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Ratios and Rates

LESSON 4-1

Problem of the Day

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

12 mi

4-1

Ratios and Rates

LESSON 4-1

Check Skills You’ll Need

(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.

3

9

1

6

15

25

4

5

45

54

2

3

4

7

7

12

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

Check Skills You’ll Need

4-1

Ratios and Rates

LESSON 4-1

Check Skills You’ll Need

Solutions

1. The least common denominator is the smallest multiple the

denominators have in common.

3

9

4

5

45

54

7

12

2. 3. 4. 5.

4-1

Convert minutes to seconds so that both

measures are in the same units.

Divide the common units.

36 s

12 min

36 s

720 s

=

Divide the numerator and denominator

by the GCF, 36.

36

720

36 ÷36

720 ÷ 36

=

1

20

=

Simplify.

1

20

The ratio of 36 seconds: 12 minutes is .

Ratios and RatesLESSON 4-1

Additional Examples

Write the ratio 36 seconds to 12 minutes in simplest form.

Quick Check

4-1

number of minutes

$4.50

30 min

Write a rate comparing cost to minutes.

=

Divide.

= $.15/min

Ratios and RatesLESSON 4-1

Additional Examples

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

The unit rate is $.15 per minute.

Quick Check

4-1

Keneesha

Write the rates

comparing

miles to gallons.

miles

gallons

267 mi

11 gal

miles

gallons

210 mi

9 gal

=

=

Divide.

23.33333333 mi/gal

24.27272727 mi/gal

Round to the

nearest tenth.

24.3 mi/gal

23.3 mi/gal

Ratios and RatesLESSON 4-1

Additional Examples

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.

4-1

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.

Ratios and RatesLESSON 4-1

Additional Examples

(continued)

Quick Check

4-1

15

Ratios and RatesLESSON 4-1

Lesson Quiz

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?

1

3

3.2 mi/h

12-oz bottle

4-1

2

1

8

3

4

7

8

, , ,

Converting UnitsLESSON 4-2

Problem of the Day

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.

4-2

Converting Units

LESSON 4-2

Check Skills You’ll Need

(For help, go to Lesson 2-5.)

1. Vocabulary Review What is the product of a number

and its reciprocal?

Find each product. Write the answer in simplest form.

10

3

1

4

4

6

5

6

- 3.
- 4. 5.

•

•

6

7

8

3

4

9

3

2

•

•

Check Skills You’ll Need

4-2

1

2

4 • 3

9 • 2

4 • 3

9 • 2

2

3

6 • 8

7 • 3

6 • 8

7 • 3

16

7

2

7

=

=

=

=

= 2

1

1

3

Converting UnitsLESSON 4-2

Check Skills You’ll Need

Solutions

1. 1 2.

3. 4.

5.

5

10 • 1

3 • 4

10 • 1

3 • 4

5

6

=

=

2

2

4 • 5

6 • 6

4 • 5

6 • 6

10

18

5

9

=

=

=

3

4-2

1 mi

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

Multiply by a conversion

factor .

0.7 mi

1

5,280 ft

1 mi

0.7 = •

5,280 ft

1 mi

(0.7)(5,280) ft

1

=

Simplify.

Divide.

= 3,696 ft

Converting UnitsLESSON 4-2

Additional Examples

Convert 0.7 mi to ft.

There are 3,696 feet in 0.7 miles.

Quick Check

4-2

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

Multiply by two ratios that

each equal one.

6.84 m

1 s

6.84 m

1 s

1 km

1000 m

60 s

1 min

= • •

Divide by the common units.

(6.48)(1)(60) km

(1)(1,000)(1) min

=

Simplify.

Use a calculator.

= 0.4104

Converting UnitsLESSON 4-2

Additional Examples

Quick Check

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.

Check for Reasonableness The answer 0.4104 km/min is close to the

estimate 0.42. The answer is reasonable.

4-2

divisible by 4.

33 qt 32 qt

32 qt

1

1 gal

4 qt

Multiply by the conversion factor.

= •

Divide by the common units.

32

4

= gallons

Simplify.

Divide.

= 8 gallons

Converting UnitsLESSON 4-2

Additional Examples

Use compatible numbers to estimate the number of gallons in 33 quarts.

1 gal

4 qt

The conversion factor for changing gallons to quarts is .

Quick Check

There are about 8 gallons in 33 quarts.

4-2

factor .

650 g

1

1 oz

28.4 g

650 g =

•

1 oz

28.4 g

(650)(1) oz

22.9 oz

28.4

Simplify. Divide using

=

a calculator.

Converting UnitsLESSON 4-2

Additional Examples

Convert 650 g to ounces.

There are about 22.9 oz in 650 g.

Quick Check

4-2

Converting Units

LESSON 4-2

Lesson Quiz

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

4-2

Solving Proportions

LESSON 4-3

Problem of the Day

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)

4-3

Solving Proportions

LESSON 4-3

Check Skills You’ll Need

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

a + 2

b + 2

1. Vocabulary Review Is the fraction in

simplest form? Explain.

