-3

Example INCOME The graph shows Mr. Jackson’s annual income between 1998 and 2006. Find the rate ofchange in Mr. Jackson’s income between 1998 and 2001.

Use the formula for the rate of change.Let (x1, y1) 5 (1998, 48,500) and (x2, y2) 5 (2001, 53,000).

}y

x2

2

–

–

y

x1

1} 5 }

532,000001

2

2

4189,95800

} Write the formula for rate of change.

5 }4,5

300} Simplify.

5 }1,5

100} Express this rate as a unit rate.

Between 1998 and 2001, Mr. Jackson’s income increased an average of $1,500 per year.

SURF For Exercises 1–3, use the graph that shows the average daily wave height as measured by an ocean buoy over a nine-day period.

1. Find the rate of change in the average daily

wave height between day 1 and day 3.





y

x

Wav

e H

eigh

t

7

9

11

13

15

1 3 5 7 9

Day

Wave Height

(7, 14)

(9, 11)

(3, 12)

(1, 8)

0

'98 '00 '02 '04 '06

50,000

45,000

55,000

60,000

65,000

Annual

Inco

me

($)

Year

Mr. Jackson's Income

y

x

1998, 48,500

2006, 57,000

2001, 53,000

0

To find the rate of change between two data points, divide the difference of the y-coordinates by the

difference of the x-coordinates. The rate of change between (x1, y1) and (x2, y2) is }x

y

2

2

2

2 y

x1

1

}.

Exercises


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Skills Practice

Rate of Change

NAME ________________________________________ DATE ______________ PERIOD _____

4-3

TEMPERATURE Use the table below that shows the high temperature ofa city for the first part of August.

1. Find the rate of change in the high temperature between August 1 and

August 5.


August 14.

3. During which of these two time periods did the high temperature rise

faster?


August 15. Then interpret its meaning.

COMPANY GROWTH Use the graph that shows the number of employees at a company between 2000 and 2008.

5. Find the rate of change in the number of

employees between 2000 and 2002.


employees between 2002 and 2005.

7. During which of these two time periods did

the number of employees grow faster?


employees between 2005 and 2008. Then interpret

its meaning.

y

x

Num

ber

of

Emplo

yees

10

20

30

40

50

02000 2002 2004 2006 2008

Year

Company Growth

(2005, 41)

(2008, 38)

(2002, 32)

(2000, 18)

Date 1 5 14 15

High Temperature (ºF) 85 93 102 102


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NAME ________________________________________ DATE ______________ PERIOD _____

Practice

Rate of Change

4-3

SNOWFALL For Exercises 1–3, use the following information.

The amount of snow that fell during five time periods is shown in the table.

Time (P.M.) 2:00 2:10 2:20 2:30 2:40

Snowfall (in.) 3.8 5.1 5.5 7.8 8.3

1. Find the rate of change in inches of

snow that fell per minute between

2:00 P.M. and 2:10 P.M.

2. Find the rate of change in inches of

snow that fell per minute between

2:30 P.M. and 2:40 P.M.

3. Make a graph of the data. During

which time period did the rate of

snowfall increase the greatest? Explain your

reasoning.

POPULATION For Exercises 4–7, use the the information below and at the right.

The graph shows the population of Washington,D.C., every ten years from 1950 to 2000.

4. Find the rate of change in population

between 1950 and 1970.

5. Between which two 10-year periods did

the population decrease at the fastest

rate?

6. Find the rate of change in population

between 1950 and 2000.

7. If the rate of change in population between 1950 and 2000 were to continue, what

would you expect the population to be in 2010? Explain your reasoning.

Population of Washington, D.C.

Popula

tion

(th

ousa

nds)

Year

600

550

500

450

400

850

800

750

700

650

01950 1960 1970 1980

Source: U.S. Census Bureau

1990 2000 2010

(1990, 607)

(2000, 572)

(1970, 757)

(1980, 638)

(1950, 802)

(1960, 764)

Snow

fall (

in.)

Time (P.M.)

109

8

7

6

5

4

32

1

02:00 2:10 2:20 2:30 2:40


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Rate of Change

NAME ________________________________________ DATE ______________ PERIOD _____

4-3

ELECTIONS For Exercises 1–3, use the table that shows the total numberof people who had voted in District 5 at various times on election day.

1. Find the rate of change in the number

of voters between 8:00 A.M. and

10:00 A.M. Then interpret its meaning.

2. Find the rate of change in the number

of voters between 10:00 A.M. and

1:00 P.M. Then interpret its meaning.

3. During which of these two time periods

did the number of people who had

voted so far increase faster? Explain

your reasoning.

4. MUSIC At the end of 2005, Candace had

47 CDs in her music collection. At the

end of 2008, she had 134 CDs. Find the

rate of change in the number of CDs in

Candace’s collection between 2005 and

2008.

5. FITNESS In 1998, the price of an annual

membership at Mr. Jensen’s health club

was $225. In 2008, the price of the

same membership was $319.50. Find

the rate of change in the price of the

annual membership between 1998 and

2008.

6. HIKING Last Saturday Fumio and Kishi

went hiking in the mountains. When

they started back at 2:00 P.M., their

elevation was 3,560 feet above sea

level. At 6:00 P.M., their elevation was

2,390 feet. Find the rate of change of

their elevation between 2:00 P.M. and

6:00 P.M. Then interpret its meaning.

Time 8:00 A.M. 10:00 A.M. 1:00 P.M. 4:30 P.M. 7:00 P.M.

Number of Voters 141 351 798 1,008 1,753


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NAME ________________________________________ DATE ______________ PERIOD _____

Enrichment4-3

Analyzing Graphs

A graph can be used to represent many real-life situations. Graphs such asthese often have time as the dimension on the horizontal axis. By analyzingthe rate of change of different parts of a graph, you can draw conclusionsabout what was happening in the real-life situation at that time.

TRAVEL The graph at the right represents the speedof a car as it travels along the road. Describe whatis happening in the graph.

• At the origin, the car is stopped.

• Where the line shows a fast, positive rate of

change, the car is speeding up.

• Then the car is going at a constant speed,

shown by the horizontal part of the graph.

• The car is slowing down where the graph

shows a negative rate of change.

• The car stops and stays still for a short time. Then it speeds up again.

The starting and stopping process repeats continually.

Analyze each graph.

1. Ashley is riding her bicycle along a scenic trail.

Describe what is happening in the graph.

2. The graph shows the depth of water in a pond as you

travel out from the shore. Describe what is

happening in the graph.

car is acceleratingmaintaining speed

car is slowing downcar is stopped

Sp

ee

d (

mp

h)

Time (min)0

Sp

eed

(ft

/s)

Time (min)0

Dep

th (

ft)

Distance from Shore (ft)0


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NAME ________________________________________ DATE ______________ PERIOD _____


Constant Rate of Change

4-4

x

y

O x

y

O x

y

O


Read the introduction at the top of page 204 in your textbook. Writeyour answers below.

1. Pick several pairs of points from those plotted and find the rate of change

between them. What is true of these rates?

Read the Lesson

2. Use the graph to create a table of values for each graph. Use the table to

find the constant rate of change for each graph.


3. A linear relationship is defined as having a constant rate of change between any two

points along the line. Look at the following points and determine if they have a linear

relationship: (0,0), (1,2), (3,5), (4,9), (5,15).


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NAME ________________________________________ DATE ______________ PERIOD _____



4-4

Exercises

The table shows the relationship between feet and seconds. Is therelationship between feet and seconds linear? If so, find theconstant rate of change. If not, explain your reasoning.

rate of change 5 }se

fceoentds

} }cchhaannggee

iinn

xy

}

5 }2

34} or 2}

34

}

The rate of change is 2}34

} feet per second.

