ratios and rates

FeatureLesson

Course 3Course 3

LessonMain

How many miles are in 21,120 yd? (Hint: 1 mi = 5,280 ft)

12 mi

LESSON 4-1LESSON 4-1

Ratios and RatesRatios and Rates

Problem of the Day

4-1

FeatureLesson

Course 3Course 3

LessonMain



(For help, go to Lesson 2-3.)

1. Vocabulary Review What is the least common denominator of two rational numbers?

Determine which rational number is greater.

2. , 3. , 4. , 5. ,39

16

1525

45

4554

23

47

712

Check Skills You’ll Need


4-1

FeatureLesson

Course 3Course 3

LessonMain

Ratios and RatesRatios and RatesLESSON 4-1LESSON 4-1

Solutions

1. The least common denominator is the smallest multiple the denominators have in common.

2. 3. 4. 5. 39

45

4554

712


4-1

FeatureLesson

Course 3Course 3

LessonMain

Write the ratio 36 seconds to 12 minutes in

simplest form.


36 s12 min

36 s720 s=

Convert minutes to seconds so that both measures are in the same units. Divide the common units.

36720

= 36 ÷ 36720 ÷ 36

Divide the numerator and denominator by the GCF, 36.

=1

20 Simplify.

The ratio of 36 seconds: 12 minutes is .1

20 Quick Check

Additional Examples

4-1

FeatureLesson

Course 3Course 3

LessonMain

Computer time costs $4.50 for 30 min. What is the

unit rate?


costnumber of minutes

$4.5030 min= Write a rate comparing cost to minutes.

= $.15/min Divide.

The unit rate is $.15 per minute.

Quick Check

Additional Examples

4-1

FeatureLesson

Course 3Course 3

LessonMain

Keneesha drove her car 267 mi using 11 gal of gas. Vanessa

drove her car 210 mi using 9 gal. Give the unit rate for each. Which car

got more miles per gallon of gas?


Keneesha’s car got more miles per gallon.

milesgallons

267 mi11 gal=

Write the rates comparing

miles to gallons.

milesgallons

210 mi9 gal=

Keneesha Vanessa

Divide. 24.27272727 mi/gal 23.33333333 mi/gal

24.3 mi/gal 23.3 mi/galRound to the nearest tenth.

Additional Examples

4-1

FeatureLesson

Course 3Course 3

LessonMain

(continued)


Check for Reasonableness 24.3 • 11 = 267.3 and 267.3 267. Also, 23.3 • 9 = 209.7 and 209.7 210. The answers are reasonable.

Quick Check

Additional Examples

4-1

FeatureLesson

Course 3Course 3

LessonMain


Express each ratio in simplest form.

1. 27 laps : 81 minutes

2. 12 minutes : 3 hours

3. Carli walked 16 miles in 5 hours. Find the unit rate.

4. A 21-oz bottle of shampoo costs $2.80. A 12-oz bottle costs $1.35. Which has the better unit rate?


13

115

3.2 mi/h

12-oz bottle

Lesson Quiz

4-1

FeatureLesson

Course 3Course 3

LessonMain

Write a fraction in lowest terms, with a single digit numerator, that is about the same as each decimal: 0.52, 0.13, 0.74, 0.88.


12 , , ,

18

34

78

Converting UnitsConverting Units

Problem of the Day

4-2

FeatureLesson

Course 3Course 3

LessonMain




1. Vocabulary Review What is the product of a number and its reciprocal?

Find each product. Write the answer in simplest form.

2. 3.

4. 5.

103

14

46 •

56

•

49

32•

67

83•



4-2

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Solutions

1. 1 2.

3. 4.

5.

10 • 13 • 4 = =

56

10 • 13 • 4

5

2

4 • 56 • 6 =

4 • 56 • 6 =

1018

59=

2

3= =

4 • 39 • 2

4 • 39 • 2

23

2

1

1

3

26 • 87 • 3 =

6 • 87 • 3 =

1671

= 227



4-2

FeatureLesson

Course 3Course 3

LessonMain

Convert 0.7 mi to ft.


There are 3,696 feet in 0.7 miles.

= (0.7)(5,280) ft

1 Simplify.

= 3,696 ft Divide.