Write each fraction in simplest form.

2. 3. 4. 5.

30

99

42

12

132

602

70

25

Check Skills You’ll Need

4-3

Solving Proportions

LESSON 4-3

Check Skills You’ll Need

Solutions

1. Yes; there is no common factor between the

numerator and denominator.

2.3.

4.5.

1

1

30

99

3 • 10

3 • 33

10

33

42

12

6 • 7

6 • 2

7

2

1

2

=

=

=

=

= 3

1

1

1

1

70

25

5 • 14

5 • 5

14

5

4

5

132

602

2 • 66

2 • 301

66

301

=

=

= 2

=

=

1

1

4-3

18

4

9

Write as a proportion.

gallons

gallons

Use number sense to find a

common multiplier.

8

18

4

9

Since = ,they form a proportion.

Solving ProportionsLESSON 4-3

Additional Examples

8

18

4

9

Do and form a proportion? Explain.

Quick Check

4-3

1

125

p

Write the

proportion .

=

Irish pounds

euros

Write the cross products.

0.7876 • p = 1 • 125

0.7876 • p

0.7876

125

0.7876

Divide each side by 0.7876.

=

Use a calculator.

125 0.7876

Solving ProportionsLESSON 4-3

Additional Examples

The fixed rate of conversion is 1 euro = 0.7876 Irish pounds. How many euros would you receive for 125 Irish pounds?

Let p = the number of euros.

You would receive 158.71 euros.

Quick Check

4-3

2. =

3. =

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

w

12

3

4

4

5

20

r

Solving ProportionsLESSON 4-3

Lesson Quiz

5

8

10

24

1. Is proportional to ? Explain.

No; the fractions are not equal.

9

25

5,000 rupees

4-3

Similar Figures and Proportions

LESSON 4-4

Problem of the Day

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

4-4

Similar Figures and Proportions

LESSON 4-4

Check Skills You’ll Need

(For help, go to Lesson 4-3.)

- Vocabulary Review What are the cross products for

10

15

2

3

= ?

Solve each proportion.

2. = 3. = 4. =

k

50

16

25

21

t

22

10

324

m

7

13

Check Skills You’ll Need

4-4

7t = 273

=

t = 39

10k = 1,100

=

k = 110

7t

7

273

7

10k

10

1,100

10

1

4

Similar Figures and ProportionsLESSON 4-4

Check Skills You’ll Need

Solutions

4.16m = 8,100; m = 506

4-4

First, check to see if corresponding angles are congruent.

AR BSAll right angles are 90°.

CTDU

Similar Figures and ProportionsLESSON 4-4

Additional Examples

Is rectangle ABCD similar to rectangle RSTU? Explain why or why not.

4-4

RS

DA

UR

AB corresponds to RS. DA corresponds to UR.

6

48

3

24

Substitute.

Write the cross products.

6 • 24 48 • 3

Simplify.

144 = 144

Similar Figures and ProportionsLESSON 4-4

Additional Examples

(continued)

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

The corresponding sides are in proportion, so rectangle ABCD

is similar to rectangle RSTU.

Quick Check

4-4

x

2.75 in.

5 in.

Set up a proportion.

=

Write the cross products.

2.75 • x = 5 • 22

2.75 x = 110

Simplify.

2.75x

2.75

110

2.75

Divide each side by 2.75.

=

x = 40

Simplify.

Similar Figures and ProportionsLESSON 4-4

Additional Examples

A stonemason’s sketch of a carving to be made

on a building includes the letter “E” shown below. If the

width of the actual letter in the arrangement is 22 in.,

what is the height?

The height of the letter is 40 inches.

Quick Check

4-4

d

12

21

Write a proportion.

=

Write the cross products.

12 • d = 21 •14

12d = 294

Simplify.

12d

12

294

12

Divide each side by 12.

=

d = 24.5

Simplify.

Similar Figures and ProportionsLESSON 4-4

Additional Examples

RST ~ PSU. Find the value of d.

The value of d is 24.5.

Quick Check

4-4

Similar Figures and Proportions

LESSON 4-4

Lesson Quiz

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

4-4

Similar Figures and Proportions

LESSON 4-4

Lesson Quiz

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.

4-4

Similarity Transformations

LESSON 4-5

Problem of the Day

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

4-5

Similarity Transformations

LESSON 4-5

Check Skills You’ll Need

(For help, go to Lesson 3-4.)

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)

Check Skills You’ll Need

4-5

A C is 3 times AC.

Since A is the

center of dilation

A = A .

A = A

A B C is the image of ABC after

a dilation with a scale factor of 3.

A B is 3 times AB.

ABC ~ A B C

Similarity TransformationsLESSON 4-5

Additional Examples

Quick Check

Find the image of ABC after a dilation with center A and a scale factor of 3.