Find the constant rate of change for the number of feet per second.Interpret its meaning.

Choose two points on the line. The vertical change from pointA to point B is 4 units while the horizontal change is 2 units.

rate of change 5 }se

f

c

e

o

e

n

t

ds} Definition of rate of change

5 }42

} Difference in feet between two points

divided by the difference in seconds

for those two points

5 2 Simplify.

The rate of change is 2 feet per second.

Find the rate of change for each line.

1. 2. 3.

The points given in each table lie on a line. Find the rate of change for the line.

4. 5.

y

xO

y

xO

y

xO

The slope of a line is the ratio of the rise, or vertical change, to the run, or horizontal change.

y

xO

rise: 4

run: 2

B

A

x 25 0 5 10

y 4 3 2 1

x 3 5 7 9

y 21 2 5 8

Example 1

Example 2

←

←

seconds 5 1 23 27

feet 22 1 4 27

N N N

1 3 1 3 1 3

N N N

2 4 2 4 2 4


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Skills Practice


NAME ________________________________________ DATE ______________ PERIOD _____

4-4

1. $10 per hour

3. No, the change in number of

magazines sold is not constant

5. 3 hours per volunteer

9. 228 F per hour; the temperature if

decreasing by 28 F per hour.

2. 48 per hour

4. No, the change in number of apples

per tree is not constant

6. decreasing 2 gallons per 50 miles; or 1

gallon per 25 miles

10. 3 lbs per person; Three pounds of meat

are needed per person

Time Temperature

9 60

10 64

11 68

12 72

Number of Students Number ofMagazines Sold

10 100

15 110

20 200

25 240

Number of Trees Number of Apples

5 100

10 120

15 150

20 130

Gas Left in Tank Miles Driven

12 0

10 50

8 100

6 150

Number of Number ofVolunteers Hours Logged

5 15

10 30

15 45

20 60

O

10

2 4 6 8 10

20

30

40

50

60

70

80

90

Tem

per

ature

Hours After Storm

O

3

21 3 54 6 8 10

6

9

12

15

18

21

Poun

d o

f M

eal

Number of People

7 9

Determine whether the relationship between the two quantities described in eachtable is linear. If so, find the constant rate of change. If not, explain your reasoning.

Find the constant rate of change for each graph and interpret its meaning.

Hours spent MoneyBabysitting Earned

1 10

3 30

5 50

7 70


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NAME ________________________________________ DATE ______________ PERIOD _____

Practice


4-4

Determine whether the relationship between the two quantitiesdescribed in each table is linear. If so, find the constant rate ofchange. If not, explain your reasoning.

1. Fabric Needed for Costumes 2. Distance Traveled on Bike Trip

For Exercises 3 and 4, refer to the graphs below.

3. 4.

Day 1 2 3 4

Distance (mi) 21.8 43.6 68.8 90.6

Number of Costumes

Fabric (yd)

2 4 6 8

7 14 21 28

a. Find the constant rate of change andinterpret its meaning.

b. Determine whether a proportionallinear relationship exists between thetwo quantities shown in the graph.Explain your reasoning.

Hawk Diving Toward Prey

Alt

itude

(ft.

)

Time (s)

100

80

60

40

20

02 4 6 8 10

x

y

Book Sales Sa

les

($)

Day

5,000

4,000

3,000

2,000

1,000

02 4 6 8 10

x

y

a. Find the constant rate of change andinterpret its meaning.

b. Determine whether a proportionallinear relationship exists between thetwo quantities shown in the graph.Explain your reasoning.


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NAME ________________________________________ DATE ______________ PERIOD _____

4-4

FLOWERS For Exercises 1 and 2, use LONG DISTANCE For Exercises 3–6, use the graph that shows the depth of the the graph that compares the costs of water in a vase of flowers over 8 days. long distance phone calls with three

different companies.

y

x

0

1

1 2 3 4 5 6 7 8 9 10

2

3

4

5

6

7

8

9

10

Depth of Water in Vase

Dep

th (

in.)

Day

1. Find the rate of change for the line. 2. Interpret the difference between depth

in inches and the day as a rate of

change.

y

x

0 1 2 3 4 5 6 7 8 9

0.50

1.00

1.50

2.00

2.50

Long Distance Charges

Cost

($)

Length of Call (minutes)

Company A

Company B

Company C

3. Interpret the difference between the cost

in dollars and the length in minutes for

Company A as a rate of change.



Company B as a rate of change.



Company C as a rate of change.

6. Which company charges the least for

each additional minute? Explain your

reasoning.


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NAME ________________________________________ DATE ______________ PERIOD _____

Enrichment4-4

Chien-Shiung Wu

American physicist Chien-Shiung Wu (1912–1997) was born in Shanghai, China. In 1936, she came to the United States to further her studies in science. She received her doctorate in physics in 1940 from the University of California, and became known as one of the world’s leading physicists. In 1975, she was awarded the National Medal of Science.

Wu is most famous for an experiment that she conducted in 1957. The outcome of the experiment was considered the most significant discovery in physics in more than seventy years. The exercise below will help you learn some facts about it.

The points given in each table lie on a line. Find the rate of change ofthe line. The word or phrase following the solution will complete thestatement correctly.

1. 2.

At the time of the experiment, Wu The site of the experiment was the

was a professor at . in Washington, D.C.

rate of change 5 3: Columbia University rate of change 5 0: National Bureau of

rate of change 5 1: Stanford University Standards

rate of change 5 3: Smithsonian

Institution

3. 4.

The experiment involved a substance In the experiment, the substance was

called . cooled to .

rate of change 5 2: carbon 14 rate of change 5 }43

}: 22738C

rate of change 5 1: cobalt 60 rate of change 5 }34

}: 2100°C

??

??

x 22 1 4 7

y 23 1 5 9

x 0 2 4 6

y 2 4 6 8

x 22 21 0 1

y 3 3 3 3

x 0 1 2 3

y 1 4 7 10


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NAME ________________________________________ DATE ______________ PERIOD _____


Solving Proportions

4-5




1. Write a ratio in simplest form that compares the cost to the number of

bottles of nail polish.

2. Suppose you and some friends wanted to buy 6 bottles of polish. Write a

ratio comparing the cost to the number of bottles of polish.

3. Is the cost proportional to the number of bottles of polish purchased?

Explain.

Read the Lesson

4. Complete the sentence: If two ratios form a proportion, then the ratios

are said to be __________.

5. Do the ratios }ab

} and }dc

} always form a proportion? Why or why not?

6. Explain how you can use cross products to solve proportions in which one

of the terms is not known.


7. For the proportion }ab

} and }dc

}, why do you think the products ad and bc are

called cross products?



Solving Proportions

NAME ________________________________________ DATE ______________ PERIOD _____

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Determine whether the pair of ratios }2204} and }

1128} forms a proportion.

Find the cross products.

Since the cross products are not equal, the ratios do not form a proportion.

Solve }1320} 5 }

7k0}.

}13

20} 5 }

7k0} Write the equation.

12 ? 70 5 30 ? k Find the cross products.

840 5 30k Multiply.

}83400

} 5 }3300k

} Divide each side by 30.

28 5 k Simplify. The solution is 28.

Determine whether each pair of ratios forms a proportion.

1. }1170}, }

152} 2. }

69

}, }11

28} 3. }

182}, }

1105}

4. }175}, }

1332} 5. }

79

}, }46

93} 6. }

284}, }

1228}

7. }47

}, }17

21} 8. }

23

05}, }

3405} 9. }

12

84}, }

34

}

Solve each proportion.

