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2-2. Proportional Reasoning. Warm Up. Lesson Presentation. Lesson Quiz. Holt Algebra 2. Warm Up Write as a decimal and a percent. 1. 2. 0.4; 40%. 1.875; 187.5%. A ( –1 , 2). B (0, –3). Warm Up Continued Graph on a coordinate plane. 3. A (–1, 2)

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2-2

Proportional Reasoning

Warm Up

Lesson Presentation

Lesson Quiz

Holt Algebra 2


Warm Up

Write as a decimal and a percent.

1.

2.

0.4; 40%

1.875; 187.5%


A(–1, 2)

B(0, –3)

Warm Up Continued

Graph on a coordinate plane.

3.A(–1, 2)

4.B(0, –3)


Warm Up Continued

5. The distance from Max’s house to the park is 3.5 mi. What is the distance in feet? (1 mi = 5280 ft)

18,480 ft


Objective

Apply proportional relationships to rates, similarity, and scale.


Vocabulary

ratio

proportion

rate

similar

indirect measurement


Recall that a ratio is a comparison of two numbers by division and a proportion is an equation stating that two ratios are equal. In a proportion, the cross products are equal.


If a proportion contains a variable, you can cross multiply to solve for the variable. When you set the cross products equal, you create a linear equation that you can solve by using the skills that you learned in Lesson 2-1.


Reading Math to solve for the variable. When you set the cross products equal, you create a linear equation that you can solve by using the skills that you learned in Lesson 2-1.

In a ÷ b = c ÷ d, b and c are the means, and a and d are the extremes. In a proportion, the product of the means is equal to the product of the extremes.


16 24 to solve for the variable. When you set the cross products equal, you create a linear equation that you can solve by using the skills that you learned in Lesson 2-1.

=

206.4 24p

14c

1624

p 12.9

=

=

=

=

p12.9

88132

2424

88c 1848

Example 1: Solving Proportions

Solve each proportion.

14 c

=

A.

B.

88132

206.4 = 24p

Set cross products equal.

88c =1848

Divide both sides.

88 88

8.6 = p

c = 21


y to solve for the variable. When you set the cross products equal, you create a linear equation that you can solve by using the skills that you learned in Lesson 2-1.77

y 77

152.5

=

=

=

=

=

1284

1284

x7

924 84y

2.5x 105

=

2.5 2.5

8484

Check It Out! Example 1

Solve each proportion.

15 2.5

A.

B.

x7

Set cross products equal.

924 = 84y

2.5x =105

Divide both sides.

11 = y

x = 42


Because percents can be expressed as ratios, you can use the proportion to solve percent problems.

Remember!

Percent is a ratio that means per hundred.

For example:

30% = 0.30 =

30

100


EXAMPLE 1 Finding the Part proportion to solve percent problems.

A. Find 20% of 60.

B. Find 210% of 8.

C. Find 4% of 36.


EXAMPLE 2 Finding the Percent proportion to solve percent problems.

A. What percent of 60 is 15?

B. What percent of 35 is 7?

C. 27 is what percent of 9?


EXAMPLE 3 Finding the Whole proportion to solve percent problems.

A. 40% of what number is 14?

B. 120% of what number is 90?

C. 48 is 15% of what number?


Example 2: Solving Percent Problems proportion to solve percent problems.

A poll taken one day before an election showed that 22.5% of voters planned to vote for a certain candidate. If 1800 voters participated in the poll, how many indicated that they planned to vote for that candidate?

You know the percent and the total number of voters, so you are trying to find the part of the whole (the number of voters who are planning to vote for that candidate).


Example 2 Continued proportion to solve percent problems.

Method 1 Use a proportion.

Method 2 Use a percent equation.

Divide the percent by 100.

Percent (as decimal) whole = part

0.2251800 = x

Cross multiply.

22.5(1800) = 100x

405 = x

Solve for x.

x = 405

So 405 voters are planning to vote for that candidate.


Check It Out! proportion to solve percent problems. Example 2

At Clay High School, 434 students, or 35% of the students, play a sport. How many students does Clay High School have?

You know the percent and the total number of students, so you are trying to find the part of the whole (the number of students that Clay High School has).


Check It Out! proportion to solve percent problems. Example 2 Continued

Method 1 Use a proportion.

Method 2 Use a percent equation.

Divide the percent by 100.

35% = 0.35

Percent (as decimal)whole = part

0.35x = 434

Cross multiply.

100(434) = 35x

x = 1240

Solve for x.

x = 1240

Clay High School has 1240 students.


A proportion to solve percent problems.rate is a ratio that involves two different units. You are familiar with many rates, such as miles per hour (mi/h), words per minute (wpm), or dollars per gallon of gasoline. Rates can be helpful in solving many problems.


Write both ratios in the form . proportion to solve percent problems.

meters

strides

600 m

482 strides

x m

1 stride

=

Example 3: Fitness Application

Ryan ran 600 meters and counted 482 strides. How long is Ryan’s stride in inches? (Hint: 1 m ≈ 39.37 in.)