Since 5,280 ft = 1 mi, use the conversion factor 5,280 ft.1 mi

0.7 = •0.7 mi

15,280 ft

1 miMultiply by a conversion factor .5,280 ft

1 mi

Quick Check

Additional Examples

4-2

FeatureLesson

Course 3Course 3

LessonMain

A rowing team completed a 2000-m course at a

rate of 6.84 m/s. Convert this rate to kilometers per minute.


The team rowed at a rate of 0.4104 km/min.

Estimate 6.84 7. Then, 7 • 60 ÷ 1000 = 0.42.

= • •6.84 m

1 s1 km

1000 m

Multiply by two ratios thateach equal one.

60 s1 min

6.84 m1 s

= Simplify.

= 0.4104 Use a calculator.

(6.48)(1)(60) km(1)(1,000)(1) min

Check for Reasonableness The answer 0.4104 km/min is close to the estimate 0.42. The answer is reasonable.

Divide by the common units.


Quick Check

Additional Examples

4-2

FeatureLesson

Course 3Course 3

LessonMain

Use compatible numbers to estimate the number

of gallons in 33 quarts.


There are about 8 gallons in 33 quarts.

The conversion factor for changing gallons to quarts is .

= •32 qt

1 1 gal4 qt

Multiply by the conversion factor.Divide by the common units.

= 8 gallons Divide.

= gallons Simplify.324

1 gal4 qt

33 qt 32 qtRound to the nearest numberdivisible by 4.


Quick Check

Additional Examples

4-2

FeatureLesson

Course 3Course 3

LessonMain

Convert 650 g to ounces.



Quick Check

There are about 22.9 oz in 650 g.

650 g =650 g

1 •1 oz

28.4 gMultiply by the conversion factor .

1 oz28.4 g

(650)(1) oz

28.4 22.9 oz

a calculator.Simplify. Divide using=

Additional Examples

4-2

FeatureLesson

Course 3Course 3

LessonMain

1. Convert 0.75 hours to seconds.

2. $150 per hour is how much per minute?

3. 69.2 cm is about how many meters?

4. Convert 12 qt to liters.


2,700 seconds

$2.50 per min

0.7 m


about 11.3L

Lesson Quiz

4-2

FeatureLesson

Course 3Course 3

LessonMain

Write each word phrase as an algebraic expression.

a. 12 times a number

b. 8 less than a number

c. twice the sum of 5 and a number


12n

n – 8

2(5 + n)

Solving ProportionsSolving Proportions

Problem of the Day

4-3

FeatureLesson

Course 3Course 3

LessonMain


Solving ProportionsSolving Proportions


1. Vocabulary Review Is the fraction in simplest form? Explain.

a + 2b + 2

Write each fraction in simplest form.

2. 3. 4. 5.3099

4212

132602

7025



4-3

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Course 3Course 3

LessonMain

Solving ProportionsSolving ProportionsLESSON 4-3LESSON 4-3

2. 3.

4. 5.

3099

3 • 103 • 33= =

1033

4212

6 • 76 • 2= =

72

1

1= 3

1

1

12

132602

2 • 662 • 301= =

1

1

66301

7025

5 • 145 • 5= =

145 = 2

1

1

45

Solutions

1. Yes; there is no common factor between the numerator and denominator.


4-3

FeatureLesson

Course 3Course 3

LessonMain


Do and form a proportion? Explain. 49

818

49

818

gallons Write as a proportion.

gallons Use number sense to find acommon multiplier.

Quick Check

49

818

Since = ,they form a proportion.

Additional Examples

4-3

FeatureLesson

Course 3Course 3

LessonMain

The fixed rate of conversion is 1 euro = 0.7876 Irish pounds.

How many euros would you receive for 125 Irish pounds?


You would receive 158.71 euros.

Let p = the number of euros.

=0.7876

1125

pWrite theproportion .

Irish poundseuros

0.7876 • p = 1 • 125 Write the cross products.

=0.7876 • p

0.7876125

0.7876Divide each side by 0.7876.

Quick Check

125 0.7876 Use a calculator.

Additional Examples

4-3

FeatureLesson

Course 3Course 3

LessonMain

Solve each proportion.