10. }5x

} 5 }12

55} 11. }

34

} 5 }1c2} 12. }

69

} 5 }1r0}

13. }1264} 5 }

1z5} 14. }

58

} 5 }1s2} 15. }

1t4} 5 }

1101}

16. }w6

} 5 }27.8} 17. }

5y

} 5 }16

7.8} 18. }

1x8} 5 }

376}

2024

1218

0→ 24 · 12 5 288→ 20 · 18 5 360

A proportion is an equation that states that two ratios are equivalent. To determine whether a pair of

ratios forms a proportion, use cross products. You can also use cross products to solve proportions.

Example 1

Example 2

Exercises


Determine whether each pair of ratios forms a proportion.

1. }58

}, }23

} 2. }73

}, }164} 3. }

68

}, }192}

4. }196}, }

161} 5. }

51

50}, }

122} 6. }

68

}, }1250}

7. }59

}, }1257} 8. }

138}, }

1616} 9. }

171}, }

1253}

10. }193}, }

1137} 11. }

432}, }

750} 12. }

67

}, }3469}


13. }142} 5 }

9y

} 14. }168} 5 }

4c

} 15. }7z

} 5 }81

42}

16. }150} 5 }

w8

} 17. }9x

} 5 }145} 18. }

260} 5 }

5y

}

19. }59

} 5 }6r

} 20. }n8

} 5 }170} 21. }

d5

} 5 }18

20}

22. }5y

} 5 }1130} 23. }

228} 5 }

3p5} 24. }

1t1} 5 }

11010

}

25. }1m.2} 5 }

35

} 26. }01

.

.95} 5 }

1a0} 27. }

37

} 5 }4k.2}

28. }6x.3} 5 }

158} 29. }

39.6} 5 }

0b.5} 30. }

11.45} 5 }

4y.2}

NAME ________________________________________ DATE ______________ PERIOD _____

Skills Practice

Solving Proportions

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Practice

Solving Proportions

NAME ________________________________________ DATE ______________ PERIOD _____

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1. }b

5} 5 }

1

8

6} 2. }

1

x

8} 5 }

1

6

0} 3. }

5

t} 5 }

1

8

2

0}

4. }1

1

1

0} 5 }

1

n

4} 5. }

2

3

.

5

5} 5 }

d

2} 6. }

3

1

.

8

5} 5 }

3

z

6}

7. }0

4

.4

.2

5} 5 }

1

p

4} 8. }

2

6

.4} 5 }

2

s

.8} 9. }

3

k

.6} 5 }

0

0

.

.

2

5}

10. CLASSES For every girl taking classes at the martial arts school, there are

3 boys who are taking classes at the school. If there are 236 students

taking classes, write and solve a proportion to predict the number of boys

taking classes at the school.

11. BICYCLES An assembly line worker at Rob’s Bicycle factory adds a seat to

a bicycle at a rate of 2 seats in 11 minutes. Write an equation relating the

number of seats s to the number of minutes m. At this rate, how long will

it take to add 16 seats? 19 seats?

12. PAINTING Lisa is painting a fence that is 26 feet long and 7 feet tall. A

gallon of paint will cover 350 square feet. Write and solve a proportion to

determine how many gallons of paint Lisa will need.


NAME ________________________________________ DATE ______________ PERIOD _____


Solving Proportions

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1. USAGE A 12-ounce bottle of shampoo

lasts Enrique 16 weeks. How long

would you expect an 18-ounce bottle

of the same brand to last him?

2. COMPUTERS About 13 out of 20 homes

have a personal computer. On a street

with 60 homes, how many would you

expect to have a personal computer?

3. SNACKS A 6-ounce package of fruit

snacks contains 45 pieces. How many

pieces would you expect in a 10-ounce

package?

4. TYPING Ingrid types 3 pages in the

same amount of time that Tanya types

4.5 pages. If Ingrid and Tanya start

typing at the same time, how many

pages will Tanya have typed when

Ingrid has typed 11 pages?

5. SCHOOL A grading machine can grade

48 multiple-choice tests in 1 minute.

How long will it take the machine to

grade 300 tests?

6. AMUSEMENT PARKS The waiting time to

ride a roller coaster is 20 minutes when

150 people are in line. How long is the

waiting time when 240 people are in

line?

7. PRODUCTION A shop produces 39

wetsuits every 2 weeks. How long will

it take the shop to produce 429

wetsuits?

8. FISH Of the 50 fish that Jim caught

from the lake, 14 were trout. The

estimated population of the lake is

7,500 fish. About how many trout

would you expect to be in the lake?


Enrichment

NAME ________________________________________ DATE ______________ PERIOD _____

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People of the United States

A national census is taken every ten years. The 1990 census revealed thatthere were about 250,000,000 people in the United States, and that about 8 out of 100 of these people were 5–13 years old. To find the number ofpeople in the United States that were 5–13 years old, use the ratio ofpeople 5–13 years old and create a proportion.

}1800} 5 }

250,0n00,000}

To solve the proportion, find cross products.

8 3 250,000,000 5 2,000,000,000 and n 3 100 5 100n

Then divide: 2,000,000,000 4 100 5 20,000,000.

In 1990, about 20,000,000 people in the United States were 5–13 years old.

Use the approximate ratios in each exercise to create a proportion,given that there were about 250,000,000 people in the United States.Then solve and choose the correct answer from the choices at the right.

1. The United States is a diverse collection of different ________ A. 200,000,000

races and ethnic origins. Asians or Pacific Islanders

accounted for about }1

300} of the population of the

United States. About how many people of Asian or

Pacific-Island origin lived in the United States?

2. African-Americans accounted for about }235} of the ________ B. 22,500,000

population of the United States. About how many

African-American people lived in the United States?

3. People of Hispanic origin accounted for about }1

900} ________ C. 7,500,000

of the population of the United States. About how

many people of Hispanic origin lived in the

United States?

4. Caucasian people accounted for about }45

} of the ________ D. 2,000,000

population of the United States. About how many

people of white or Caucasian origin lived in the

United States?

5. People of American-Indian, Eskimo, or Aluet origin ______ E. 30,000,000

accounted for about }1,0

800} of the population of the

United States. About how many people of

American-Indian, Eskimo, or Aluet origin lived in

the United States?

4-5

Exercises

Example


NAME ________________________________________ DATE ______________ PERIOD _____

Scientific Calculator Activity

Solving Proportions

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A calculator can be used to solve the equation ad 5 bc, which resultsfrom forming the cross products of the proportion

}

a

b} 5 }

d

c} .

A car uses 18.4 gallons of gasoline on a 432.5-mile trip.

How many miles will it travel on 22 gallons?

Write a proportion. }4

1

3

8

2

.4

.5} 5 }

2

d

2}

Find the cross products. 18.4 3 d 5 432.5 3 22

d 5 }432

1

.5

8.

3

4

22}

Enter: 432.5 22 18.4 517.1195652

The car will travel about 517 miles on 22 gallons of gasoline.

Solve each proportion. Round answers to the nearest tenth.

1. }9

4

.

.

4

2} 5 }

8

y

.5} 2. }

6

4

5

9

8

9} 5 }

19

g

8} 3. }

87

k

.2} 5 }

9

8

8

4

.3

.2

6}

4. }96

r

.2} 5 }

5

7

6

4

.

.

4

3} 5. }

9,86

p

25.5} 5 }

7

8

8

6

5

5

.

.

2

2} 6. }

2

7

1

5

.

.

5

2} 5 }

64

w

.1}

Solve.

ENTER

53

7. Four out of five dentists surveyed

recommended a certain toothpaste. If

330 dentists were surveyed, how many

recommended that toothpaste?

9. In the 2004 presidential election, George

W. Bush received 51 out of every 100 votes.

If he received about 62,000,000 votes,

about how many people voted?

8. In a class of 45 students, eight out of ten

students received an A. How many got

an A?