Use a proportion to find the length of his stride in meters.

600 = 482x

Find the cross products.

x ≈ 1.24 m


is the conversion factor. proportion to solve percent problems.

39.37 in.

1 m

1.24 m

1 stride length

39.37 in.

1 m

49 in.

1 stride length

 ≈

Example 3: Fitness Application continued

Convert the stride length to inches.

Ryan’s stride length is approximately 49 inches.


Write both ratios in the form . proportion to solve percent problems.

meters

strides

400 m

297 strides

x m

1 stride

=

Check It Out! Example 3

Luis ran 400 meters in 297 strides. Find his stride length in inches.

Use a proportion to find the length of his stride in meters.

400 = 297x

Find the cross products.

x ≈1.35 m


is the conversion factor. proportion to solve percent problems.

39.37 in.

1 m

1.35 m

1 stride length

39.37 in.

1 m

53 in.

1 stride length

 ≈

Check It Out! Example 3 Continued

Convert the stride length to inches.

Luis’s stride length is approximately 53 inches.


Reading Math proportion to solve percent problems.

The ratio of the corresponding side lengths of similar figures is often called the scale factor.

Similar figures have the same shape but not necessarily the same size. Two figures are similar if their corresponding angles are congruent and corresponding sides are proportional.


Step 1 proportion to solve percent problems.Graph ∆XYZ. Then draw XB.

Example 4: Scaling Geometric Figures in the Coordinate Plane

∆XYZ has vertices X(0, 0), Y(–6, 9) and Z(0, 9).

∆XAB is similar to∆XYZ with a vertex at B(0, 3).

Graph ∆XYZ and ∆XAB on the same grid.


= proportion to solve percent problems.

=

height of ∆XAB width of ∆XAB

height of ∆XYZ width of ∆XYZ

3x

9 6

Example 4 Continued

Step 2To find the width of ∆XAB, use a proportion.

9x = 18, so x = 2


Z proportion to solve percent problems.

Y

A

B

X

Example 4 Continued

Step 3

To graph ∆XAB, first find the coordinate of A.

The width is 2 units, and the height is 3 units, so the coordinates of A are (–2, 3).


Step 1 proportion to solve percent problems.Graph ∆DEF. Then draw DG.

Check It Out! Example 4

∆DEF has vertices D(0, 0), E(–6, 0) and F(0, –4).

∆DGH is similar to ∆DEF with a vertex at G(–3, 0).

Graph ∆DEF and ∆DGH on the same grid.


3 proportion to solve percent problems.

x

=

=

6

4

width of ∆DGH height of ∆DGH

width of ∆DEF height of ∆DEF

Check It Out! Example 4 Continued

Step 2To find the height of ∆DGH, use a proportion.

6x = 12, so x = 2


G proportion to solve percent problems.(–3, 0)

D(0, 0)

E(–6, 0)

H(0, –2)

F(0,–4)

Check It Out! Example 4 Continued

Step 3

To graph ∆DGH, first find the coordinate of H.

The width is 3 units, and the height is 2 units, so the coordinates of H are (0, –2).


= proportion to solve percent problems.

h ft

9 ft

6 ft

6

22

Shadow of tree

Height of tree

Shadow of house

Height of house

22 ft

=

9

h

Example 5: Nature Application

The tree in front of Luka’s house casts a 6-foot shadow at the same time as the house casts a 22-fot shadow. If the tree is 9 feet tall, how tall is the house?

Sketch the situation. The triangles formed by using the shadows are similar, so Luka can use a proportion to find h the height of the house.

6h = 198

h = 33

The house is 33 feet high.


= proportion to solve percent problems.

h ft

6 ft

20 ft

20

90

Shadow of climber

Height of climber

=

90 ft

6

h

Shadow of tree

Height of tree

Check It Out! Example 5

A 6-foot-tall climber casts a 20-foot long shadow at the same time that a tree casts a 90-foot long shadow. How tall is the tree?

Sketch the situation. The triangles formed by using the shadows are similar, so the climber can use a proportion to find h the height of the tree.

20h = 540

h = 27

The tree is 27 feet high.


Lesson Quiz: Part I proportion to solve percent problems.

  • Solve each proportion.

  • 2.

  • 3. The results of a recent survey showed that 61.5% of those surveyed had a pet. If 738 people had pets, how many were surveyed?

  • 4. Gina earned $68.75 for 5 hours of tutoring. Approximately how much did she earn per minute?

g = 42

k = 8

1200

$0.23


X proportion to solve percent problems.

A

B

Z

Y

Lesson Quiz: Part II

5. ∆XYZ has vertices, X(0, 0), Y(3, –6), and Z(0, –6). ∆XAB is similar to ∆XYZ, with a vertex at B(0, –4). Graph ∆XYZ and ∆XAB on the same grid.


Lesson Quiz: Part III proportion to solve percent problems.

6. A 12-foot flagpole casts a 10 foot-shadow. At the same time, a nearby building casts a 48-foot shadow. How tall is the building?

57.6 ft


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