2. =

3. =

4. Suppose the exchange rate for dollars to Indian rupees is 0.02. How many rupees should you receive for $100?

w12

34

20r

45


9

25

5,000 rupees

1. Is proportional to ? Explain.58

1024

No; the fractions are not equal.

Lesson Quiz

4-3

FeatureLesson

Course 3Course 3

LessonMain

A football team scored 38 points in a game. They scored 3 points for a field goal and 7 points for each touchdown with an extra point. How many field goals did they make? How many touchdowns?

1 field goal and 5 touchdowns or 2 touchdowns and 8 field goals


Similar Figures and ProportionsSimilar Figures and Proportions

Problem of the Day

4-4

FeatureLesson

Course 3Course 3

LessonMain




1. Vocabulary Review What are the cross products for 1015

23

Solve each proportion.

2. = 3. = 4. = 713

21 t

k50

2210

1625

324 m


= ?


4-4

FeatureLesson

Course 3Course 3

LessonMain

Similar Figures and ProportionsSimilar Figures and ProportionsLESSON 4-4LESSON 4-4

Solutions

7t = 273

=

t = 39

7t7

2737

14

10k = 1,100

=

k = 110

10k10

1,10010

4. 16m = 8,100; m = 506


4-4

FeatureLesson

Course 3Course 3

LessonMain

Is rectangle ABCD similar to rectangle RSTU? Explain why or

why not.


Additional Examples

4-4

First, check to see if corresponding angles are congruent. A R B S All right angles are 90°.

C T D U

FeatureLesson

Course 3Course 3

LessonMain

(continued)


Next, check to see if corresponding sides are in proportion.

Quick Check

Additional Examples

4-4

The corresponding sides are in proportion, so rectangle ABCD is similar to rectangle RSTU.

ABRS

DAUR

AB corresponds to RS. DA corresponds to UR.

648

324

Substitute.

6 • 24 48 • 3 Write the cross products.

144 = 144 Simplify.

FeatureLesson

Course 3Course 3

LessonMain

A stonemason’s sketch of a carving to be madeon a building includes the letter “E” shown below. If thewidth of the actual letter in the arrangement is 22 in.,what is the height?


2.75 • x = 5 • 22 Write the cross products.

Simplify.2.75 x = 110

Divide each side by 2.75.2.75x2.75

110 2.75

=

x = 40 Simplify.

Set up a proportion.2.75 in.5 in.

22 in.x=

Quick CheckThe height of the letter is 40 inches.

Additional Examples

4-4

FeatureLesson

Course 3Course 3

LessonMain

RST ~ PSU. Find the value of d.


12 • d = 21 •14 Write the cross products.

Simplify.12d = 294

Divide each side by 12.12d12

29412

=

d = 24.5 Simplify.

Write a proportion.14d

1221

=

Quick Check

The value of d is 24.5.

Additional Examples

4-4

FeatureLesson

Course 3Course 3

LessonMain


1. Are the triangles similar? Explain.

2. A model of a building is 18 in. tall and 24 in. wide. The building is 30 ft tall. How wide is the building?


No; their sides are not proportional.

40 ft

Lesson Quiz

4-4

FeatureLesson

Course 3Course 3

LessonMain


3. In the figure at the right, MNO ~ LNP. Find the value of a.

18

4. If all the lengths in Exercise 3 are doubled, are the triangles still similar? Explain why or why not.

Yes; corresponding values are multiplied by the same factor.

Lesson Quiz

4-4

FeatureLesson

Course 3Course 3

LessonMain

There are three different 1-digit numbers greater than zero and all odd. Their sum is 15. What are the numbers?

3, 5, 7 or 1, 5, 9


Similarity TransformationsSimilarity Transformations

Problem of the Day

4-5

FeatureLesson

Course 3Course 3

LessonMain


Similarity TransformationsSimilarity Transformations


1. Vocabulary Review The first coordinate in an ordered pair is the ? -coordinate.

Graph each point on a coordinate plane.