10. CHALLENGE There are 3 boys for every

2 girls in the class. If there are 30

students in the class, how many are

boys?

Example


NAME ________________________________________ DATE ______________ PERIOD _____

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It takes a worker 4 minutes to stack 2 rows of 8 boxes in awarehouse. How long will it take to stack 8 rows of 8 boxes? Use the draw adiagram strategy to solve the problem.

Understand After 4 minutes, a worker has stacked a 2 rows of 8 boxes. At this rate, howlong would it take to stack 8 rows of boxes?

Plan Draw a diagram showing the level of boxes after 4 minutes.

Solve 2 rows of 8 boxes 5 4 minutes8 rows 5 4 3 2 rows, so multiply the time by 4.4 3 4 minutes 5 16 minutes

Check

8 boxes 3 2 rows of boxes 5 16 boxes Multiply to find the total number of boxes in the stack.

4 minutes 4 16 boxes 5 0.25 min. per box Divide the number of minutes by the number of boxes.8 boxes 3 8 rows of boxes 5 64 boxes Multiply to find the number of boxes in the new stack.

64 boxes 3 0.25 min. 5 16 minutes Multiply the number of boxes by the time per box.

It will take 16 minutes to stack an 8 3 8 wall of boxes.

For Exercises 1–4, use the draw a diagram strategy to solve the problem.

1. GAS A car’s gas tank holds 16 gallons. After filling it for 20 seconds, the

tank contains 2.5 gallons. How many more seconds will it take to fill the

tank?

2. TILING It takes 96 tiles to fill a 2-foot by 3-foot rectangle. How many tiles

would it take to fill a 4-foot by 6-foot rectangle?

3. BEVERAGES Four juice cartons can fill 36 glasses of juice equally. How many

juice cartons are needed to fill 126 glasses equally?

4. PACKAGING It takes 5 large shipping boxes to hold 120 boxes of an action

figure. How many action figures would 8 large shipping boxes hold?

Exercises


Problem-Solving Investigation: Draw a Diagram



1. AQUARIUM An aquarium holds 60 gallons of water. After 6 minutes, the

tank has 15 gallons of water in it. How many more minutes will it take to

fill the tank?

2. TILING Meredith has a set of ninety 1-inch tiles. If she starts with one tile,

then surrounds it with a ring of tiles to create a larger square, how many

surrounding rings can she make before she runs out of tiles?

3. SEWING Judith has a 30-yard by 1-yard roll of fabric. She needs to use 1.5

square yards to create one costume. How many costumes can she create?

4. DRIVING It takes 3 gallons of gas to drive 102 miles. How many miles can

be driven on 16 gallons of gas?

5. PACKING Hector can fit 75 compact discs into 5 boxes. How many compact

discs can he fit into 14 boxes?

NAME ________________________________________ DATE ______________ PERIOD _____

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Skills Practice



Use the draw a diagram strategy tosolve Exercises 1 and 2.

1. SWIMMING Jon is separating the widthof the swimming pool into equal-sizedlanes with rope. It took him 30 minutesto create 6 equal-sized lanes. How longwould it take him to create 4 equal-sized lanes in a similar swimming pool?

2. TRAVEL Two planes are flying from SanFrancisco to Chicago, a distance of 1,800miles. They leave San Francisco at thesame time. After 30 minutes, one planehas traveled 25 more miles than theother plane. How much longer will ittake the slower plane to get to Chicagothan the faster plane if the faster planeis traveling at 500 miles per hour?

Use any strategy to solve Exercises 3–6.Some strategies are shown below.

3. TALENT SHOW In a solo singing andpiano playing show, 18 people sang and14 played piano. Six people both sangand played piano. How many peoplewere in the singing and piano playingshow?

4. LETTERS Suppose you have three stripsof paper as shown. How many capitalletters of the alphabet could you formusing one or more of these three stripsfor each letter? List them according tothe number of strips.

5. CLOTHING A store has 255 wool ponchosto sell. There are 112 adult-sizedponchos that sell for $45 each. The restare kid-sized and sell for $32 each. Ifthe store sells all the ponchos, howmuch money will the store receive?

6. DINOSAURS Brad mad a model of aStegosaurus. If you multiply the model'slength by 8 and subtract 4, you will findthe length of an average Stegosaurus. Ifthe actual Stegosaurus is 30 ft long, howlong is Brad’s model.

NAME ________________________________________ DATE ______________ PERIOD _____

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Mixed Problem Solving

PROBLEM-SOLVING STRATEGIES

• Work backward.

• Look for a pattern.

• Use a Venn diagram.

• Draw a diagram.

Practice



NAME ________________________________________ DATE ______________ PERIOD _____

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1. TILING Kelly is using 3-inch square tiles

to cover a 4-foot by 2-foot area. The

tiles are 0.5 inches tall. If the tiles were

stacked on top of each other to create a

tower, how many inches tall would the

tower be?

2. AQUARIUM An aquarium holds 42

gallons of water. After 2 minutes, the

aquarium has 3 gallons of water in it.

How many more minutes will it take to

completely fill the aquarium?

3. FABRIC It takes Lucy 7 minutes to cut a

20-yard by 1-yard roll of fabric into 14

equal pieces. How many minutes would

it take her to cut the fabric into 25

equal pieces?

4. SPORTS The width of a soccer field is 12

feet more than }2

3} of its length. If the

field is 96 feet long, what is its

perimeter?

5. BEVERAGES It requires 4 gallon jugs of

water to fill 104 glasses equally. How

many gallons jugs are required to fill

338 glasses equally?

6. GAS It takes Richard 48 seconds to fill

his gas tank with 3 gallons of gas. If

the tank holds 14 gallons, how many

more seconds will it take to fill it

completely?





NAME ________________________________________ DATE ______________ PERIOD _____


Similar Polygons

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Complete the Mini Lab at the top of page 218 in your textbook.


1. Compare the angles of the triangles by matching them up. Identify the

angle pairs that have equal measure.

2. Express the ratios }D

LK

F}, }

E

JK

F}, and }

D

L

E

J} to the nearest tenth.

3. What do you notice about the ratios of the matching sides of matching

triangles?

Read the Lesson

4. Complete the sentence: If two polygons are similar, then their

corresponding angles are __________, and their corresponding sides are

__________.

5. If two polygons have corresponding angles that are congruent, does that

mean that the polygons are similar? Why or why not?

6. If the sides of one square are 3 centimeters and the sides of another

square are 9 centimeters, what is the ratio of corresponding sides from

the first square to the second square?


7. Look up the everyday definition of the word similar in a dictionary. How

does the definition relate to what you learned in this lesson?


NAME ________________________________________ DATE ______________ PERIOD _____


Similar Polygons

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Determine whether nABC is similar to nDEF. Explain your reasoning.

/A > /D, /B > /E, /C > /F,

}DAB

E} 5 }

46

} or }23

}, }BE

CF} 5 }

69

} or }23

}, }DAC

F} 5 }

182} or }

23

}

The corresponding angles are congruent, and the corresponding sides are proportional.

Thus, nABC is similar to nDEF.

Given that polygon KLMN , polygon PQRS, write a proportion tofind the measure of PwQw. Then solve.

The ratio of corresponding sides from polygon KLMN to

polygon PQRS is }43

}. Write a proportion with this scale

factor. Let x represent the measure of PwQw.

}PK

QL} 5 }

43

} KwLw corresponds to PwQw. The scale factor is }43

}.

}5x

} 5 }43

} KL 5 5 and PQ 5 x

5 ? 3 5 x ? 4 Find the cross products.

}145} 5 }

44x} Multiply. Then divide each side by 4.

3.75 5 x Simplify.