4-5

Step 1 Multiply the x- and

y-coordinates of each point by .

Step 2 Graph the image.

1

2

1

2

K (–2, –1) K (–1, – )

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

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

N (4, –1) N (2, – )

1

2

Similarity TransformationsLESSON 4-5

Additional Examples

Quick Check

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).

1

2

4-5

4

3

2

PQ

PQ

image

original

= = = 1.5

Similarity TransformationsLESSON 4-5

Additional Examples

The figure below PQR shows the outline of a

playing field. A city planner dilates the design to show

the area available for community youth to play sports.

Find the scale factor. Is it an enlargement or a reduction?

The scale factor is 1.5.

The dilation is an enlargement.

Quick Check

4-5

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

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

1

3

, reduction

Similarity TransformationsLESSON 4-5

Lesson Quiz

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.

1

5

Figure ABCD shows the outline of a porch. The figure

A′B′C′D′ is the outline of a table formed by dilating

ABCD. Find the scale factor. Is it an enlargement

or a reduction?

4-5

Scale Models and Maps

LESSON 4-6

Problem of the Day

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.

4-6

Scale Models and Maps

LESSON 4-6

Check Skills You’ll Need

(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

Check Skills You’ll Need

4-6

Scale Models and Maps

LESSON 4-6

Check Skills You’ll Need

Solutions

1. multiplication 2. 12.8

3. 44.84 4. 39.6

5. 84.5

4-6

2

Let = the actual length of the cellar.

1

2

blueprint measure (in.)

actual measure (ft)

blueprint length (in.)

actual length (ft)

4

=

8

Scale Models and MapsLESSON 4-6

Additional Examples

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?

First, find the actual length of the cellar.

4-6

2

• = 8 • 4

Write the cross

products.

1

2

Simplify.

= 32

1

2

32

Divide each side

by .

=

1

2

1

2

1

2

Simplify.

=

64

Scale Models and MapsLESSON 4-6

Additional Examples

(continued)

4-6

2

blueprint measure (in.)

actual measure (ft)

blueprint length (in.)

actual length (ft)

3

=

8

w

Scale Models and MapsLESSON 4-6

Additional Examples

(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.

4-6

2

• w = 8 • 3

Write the cross

products.

1

2

Simplify.

w = 24

1

2

w

24

Divide each side

by .

=

1

2

1

2

1

2

Simplify.

w =

48

Scale Models and MapsLESSON 4-6

Additional Examples

(continued)

Quick Check

The width of the actual room is 48 ft.

4-6

actual (km)

1

map (cm)

actual (km)

7.5

Set up a proportion.

=

d

50

1 • d = 50 • 7.5

Write the cross products.

d = 375

Simplify.

Scale Models and MapsLESSON 4-6

Additional Examples

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?

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

The actual distance from El Paso, Texas to Chihuahua, Mexico

is 375 kilometers.

Quick Check

4-6

Scale Models and Maps

LESSON 4-6

Lesson Quiz

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

4-6

Scale Models and Maps

LESSON 4-6

Lesson Quiz

For Exercises 3–4, use the diagram.

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

1 in. = 36 ft

4. Find the actual length of the court.

81 ft

4-6

Similarity and Indirect Measurement

LESSON 4-7

Problem of the Day

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

4-7

Similarity and Indirect Measurement

LESSON 4-7

Check Skills You’ll Need

(For help, go to Lesson 4-4.)

1. Vocabulary Review Similar figures have the

same ? but not necessarily the same size.

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

Check Skills You’ll Need

4-7

student’s height

length of flagpole’s shadow

length of student’s shadow

Words

Let h = the flagpole’s height.

=

h

6

51

17

Proportion

=

17h = 6 • 51 Write the cross products.

17h

17

6 • 51

17

=

Divide each side by 17.

h = 18 Simplify.

Similarity and Indirect MeasurementLESSON 4-7

Additional Examples

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.

Quick Check

The height of the flagpole is 18 ft.

4-7

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

ED corresponds to AB.

CD corresponds to CB.

ED

AB

CD

CB

=

d

416

141

312

Substitute.

=

Write the cross products.

312 • d = 416 • 141

Similarity and Indirect MeasurementLESSON 4-7

Additional Examples

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

4-7

312 lengths of corresponding sides.d 58,656

312 312

Divide each side by 312.

=

Use a calculator.

58,656 312 188

Similarity and Indirect MeasurementLESSON 4-7

Additional Examples

(continued)

Simplify.

312d = 58,656

The length d is 188 m.

Quick Check

4-7

Similarity and Indirect Measurement lengths of corresponding sides.

LESSON 4-7

Lesson Quiz

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

4-7

Similarity and Indirect Measurement lengths of corresponding sides.

LESSON 4-7

Lesson Quiz

Use the diagram for Exercise 3. EFG ~ JHG

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

4-7

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