2. A(3, 6) 3. B(–2, 7)

4. C(5, –1) 5. D(–3, 0)



4-5

FeatureLesson

Course 3Course 3

LessonMain

Similarity TransformationsSimilarity TransformationsLESSON 4-5LESSON 4-5

Solutions

1. 2-5.x


4-5

FeatureLesson

Course 3Course 3

LessonMain

Find the image of ABC after a dilation with center A and a

scale factor of 3.


A B is 3 times AB.

A C is 3 times AC.

Since A is the center of dilation

A = A .

A = A

Quick Check

A B C is the image of ABC after a dilation with a scale factor of 3. ABC ~ A B C

Additional Examples

4-5

FeatureLesson

Course 3Course 3

LessonMain

Find the coordinates of the image of quadrilateral KLMN after

a dilation with a scale factor of . Quadrilateral KLMN has vertices

K (–2, –1), L (0, 2), M (4, 2), and N (4, –1).


12

Step 1 Multiply the x- and

y-coordinates of each point by . 12

Step 2 Graph the image.

K (–2, –1) K (–1, – )

L (0, 2) L (0, 1)

M (4, 2) M (2, 1)

N (4, –1) N (2, – )

12

12

Quick Check

Additional Examples

4-5

FeatureLesson

Course 3Course 3

LessonMain


The figure below PQR shows the outline of aplaying field. A city planner dilates the design to showthe area available for community youth to play sports.Find the scale factor. Is it an enlargement or a reduction?

Quick Check

The scale factor is 1.5.

The dilation is an enlargement.

= = = 1.5imageoriginal

P Q PQ

64

32

Additional Examples

4-5

FeatureLesson

Course 3Course 3

LessonMain

ABC has coordinates A(0, 0), B(10, 0), and C(5, 5). Find the coordinates of the image of ABC after a dilation with each scale factor.

1.

2. 4

3.


15

A (0, 0), B (2, 0), C (1, 1)

A (0, 0), B (40, 0), C (20, 20)

Figure ABCD shows the outline of a porch. The figureA′B′C′D′ is the outline of a table formed by dilatingABCD. Find the scale factor. Is it an enlargementor a reduction?

, reduction13

Lesson Quiz

4-5

FeatureLesson

Course 3Course 3

LessonMain

Mirror primes are pairs of prime numbers in which the digits are reversed, such as 13 and 31. Find all the mirror primes less than 100.

13 and 31, 17 and 71, 37 and 73, and 79 and 97; 11 is its own mirror image.


Scale Models and MapsScale Models and Maps

Problem of the Day

4-6

FeatureLesson

Course 3Course 3

LessonMain


Scale Models and MapsScale Models and Maps

(For help, go to the Skills Handbook page 632.)

1. Vocabulary Review A product is the result of which operation?

Multiply.

2. 4 3.2 3. 7.6 5.9

4. 1.8 22 5. 13 6.5



4-6

FeatureLesson

Course 3Course 3

LessonMain

Scale Models and MapsScale Models and MapsLESSON 4-6LESSON 4-6

Solutions

1. multiplication 2. 12.8

3. 44.84 4. 39.6

5. 84.5


4-6

FeatureLesson

Course 3Course 3

LessonMain

On a blueprint, the cellar is 4 in. by 3 in. The scale is

in. = 8 ft. What are the length and width of the actual cellar?


12

First, find the actual length of the cellar.

Let = the actual length of the cellar.

blueprint measure (in.)actual measure (ft)

128

=4 blueprint length (in.)

actual length (ft)

Additional Examples

4-6

FeatureLesson

Course 3Course 3

LessonMain

(continued)


12

= 32 Simplify.

= 64 Simplify.

12

12

=12

32 Divide each side

by .12

12

• = 8 • 4 Write the cross products.

Additional Examples

4-6

FeatureLesson

Course 3Course 3

LessonMain

(continued)


The length of the actual room is 64 ft.

Next, find the actual width of the cellar.

Let w = the actual width of the cellar.

blueprint measure (in.)actual measure (ft)

128

=3 blueprint length (in.)

actual length (ft)w

Additional Examples

4-6

FeatureLesson

Course 3Course 3

LessonMain

(continued)


w = 48 Simplify.

12

w = 24 Simplify.

12

• w = 8 • 3 Write the cross products.