1. Determine whether the polygons 2. The triangles below are similar. Write a

below are similar. Explain your proportion to find each missing measure.

reasoning. Then solve.

12

6 4

815

x11

5

124

Kx

3

5

N

LP Q

S RM

4

B

8

E

CA D F

646

12

9

Two polygons are similar if their corresponding angles are congruent and their corresponding side

measures are proportional.

Example 1

Example 2

Exercises


Skills Practice

Similar Polygons

NAME ________________________________________ DATE ______________ PERIOD _____

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Determine whether each pair of polygons is similar. Explain.

1. 2.

3. 4.

Each pair of polygons is similar. Write a proportion to find each missing measure.Then solve.

5. 6.

7. 8.

x

6

9.6

14

22.4

9.6

6

10

6 6.5

1.5

3.63.9

x

x

1.8

2.6

6.5

8 8

2.6

44

x

6 7

7

3

3.53.520

8

14

10

74

151

101

150

100

15

8

10

12


NAME ________________________________________ DATE ______________ PERIOD _____

Practice

Similar Polygons

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Determine whether each pair of polygons is similar. Explain.

1. 2.

Each pair of polygons is similar. Write and solve a proportion to findeach missing measure.

3. 4.

5. 6.

7. TILES A blue rectangular tile and a red rectangular tile are similar. The

blue tile has a length of 10 inches and a perimeter of 30 inches. The red

tile has a length of 6 inches. What is the perimeter of the red tile?

5

13

17

8

15

12

45

22.87.6

5

8

1515

24

4

4

10

x5.6

6

4

4.5

4

6

x

3.5

5

8

20

14

x

12

6

6 9

3 18

18x



Similar Polygons

NAME ________________________________________ DATE ______________ PERIOD _____

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1. JOURNALISM The editor of the school

newspaper must reduce the size of a

graph to fit in one column. The original

graph is 2 inches by 2 inches, and the

scale factor from the original to the

reduced graph is 8:3. Find the

dimensions of the graph as it will

appear in one column of the newspaper.

2. PHOTOCOPIES Lydia plans to use a

photocopy machine to increase the size

of a small chart that she has made as

part of her science project. The original

chart is 4 inches by 5 inches. If she

uses a scale factor of 5:11, will the

chart fit on a sheet of paper 8}12

} inches

by 11 inches? Explain.

3. MICROCHIPS The image of a microchip

in a projection microscope measures 8

inches by 10 inches. The width of the

actual chip is 4 millimeters. How long

is the chip?

4. PROJECTIONS A drawing on a

transparency is 11.25 centimeters wide

by 23.5 centimeters tall. The width of

the image of the drawing projected onto

a screen is 2.7 meters. How tall is the

drawing on the screen?

5. GEOMETRY Polygon ABCD is similar to

polygon FGHI. Each side of polygon

ABCD is 3}14

} times longer than the

corresponding side of polygon FGHI.

Find the perimeter of polygon ABCD.

6. KITES A toy company produces two

kites whose shapes are geometrically

similar. Find the length of the missing

side of the smaller kite.

25 in.25 in.

30 in.

22.5 in.

30 in.

xB

A

D

I

H

F

G

C

3 in.

2 in.

5 in.

3 in.


NAME ________________________________________ DATE ______________ PERIOD _____

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Similar and Congruent Figures

If a 4-inch by 5-inch photograph is enlarged to an 8-inch by 10-inchphotograph, the photographs are said to be similar. They have the sameshape, but they do not have the same size. If a 4-inch by 5-inch photographis duplicated and a new 4-inch by 5-inch photograph is made, thephotographs are said to be congruent. They have the same size and shape.

In each exercise, identify which of the triangles are congruent toeach other and identify which of the shapes are similar to each other.Use the symbol for > congruence and the symbol , for similarity.

1. 2.

3. 4.

5.

a

b

c

d

f

a

b

c

d

AB

C

D

J L

H NK M

A B C D

H K LM N

P

Q

R T

AB

T

C

D

Q

G

H

J

LKM

NP

Example


TI-83/84 Plus Activity

Similar Polygons

NAME ________________________________________ DATE ______________ PERIOD _____

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You can construct similar polygons on a TI-83/84 Plus graphing calculator. After entering the

coordinates of one polygon, construct a similar polygon by multiplying or dividing each of the

x- and y-values by the same amount. First, clear all lists by pressing [MEM] 4 .

Display a polygon with vertices at (1, 7),

(1, 1), (5, 1), and (5, 7) on a TI-83/84 Plus graphing calculator.

Then, construct a similar polygon by multiplying all the x- and

y-values by 2.

Step 1 Enter:

Enter each x-value under L1 and y-value under L2.

Repeat the first value in each list at the bottom of each

list.

Multiply each x-value by 2. Enter the new values under

L3. Do the same for the y-values and enter the new

values under L4.

Step 2 Press and enter the settings as shown.

Step 3 Enter: [STAT PLOT]

[L1]

[L2]

[L3] [L4]

Use a graphing calculator to construct the following polygons. Construct a similarpolygon for each shape.

1. Construct a polygon with vertices at (50, 75), (40, 35), (25, 40), (10, 60), and (20, 70).

Construct a similar polygon by dividing both the x-and y-values by 5. Press 9 to

set the window. Write the coordinates of the second polygon.

(10, 15), (8, 7), (5, 8), (2, 12), (4, 14)2. Construct a polygon with vertices at (2, 6), (12, 6), (12, 12), and (2, 12). Construct a

similar polygon by multiplying the x-values and the y-values by 3. Write the

coordinates of the second polygon.

ZOOM

GRAPH2nd2ndENTER

ENTERENTER

2nd2ndENTER

ENTERENTER2nd

WINDOW

ENTERSTAT

ENTER2nd

[0, 20] scl: 1 by [0, 15] scl: 1


NAME ________________________________________ DATE ______________ PERIOD _____


Dilations

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Complete the Mini Lab at the top of page 225 in your textbook.


1. Measure and compare corresponding lengths on the original figure and

the new figure. Describe the relationship between these measurements?

How does this relate to the change in grid size?

2. MAKE A CONJECTURE What size squares should you use to create a

version of the original figure with dimensions that are four times the

corresponding lengths on the original? Explain.

Read the Lesson

3. If you are given the coordinates of a figure and the scale factor of a

dilation of that figure, how can you find the coordinates of the new

figure?

4. When you graph a figure and its image after a dilation, how can you

check your work?


5. Complete the table below to help you remember the effects of different

scale factors.

If the scale factor is Then the dilation is

between 0 and 1

greater than 1

equal to 1



Dilations

NAME ________________________________________ DATE ______________ PERIOD _____

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Graph nABC with vertices A(22, 21),

B(2, 3), and C(2, 21). Then graph its

image nA9B9C9after a dilation with a

scale factor of }32

}.

A(22, 21) → 122 ? }32

}, 21 ? }32

}2 → A9123, 21}1

2}2

B(2, 3) → 12 ? }32

}, 3 ? }32

}2 → B913, 4}12

}2C(2, 21) → 12 ? }

32

}, 21 ? }32

}2 → C913, 21}1

2}2

Segment M9N9 is a dilation of segment MN. Find the scale factor of the dilation, and classify it as an enlargement or a reduction.

Write the ratio of the x- or y-coordinate of one vertex of thedilated figure to the x- or y-coordinate of the correspondingvertex of the original figure. Use the x-coordinates of N(1, 22)and N9(2, 24).

5 }21

} or 2

The scale factor is 2. Since the image is larger than the originalfigure, the dilation is an enlargement.

1. Polygon ABCD has vertices A(2, 4), B(21, 5), C(23, 25), and

D(3, 24). Find the coordinates of its image after a dilation

with a scale factor of }12

}. Then graph polygon ABCD and its

dilation.

