Holt Algebra 2

2-2 Proportional Reasoning

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

Holt Algebra 2

2-2 Proportional Reasoning

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

2-2 Proportional Reasoning

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

2-2 Proportional Reasoning

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

2-2 Proportional Reasoning

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

2-2 Proportional Reasoning

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 number of students that play a sport, so you are trying to find the number of students that Clay High School has.

Check It Out! Example 2

Holt Algebra 2

2-2 Proportional Reasoning

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

2-2 Proportional Reasoning

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

2-2 Proportional Reasoning

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

2-2 Proportional Reasoning

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

2-2 Proportional Reasoning

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

2-2 Proportional Reasoning

Example 5: Nature Application

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?

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

=10

12 h48

=Shadow of pole Height of pole

Shadow of building Height of building

10h = 576

h = 57.6

The building is 57.6 feet high.

12 ft

10 ft

h ft

48 ft

Holt Algebra 2

2-2 Proportional Reasoning

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

2-2 Proportional Reasoning

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

Check It Out! Example 5

Holt Algebra 2

2-2 Proportional Reasoning

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

2-2 Proportional Reasoning

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

2-2 Proportional Reasoning

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