Holt Algebra 2 2-2 Proportional Reasoning 2-2 Proportional Reasoning Holt Algebra 2 Warm Up Warm Up...

Holt Algebra 2

2-2 Proportional Reasoning 2-2 Proportional Reasoning

Holt Algebra 2

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Algebra 2

2-2 Proportional Reasoning

Warm UpWrite as a decimal and a percent.

1.

2.

0.4; 40%

1.875; 187.5%

Holt Algebra 2


Warm Up Continued

Graph on a coordinate plane.

3. A(–1, 2)

4. B(0, –3) A(–1, 2)

B(0, –3)

Holt Algebra 2


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

Holt Algebra 2


Apply proportional relationships to rates, similarity, and scale.

Objective

Holt Algebra 2


ratioproportionratesimilarindirect measurement

Vocabulary

Holt Algebra 2


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.

Holt Algebra 2


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.

Holt Algebra 2


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.

Reading Math

Holt Algebra 2


Solve each proportion.

Example 1: Solving Proportions

A.

206.4 = 24p Set cross products equal.

=

=16 24 p 12.9

16 24 p 12.9

206.4 24p 24 24 Divide both sides.

8.6 = p

14 c 88 132

=

=

=

B.

14 c 88 132

88c = 1848

=88c 184888 88

c = 21

Holt Algebra 2


Solve each proportion.

A.

924 = 84y Set cross products equal.

=

=

y 77 12 84

y 77 12 84

Divide both sides.

11 = y

15 2.5 x 7

=

924 84y 84 84

=

=

B.

15 2.5 x 7

2.5x =105

=2.5x 1052.5 2.5

x = 42

Check It Out! Example 1

Holt Algebra 2


Percent is a ratio that means per hundred.

For example:

30% = 0.30 =

Remember!

30100

Because percents can be expressed as ratios, you can use the proportion

to solve percent problems.

Holt Algebra 2


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?

Example 2: Solving Percent Problems

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).

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Example 2 Continued

Method 1 Use a proportion.

Cross multiply.

Solve for x.

So 405 voters are planning to vote for that candidate.

Method 2 Use a percent equation.22.5% 0.225 Divide the percent

by 100.

Percent (as decimal) whole = part

0.225 1800 = x

405 = x

x = 405

22.5(1800) = 100x

Holt Algebra 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).


Holt Algebra 2


Check It Out! Example 2 Continued

Method 1 Use a proportion.

Cross multiply.

Solve for x.

Clay High School has 1240 students.

Method 2 Use a percent equation.

Divide the percent by 100.

0.35x = 434

35% = 0.35

x = 1240

Percent (as decimal) whole = part

x = 1240

100(434) = 35x

Holt Algebra 2


A 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.

Holt Algebra 2


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

Example 3: Fitness Application

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

Find the cross products.

600 m 482 strides

x m 1 stride=

600 = 482x

x ≈ 1.24 m

Write both ratios in the form .

metersstrides

Holt Algebra 2


Example 3: Fitness Application continued

Convert the stride length to inches.

Ryan’s stride length is approximately 49 inches.

is the conversion factor. 39.37 in.1 m

≈ 1.24 m1 stride length

39.37 in.1 m

49 in.1 stride length

Holt Algebra 2


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


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

x ≈ 1.35 m

400 = 297x Find the cross products.

400 m 297 strides

x m 1 stride=

Write both ratios in the form .

metersstrides

Holt Algebra 2


Convert the stride length to inches.

Luis’s stride length is approximately 53 inches.


is the conversion factor. 39.37 in.1 m

≈ 1.35 m1 stride length

39.37 in.1 m

53 in.1 stride length

Holt Algebra 2


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.

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

Reading Math

Holt Algebra 2


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.

Step 1 Graph ∆XYZ. Then draw XB.

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Example 4 Continued

= height of ∆XAB width of ∆XAB

height of ∆XYZ width of ∆XYZ

=3 x

9 6

9x = 18, so x = 2

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

Holt Algebra 2


Example 4 Continued

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

BA

X

YZ

Step 3 To graph ∆XAB, first find the coordinate of A.

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∆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.


Step 1 Graph ∆DEF. Then draw DG.

Holt Algebra 2


= width of ∆DGH height of ∆DGH

width of ∆DEF height of ∆DEF


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

6x = 12, so x = 2

=36 4

x

Holt Algebra 2


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


G(–3, 0)D(0, 0)

H(0, –2)

● ●

●

●●

●

E(–6, 0)

F(0,–4)

Step 3

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

Holt Algebra 2


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.

=6

9 h22

=Shadow of tree Height of tree

Shadow of house Height of house

6h = 198

h = 33

The house is 33 feet high.

9 ft

6 ft

h ft

22 ft

Holt Algebra 2


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.

= 20 6 h

90=

Shadow of climber Height of climber

Shadow of treeHeight of tree

20h = 540

h = 27

The tree is 27 feet high.

6 ft

20 ft

h ft

90 ft


Holt Algebra 2


Lesson Quiz: Part ISolve each proportion.

1. 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?

k = 8 g = 42

$0.23

1200

Holt Algebra 2


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.

YZ

AB

X

Lesson Quiz: Part II

Holt Algebra 2


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

Lesson Quiz: Part III

Holt Algebra 2 2-2 Proportional Reasoning 2-2 Proportional Reasoning Holt Algebra 2 Warm Up Warm Up...

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