2. Segment P9Q9 is a dilation of segment PQ. Find the scale

factor of the dilation, and classify it as an enlargement or a

reduction.

x-coordinate of point N9}}}x-coordinate of point N

y

xO

C

B

B9

A9 C9A

y

xO

M9M

N

N9

y

xO

y

xO

PP9

Q9 Q

Example 1

Example 2

Exercises

The image produced by enlarging or reducing a figure is called a dilation.


NAME ________________________________________ DATE ______________ PERIOD _____

Skills Practice

Dilations

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Find the coordinates of the vertices of triangle A9B9C9 after triangleABC is dilated using the given scale factor. Then graph triangle ABC

and its dilation.

1. A(1, 1), B(1, 3), C(3, 1); scale factor 3 2. A(22, 22), B(21, 2), C(2, 1); scale factor 2

3. A(24, 6), B(2, 6), C(0, 8); scale factor }12

} 4. A(23, 22), B(1, 2), C(2, 23); scale factor 1.5

Segment P9Q9 is a dilation of segment PQ. Find the scale factor of the dilationand classify it as an enlargement or a reduction.

5. 6.

7. 8.

y

xO

y

xO

y

xO

y

xO

y

xO

P9

Q9

P

Q

y

xO

P9

Q9

P

Q

y

xO

P9

P

Q

Q9

y

xO

P9

P

Q

Q9


Practice

Dilations

NAME ________________________________________ DATE ______________ PERIOD _____

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Draw the image of the figure after the dilation with the given centerand scale factor.

1. center: C, scale factor: 2 2. center: N, scale factor: }12

}

Find the coordinates of the vertices of polygon F9G9H9J9 afterpolygon FGHJ is dilated using the given scale factor. Then graphpolygon FGHJ and polygon F9G9H9J9.

3. F(22, 2), G(2, 3), H(3, 22), J(21, 23); 4. F(22, 2), G(2, 4), H(3, 23), J(24, 24);

scale factor scale factor 2

In the exercises below, figure R9S9T9 is a dilation of figure RST andfigure A9B9C9D9 is a dilation of figure ABCD. Find the scale factor ofeach dilation and classify it as an enlargement or as a reduction.

5. 6.

7. GLASS BLOWING The diameter of a vase is now 4 centimeters. If the diameter increases

by a factor of }7

3}, what will be the diameter then?

3}4

C

B

A

x

y

R

R

S

S

TT x

y

‘

‘

‘

O

A

B

D

D

C

x

y

‘ C‘

B‘A‘

O

x

y

O 4

4

8

8

-4

-4

-8

-8

N

L

P

O

M


NAME ________________________________________ DATE ______________ PERIOD _____


Dilations

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1. EYES Dave’s optometrist used medicine

to dilate his eyes. Before dilation, his

pupils had a diameter of

4.1 millimeters. After dilation,

his pupils had a diameter of

8.2 millimeters. What was the

scale factor of the dilation?

2. BIOLOGY A microscope increases the

size of objects by a factor of 8. How

large will a 0.006 millimeter

paramecium appear?

3. PHOTOGRAPHY A photograph was

enlarged to a width of 15 inches. If the

scale factor was }32

}, what was the width

of the original photograph?

4. MOVIES Film with a width of

35 millimeters is projected onto a screen

where the width is 5 meters. What is

the scale factor of this enlargement?

5. PHOTOCOPYING A 10-inch long copy of

a 2.5-inch long figure needs to be made

with a copying machine. What is the

appropriate scale factor?

6. MODELS A scale model of a boat is

going to be made using a scale of }510}.

If the original length of the boat is

20 meters, what is the length of the

model?

7. MODELS An architectural model is

30 inches tall. If the scale used to build

the model is }1

120}, what is the height of

the actual building?

8. ADVERTISING An advertiser needs a

4-inch picture of a 14-foot automobile.

What is the scale factor of the

reduction?


Enrichment

NAME ________________________________________ DATE ______________ PERIOD _____

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Dilation and Area

A dilation of a shape creates a new shape that is similar to theoriginal. The ratio of the new image to the original is called thescale factor.

Plot and draw each shape. Then perform the dilation ofeach shape using a scale factor of two. After the new imagehas been drawn, determine the area of both the originalshape and its dilation.

1. A(2, 1), B(7, 1), C(4, 4)

Area of original ________

Area of dilation ________

2. circle with radius (1, 2) to (4, 2)

Use 3.14 for p.

Area of original ________

Area of dilation ________

3. R(23, 2), E(23, 22), C(4, 22), I(4, 2) 4. S(23, 3), Q(23, 23), A(3, 23), R(3, 3)

Area of original ________ Area of original ________

Area of dilation ________ Area of dilation ________

5. What general statement can be made about the area of a figure when

compared to its area after being dilated by scale factor 2?

y

xO

Exercises


NAME ________________________________________ DATE ______________ PERIOD _____

TI-73 Activity

Dilations

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Use Lists and Plots to draw figures and their dilations. Enter thecoordinates of one polygon. Then multiply those coordinates by ascale factor. Plot both sets of coordinates.

Graph nABC with vertices at A(1, 1), B(3, 5),and C(2, 8). Graph an image of nABC

with the scale factor of 3.

Step 1 Clear all lists and open the List feature.

[MEM] 6

Step 2 In L1, enter the x-coordinates of the vertices.In L2, enter the y-coordinates. Repeat the first ordered pair at the end of the list. Pressafter each value.

Step 3 In L3, enter a formula to multiply the x-coordinates in L1 by 3.

[TEXT] “Done [STAT] 1

3 [TEXT] “Done

Step 4 In L4, enter a formula to multiply the y-coordinates in L2 by 3.

[TEXT] “Done [STAT] 2 3 [TEXT]

“Done

Step 5 Create two plots.

[PLOT] 1 Turn on Plot 1. Choose line graph. Choose L1 and L2.

[PLOT] 2 Turn on Plot 2. Choose line graph. Choose L3 and L4.

Step 6 Set the graph window and display the plots.

7

Use your calculator to draw the polygons and their dilations. Sketch thedrawings. Label the vertices and coordinates.

1. A(1, 2), B(3, 4), C(4, 3) 2. A(4, 8), B(16, 16), C(12, 4)

Scale factor: 5 Scale factor: }1

4}

3. A(1, 6), B(6, 4), C(4, 2) 4. A(1, 1), B(1, 3), C(3, 3), D(3, 1)

Scale factor: 3 Scale factor: 4

5. A(1.8, 9), B(12, 0.9), C(6, 18), D(21, 18)

Scale factor: }1

3}

ZOOM

2nd

2nd

ENTERENTER

2nd2nd2nd

ENTERENTER2nd

2nd2nd

ENTER

LISTENTER2nd

Example


NAME ________________________________________ DATE ______________ PERIOD _____


Indirect Measurement

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1. What appears to be true about the corresponding angles in the two

triangles?

2. If the corresponding sides are proportional, what could you conclude

about the triangles?

Read the Lesson

3. Complete the following sentence.

When you solve a problem using shadow reckoning, the objects being

compared and their shadows form two sides of _______________.

4. Suppose that you are standing near a building and you see the shadows

cast by you and the building. If you know the length of each of these

shadows and you know how tall you are, write a proportion in words that

you can use to find the height of the building.

5. STATUE If a statue casts a 6-foot shadow and a 5-foot mailbox casts a

4-foot shadow, how tall is the statue?


6. Work with a partner. Have your partner draw two triangles that are

similar with the lengths of two corresponding sides labeled and the

length of one additional side labeled. Tell your partner how to write a

proportion to solve for the length of the side corresponding to the

additional side labeled.