12

12

=12

24 Divide each side

by .12

w

The width of the actual room is 48 ft. Quick Check

Additional Examples

4-6

FeatureLesson

Course 3Course 3

LessonMain

The map distance from El Paso, Texas, to Chihuahua, Mexico, measures about 7.5 cm. The scale is 1 cm = 50 km. What is the actual distance?


1 • d = 50 • 7.5 Write the cross products.

d = 375 Simplify.

The actual distance from El Paso, Texas to Chihuahua, Mexicois 375 kilometers.

Let d be the actual distance from El Paso, Texas to Chihuahua, Mexico.

map (cm)actual (km)

150

=7.5 map (cm)

actual (km)dSet up a proportion.

Quick Check

Additional Examples

4-6

FeatureLesson

Course 3Course 3

LessonMain

1. A 6-ft man is designing a new chair that would make him feel like a 2.5-ft child. The seat of a normal chair is 1.5 ft high. How high should he make the seat in his new chair?

2. A map scale shows 4 cm to represent 6 km. Two intersections measure 1 cm apart on the map. What is the actual distance?


3.6 ft

1.5 km

Lesson Quiz

4-6

FeatureLesson

Course 3Course 3

LessonMain


3. A tennis court is 36 ft wide. A drawing of the court is 2 in. long and 1 in. wide. Find the scale used.1

4

4. Find the actual length of the court.

1 in. = 36 ft

81 ft

Lesson Quiz

4-6

For Exercises 3–4, use the diagram.

FeatureLesson

Course 3Course 3

LessonMain

A rectangular field is 120 yd long and 53 yd 1 ft wide. How much longer is the field than it is wide?

66 yd 2 ft


Similarity and Indirect MeasurementSimilarity and Indirect Measurement

Problem of the Day

4-7

FeatureLesson

Course 3Course 3

LessonMain


Similarity and Indirect MeasurementSimilarity and Indirect Measurement


1. Vocabulary Review Similar figures have the same ? but not necessarily the same size.

2. If ABC ~ XYZ, which angle is congruent to B?



4-7

FeatureLesson

Course 3Course 3

LessonMain

Solutions

1. shape

2. Y

Similarity and Indirect MeasurementSimilarity and Indirect MeasurementLESSON 4-7LESSON 4-7


4-7

FeatureLesson

Course 3Course 3

LessonMain

When a 6-ft student casts a 17-ft shadow, a flagpole casts a shadow that is 51 ft long. Find the height of the flagpole.


Set up a proportion for the similar triangles.

17h = 6 • 51 Write the cross products.

h = 18 Simplify.

Divide each side by 17.17h17

6 • 5117=

The height of the flagpole is 18 ft.

Words

Let h = the flagpole’s height.

Proportion

flagpole’s heightstudent’s height

length of flagpole’s shadowlength of student’s shadow=

h6

5117=

Quick Check

Additional Examples

4-7

FeatureLesson

Course 3Course 3

LessonMain

In the figure below, ABC ~ EDC. Find d.


=

Use similar triangles to set up a proportion involving the lengths of corresponding sides.

EDAB

CDCB

ED corresponds to AB. CD corresponds to CB.

=d

416141312

Substitute.

312 • d = 416 • 141 Write the cross products.

Additional Examples

4-7

FeatureLesson

Course 3Course 3

LessonMain

(continued)


The length d is 188 m.

312d = 58,656 Simplify.

312d 58,656 312 312= Divide each side by 312.

58,656 312 188 Use a calculator.

Quick Check

Additional Examples

4-7

FeatureLesson

Course 3Course 3

LessonMain

1. A 5-ft tall student casts a 12-ft shadow. A tree casts a 27-ft shadow. How tall is the tree?

2. A 6-ft man casts a 9-ft shadow. A sculpture casts a 45-ft shadow. How tall is the sculpture?


11.25 ft tall

30 ft

Lesson Quiz

4-7

FeatureLesson

Course 3Course 3

LessonMain


3. The diagram shows an outline of a village green EFG next to a small park JHG. The length of JH is 47.4 m, FG is 31 m, and HG is 15.8 m. Find the length of EF.

93 m

Use the diagram for Exercise 3. EFG ~ JHG

Lesson Quiz

4-7

ratios and rates

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