Proportion

1-1

TOPIC 11

PROPORTION

1-2

• A proportion is an equation that equates two ratios.

• For example, if the ratio is equal to the ratio , then the following proportion can be written:

means extremes

a and d are the extremes of the proportion and b and c are the means of the proportion.

What is a Proportion?

d

c

b

a

b

a

d

c

1-3

• 1. Cross Product Property: The product of the extremes equals the product of the means.

• If , then ad = bc.

• 2. Reciprocal Property: If two ratios are equal, then their reciprocals are also equal.

• If , then

Properties of Proportions

d

c

b

a

d

c

b

a

c

d

a

b

1-4

• a)

Use the reciprocal property to bring the variable to the top.

Multiply each side by 6.

Simplify

• b)

Use the cross product property.

Use the Distributive Property to simplify the expression.

Subtract 4s from both sides and add 50 to both sides.

Divide by 6 and simplify.

Example

x

6

14

9

104

5 ss

69

14 x

x

9

146

x3

28

4510 ss

ss 45010

506 s

3

25

6

50s

1-5

1-6

1-7

SOLVE THE PROPORTIONS BELOW

1.

2.

3.

4.

5

3

21

n

3

712

n

5

13

15

n

10

n 16

2

105

3

3

3

n

3

3

n 84

3

5

5

n 195

5

2

2

n 160

2

35 n

28n

39n

80n

1-8

Name Correct and Incorrect Proportion for the Illustration Below:

1-9

1-10

Solve each proportion.

b30

39

=1.5635

y5

=2.

412

p9

=3.56m

2826

=4.

b = 10 y = 8

p = 3 m = 52

5. Jody attends a class with a ratio of men to women at 5 to 2. If there are 20 men, how many women are in

the class?

6. A recipe calls for 2 cups of sugar and 5 cups of flour. How much flour is needed if 5 1/2 cups of sugar are

used?7. On a map one inch represents 150 miles. A city is 3 inches from home on the map. How far away is the

city?

1-11

TOPIC 12SCALE

DRAWING

1-12

A scale drawing is a two-dimensional drawing of an object that is proportional to the object.

A scale gives the ratio of the dimensions in the drawing to the dimensions of the object. All dimensions are reduced or enlarged using the same scale. Scales can use the same units or different units.

A scale model is a three-dimensional model that is proportional to the object.

1-13

Under a 1000:1 microscope view, an amoeba appears to have a length of 8 mm. What is its actual length?

Additional Example 1: Finding Actual Measurements

Write a proportion using the scale. Let x be the actual length of the amoeba.

1000 x = 1 8 The cross products are equal.

x = 0.008

The actual length of the amoeba is 0.008 mm.

1000 1 = 8 mm

x mm

Solve the proportion.

1-14

Under a 10,000:1 microscope view, a fiber appears to have length of 1 mm. What is its actual length?

Check It Out! Example 1

Write a proportion using the scale. Let x be the actual length of the fiber.

10,000 x = 1 1 The cross products are equal.

x = 0.0001

The actual length of the fiber is 0.0001 mm.

10,000 1 = 1 mm

x mm

Solve the proportion.

1-15

A. The length of an object on a scale drawing is 2 cm, and its actual length is 8 m. The scale is 1 cm: __ m. What is the scale?

Additional Example 2: Using Proportions to Find Unknown Scales

1 cmx m = 2 cm

8 m

1 8 = x 2 Find the cross products.

8 = 2x

Divide both sides by 2.

The scale is 1 cm:4 m.

4 = x

Set up proportion using scale length .actual length

1-16

The scale a:b is read “a to b.” For example, the scale 1 cm:4 m is read “one centimeter to four meters.”

Reading Math

1-17

The length of an object on a scale drawing is 4 cm, and its actual length is 12 m. The scale is 1 cm: __ m. What is the scale?

Check It Out! Example 2

1 cmx m = 4 cm

12 m Set up proportion using scale length .actual length

1 12 = x 4 Find the cross products.

12 = 4x

Divide both sides by 4.

The scale is 1 cm:3 m.

3 = x

1-18

The ratio of a length on a scale drawing or model to the corresponding length on the actual object is called the scale factor.

When finding a scale factor, you must use the same measurement units. You can use a scale factor to find unknown dimensions.

1-19

A model of a 27 ft tall house was made using a scale of 2 in.:3 ft. What is the height of the model?

Additional Example 3: Using Scale Factors to Find Unknown Dimensions

Find the scale factor.

The scale factor for the model is . Now set up a proportion. 118

2 in.3 ft

= 2 in.36 in.

= 1 in.18 in.

= 118

324 = 18h

Convert: 27 ft = 324 in.

Find the cross products.

18 = h

The height of the model is 18 in.Divide both sides by 18.

118

= h in.324 in.

1-20

A model of a 24 ft tall bridge was made using a scale of 4 in.:2 ft. What is the height of the model?

Check It Out! Example 3

Find the scale factor.

The scale factor for the model is . Now set up a proportion. 16

4 in.2 ft

= 4 in.24 in.

= 1 in.6 in.

= 16

288 = 6h

Convert: 24 ft = 288 in.

Find the cross products.

48 = h

The height of the model is 48 in.Divide both sides by 6.

16

= h in.288 in.

1-21

A DNA model was built using the scale 5 cm: 0.0000001 mm. If the model of the DNA chain is 20 cm long, what is the length of the actual chain?

Additional Example 4: Life Science Application

The scale factor for the model is 500,000,000. This means the model is 500 million times larger than the actual chain.

5 cm 0.0000001 mm

50 mm 0.0000001 mm= = 500,000,000

Find the scale factor.

1-22

Additional Example 4 Continued

500,000,000 1

20 cm x cm= Set up a proportion.

500,000,000x = 1(20)

x = 0.00000004

The length of the DNA chain is 4 10-8 cm.

Find the cross products.

Divide both sides by 500,000,000.

1-23

A model was built using the scale 2 cm:0.01 mm. If the model is 30 cm long, what is the length of the actual object?

Check It Out! Example 4

The scale factor for the model is 2,000. This means the actual object is two thousand times larger than the model.

2 cm 0.01 mm

20 mm 0.01 mm= = 2,000

Find the scale factor.

1-24

Check It Out! Example 4 Continued

2,000 1

30 cm x cm= Set up a proportion.

2,000x = 1(30)

x = 0.015

The length of the actual object is 1.5 10-2 cm.

Find the cross products.

Divide both sides by 2,000.

1-25

MORE APPLICATIONS OF SCALE DRAWING

1-26

1-27

1-28

1-29

1-30

1-31

1. Using a in. = 1 ft scale, how long would a

drawing of a 22 ft car be?

2. What is the scale of a drawing in which a 9 ft and the wall is

6 cm long?

3. The height of a person on a scale drawing is 4.5 in. The

scale is 1:16. What is the actual height of the person?

Lesson Quiz

5.5 in.

1 cm = 1.5 ft

72 in.

14

1-32