NAME ________________________________________ DATE ______________ PERIOD _____



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→

→

5

LIGHTING George is standing next to alightpole in the middle of the day. George’s shadow is 1.5 feet long, and the lightpole’s shadow is 4.5 feet long. If George is 6 feet tall, how tall is the lightpole?

Write a proportion and solve.

George’s shadow → 1.5 6 George’s height

lightpole’s shadow → 4.5 h lightpole’s height

1.5 ? h 5 4.5 ? 6 Find the cross products.

1.5h 5 27 Multiply.

}11.5.5h

} 5 }12.75} Divide each side by 1.5.

h 5 18 Simplify.

The lightpole is 18 feet tall.

1. MONUMENTS A statue casts a shadow 30 feet

long. At the same time, a person who is 5 feet

tall casts a shadow that is 6 feet long. How

tall is the statue?

2. BUILDINGS A building casts a shadow 72 meters

long. At the same time, a parking meter that is

1.2 meters tall casts a shadow that is 0.8 meter

long. How tall is the building?

3. SURVEYING The two

triangles shown in the

figure are similar. Find

the distance d across

Red River.

0.9 km

1 km

1.8 km

h ft

30 ft

5 ft

6 ft

Indirect measurement allows you to find distances or lengths that are difficult to measure directly

using the properties of similar polygons.

h ft

6 ft

4.5 ft 1.5 ft

Example

Exercises


Skills Practice


NAME ________________________________________ DATE ______________ PERIOD _____

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Write a proportion and solve the problem.

1. HEIGHT How tall is Becky? 2. FLAGS How tall is the flagpole?

3. BEACH How deep is the water 4. ACCESSIBILITY How high is the ramp

50 feet from shore? when it is 2 feet from the building?

(Hint: nABE , nACD)

5. AMUSEMENT PARKS The triangles in 6. CLASS CHANGES The triangles in the figure

the figure are similar. How far is the are similar. How far is the entrance to

water ride from the roller coaster? the gymnasium from the band room?

Round to the nearest tenth.

45 m

Information Booth

Water Ride

Ferris Wheel

Roller Coaster

Park Entrance21 m

10 md m

30 m

25 m

Tennis Court

Band Room

Gymnasium

Cafeteria

Fountain

32 m

d m

2 ft

h ftA

BC

DE20 ft

2.5 fth ft

50 ft8 ft

1 ft

9 ft

h ft

6 ft1

2

31 ft1

2

7 ft

3 ft

h ft

4 ft


NAME ________________________________________ DATE ______________ PERIOD _____

Practice


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In Exercises 1-4, the triangles are similar. Write a proportion andsolve the problem.

1. TREES How tall is Yori? 2. TREASURE HUNT How far is it from

the hut to the gold coins?

3. LAKE How deep is the water 31.5 feet 4. SURVEYING How far is it across the

from the shore? (Hint: nABC , nADE) pond? (Hint: nRST , nRUV)

For Exercise 5, draw a diagram of the situation. Then write a proportion andsolve the problem.

5. ARCH The Gateway Arch in St. Louis,

Missouri, is 630 feet tall. Suppose a

12-foot tall pole that is near the Arch

casts a 5-foot shadow. How long is the

Arch’s shadow?

5 ft

20 ft

25 ft

h

2 ft

6 ft31.5 ft

d ft

E

DB

C

A

162.5 m156 m

325 m

R

S

V U

T

d m

15 yd

18 yd

12 yd

x yd

Gold Coins

Shovel

Hut

Silver Coins

Jewels




NAME ________________________________________ DATE ______________ PERIOD _____

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1. HEIGHT Paco is 6 feet tall and casts a

12-foot shadow. At the same time,

Diane casts an 11-foot shadow. How tall

is Diane?

2. LIGHTING If a 25-foot-tall house casts a

75-foot shadow at the same time that a

streetlight casts a 60-foot shadow, how

tall is the streetlight?

3. FLAGPOLE Lena is 5}12

} feet tall and casts

an 8-foot shadow. At the same time, a

flagpole casts a 48-foot shadow. How

tall is the flagpole?

4. LANDMARKS A woman who is 5 feet

5 inches tall is standing near the

Space Needle in Seattle, Washington.

She casts a 13-inch shadow at the same

time that the Space Needle casts a

121-foot shadow. How tall is the Space

Needle?

5. NATIONAL MONUMENTS A 42-foot

flagpole near the Washington

Monument casts a shadow that is

14 feet long. At the same time, the

Washington Monument casts a shadow

that is 185 feet long. How tall is the

Washington Monument?

6. ACCESSIBILITY A ramp slopes upward

from the sidewalk to the entrance of a

building at a constant incline. If the

ramp is 2 feet high when it is 5 feet

from the sidewalk, how high is the

ramp when it is 7 feet from the

sidewalk?

5 ft

2 ft


NAME ________________________________________ DATE ______________ PERIOD _____

Enrichment4-9

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A proportion can be used to determine the heightof tall structures if three variables of the proportionare known. The three known variables are usuallythe height a of the observer, the length b of theobserver’s shadow, and the length d of thestructure’s shadow. However, a proportion can besolved given any three of the four variables.

This chart contains information about various observers and tallbuildings. Use proportions and your calculator to complete the chartof tall buildings of the world.

a

b

c

d

ab

cd

5

Height ofObserver

Length ofShadow

BuildingLocation

Height ofBuilding

Length ofShadow

1. 5 ft 8 in.Natwest,London

80 ft

2. 4 ft 9 in.Columbia SeafirstCenter, Seattle

954 ft 318 ft

3. 5 ft 10 in. 14 in.Wachovia Building,Winston-Salem

410 ft

4. 15 in.Waterfront Towers,Honolulu

400 ft 83 ft 4 in.

5. 6 ftCN Tower,Toronto

1,821 ft 303 ft 6 in.

6. 5 ft 9 in. 1 ft 11 in.Gateway Arch,St. Louis

630 ft

7. 5 ft 3 in. 1 ft 9 in.Eiffel Tower,Paris

328 ft

8. 8 in.Texas CommerceTower, Houston

1,002 ft 167 ft

9. 5 ft 6 in. 11 in.John HancockTower, Boston

131 ft 8 in.

10. 5 ft 4 in. 8 in.Sears Tower,Chicago

181 ft 9 in.

11. 5 ft 6 in.Barnett Tower,Jacksonville

631 ft 105 ft 2 in.


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NAME ________________________________________ DATE ______________ PERIOD _____


Scale Drawings and Models

4-10




1. How many inches tall is the photo of the statue?

2. The actual height of the statue is 19 feet. Write a ratio comparing the

photo height to the actual height.

3. Simplify the ratio you found and compare it to the scale shown below the

photo.

Read the Lesson

4. Give another example of a scale drawing or scale model that is different

from the examples of scale drawings and scale models given on page 236

in your textbook.

5. Complete the sentence: distances on a scale model are __________ to

distances in real life.

6. What is the scale factor for a model if part of the model that is 4 inches

corresponds to a real-life object that is 16 inches?


7. Make a scale drawing of a room, such as your classroom or your bedroom.

Select an appropriate scale so that your drawing is a reasonable size. Be

sure to indicate your scale on your drawing. Use another piece of paper if

necessary.


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NAME ________________________________________ DATE ______________ PERIOD _____

4-10

Exercises

Example 1 INTERIOR DESIGN A designer has made

a scale drawing of a living room for one of her

clients. The scale of the drawing is 1 inch 5 1}13

}

feet. On the drawing, the sofa is 6 inches long.

Find the actual length of the sofa.

Let x represent the actual length of the sofa. Write and solve a proportion.

The actual length of the sofa is 8 feet.

Find the scale factor for the drawing in Example 1.

Write the ratio of 1 inch to 1}13

} feet in simplest form.

5 }116

iinn.@.@}

Convert 1}13

} feet to inches.

The scale factor is }116} or 1:16. This means that each distance on the drawing is }

116} the

actual distance.

LANDSCAPING Yutaka has made a scale drawingof his yard. The scale of the drawing is1 centimeter 5 0.5 meter.

1. The length of the patio is 4.5 centimeters in the

drawing. Find the actual length.

2. The actual distance between the water faucet and

the pear tree is 11.2 meters. Find the corresponding

distance on the drawing.

3. Find the scale factor for the drawing.

1 in.}

1}13

} ft

To find the scale factor for scale drawings and models, write the ratio given by the scale in simplest

form.

1 in. 6 in.

1 ft1

3

x ft5

1 · x 5 1 · 61

3

x 5 8

Find the cross products.

Simplify.

Drawing Scale Actual Length

1 in. = 1 ft

Sofa

13

1 cm = 0.5 m

Patio

PondPath

Water Faucet

PearTree

Garden

Distances on a scale drawing or model are proportional to real-life distances. The scale is determined

by the ratio of a given length on a drawing or model to its corresponding actual length.

Example 2


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NAME ________________________________________ DATE ______________ PERIOD _____

Skills Practice


4-10

ARCHITECTURE The scale on a set of architectural drawings for a houseis 1.5 inches 5 2 feet. Find the length of each part of the house.

7. What is the scale factor of these drawings?

TOWN PLANNING For Exercises 8–11, use the following information.

As part of a downtown renewal project, businesses have constructeda scale model of the town square to present to the city commissionfor its approval. The scale of the model is 1 inch 5 7 feet.

8. The courthouse is the tallest building in the town square. If it is

5}12

} inches tall in the model, how tall is the actual building?

9. The business owners would like to install new lampposts that are

each 12 feet tall. How tall are the lampposts in the model?

10. In the model, the lampposts are 3}37

} inches apart. How far apart

will they be when they are installed?

11. What is the scale factor?

12. MAPS On a map, two cities are 6}12

} inches apart. The actual distance

between the cities is 104 miles. What is the scale of the map?

Room Drawing Length Actual Length

1. Living Room 15 inches

2. Dining Room 10.5 inches

3. Kitchen 12}34

} inches

4. Laundry Room 8}14

} inches

5. Hall 13}78

} inches

6. Garage 16.5 inches


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Practice


NAME ________________________________________ DATE ______________ PERIOD _____

4-10

LANDSCAPE PLANS For Exercises 1–4, use the drawing and an inch rulerto find the actual length and width of each section of the park.Measure to the nearest eighth of an inch.

1. Playground

2. Restrooms

3. Picnic Area

4. What is the scale factor of the

park plan? Explain its meaning.

5. SPIDERS The smallest spider, the Patu marples of Samoa, is 0.43 millimeter long.

A scale model of this spider is 8 centimeters long. What is the scale of the model?

What is the scale factor of the model?

6. ANIMALS An average adult giraffe is 18 feet tall. A newborn giraffe is about 6 feet tall.

Kayla is building a model of a mother giraffe and her newborn. She wants the model to

be no more than 17 inches high. Choose an appropriate scale for a model of the giraffes.

Then use it to find the height of the mother and the height of the newborn giraffe.

7. TRAVEL On a map, the distance between Charleston and Columbia, South Carolina, is 5

inches. If the scale of the map is }7

8} inch 5 20 miles, about how long would it take the

Garcia family to drive from Charleston to Columbia if they drove 60 miles per hour?

Lawn

Picnic

Area

Dog

Run

Playground

Key1 in. 5 68 ft

Restrooms


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NAME ________________________________________ DATE ______________ PERIOD _____



4-10

CAMPUS PLANNING For Exercises 1–3, use thefollowing information.

The local school district has made a scale modelof the campus of Engels Middle School includinga proposed new building. The scale of the modelis 1 inch 5 3 feet.

1. An existing gymnasium is 8 inches tall

in the model. How tall is the actual

gymnasium?

2. The new building is 22.5 inches from

the gymnasium in the model. What

will be the actual distance from the

gymnasium to the new building if it

is built?

3. What is the scale factor of the model? 4. MAPS On a map, two cities are 5}34

}

inches apart. The scale of the map is

}12

} inch 5 3 miles. What is the actual

distance between the towns?

5. TRUCKS The bed of Jerry’s pickup truck

is 6 feet long. On a scale model of the

truck, the bed is 8 inches long. What is

the scale of the model?

6 ft

6. HOUSING Marta is making a scale

drawing of her apartment for a school

project. The apartment is 28 feet wide.

On her drawing, the apartment is

7 inches wide. What is the scale of

Marta’s drawing?

28 ft

Gym

nas

ium

Par

king

Academic

Building

View of Campus from Above

New

Building


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NAME ________________________________________ DATE ______________ PERIOD _____

4-10

Scale Drawings

The figure at the right has an area of 6 square units. If the figurerepresented a map, and was drawn to a scale of 1 unit 5 3 feet, thelengths of the sides would be 6 ft and 9 ft. So, the figure wouldrepresent an area of 54 square feet.

The ratio of the actual area of the figure to the scale area of thefigure can be expressed as a ratio.

}ascctaulaelaarreeaa

} 5 }564} 5 }

19

} or 1 to 9

Find the actual area and the scale area of these figures. Thendetermine the ratio of actual area to scale area.

1. Scale: 1 unit 5 4 ft 2. Scale: 1 unit 5 50 cm

actual area actual area

scale area scale area

ratio ratio

3. Scale: 1 unit 5 8 mi 4. Scale: 1 unit 5 12 m



ratio ratio

5. Scale: 1 unit 5 18 in. 6. Scale: 1 unit 5 6 km



ratio ratio


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NAME ________________________________________ DATE ______________ PERIOD _____

Spreadsheet Activity

Scale Drawings

4-10

Example

You can use a spreadsheet to calculate the actual measurements froma scale drawing.

The table shows the measurements on anarchitect’s drawing for various rooms in a house. If 2centimeters represents 1.5 feet, find the actualmeasurements for each room.

Step 1 Use the first column for all of the scale measurements.Use the formula bar to enter the numbers. Then press ENTER to move to the next cell.

Step 2 In cell B1, enter an equals sign followed by A1*1.5/2.Then press ENTER to return the actual size of the living room.

Step 3 Click on the bottom right side of cell B1 and drag the formula down through cell B6 to return the actual measurements for the other rooms.

The actual measurements for all of the rooms are 22.5 feet, 18.75 feet,30 feet, 33.75 feet, 18.75 feet,and 31.5 feet.

Find the actual measurements for each scale measurement if 1 inch 5 2 feet.

1. 5 in. 2. 3 in. 3. 4 in. 4. 2 in.

5. 3.5 in. 6. 6.7 in. 7. 8.5 in. 8. 5.2 in.

9. 6.5 in. 10. 11 in. 11. 9 in. 12. 12 in.

13. 13 in. 14. 17 in. 15. 22.1 in. 16. 25.4 in.

17. 16.25 in. 18. 15.5 in.

RoomScale

Measurement

Living 30 cm

Bedroom 1 25 cm

Bedroom 2 40 cm

Master 45 cm

Dining 25 cm

Kitchen 42 cm

A1

34567

2

B C30

18.75

30

18.75

31.5

22.5

33.75

25

40

45

25

42

Sheet 1 Sheet 2 Sheet 3

Exercises

Documents

4-1 Study Guide and Intervention 8 Chapter 4... · Study Guide and Intervention Ratios and Rates 4-1 ... Express this ratio in simplest form. 2. ... For each exercise below, rates