Proportional Reasoning

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Proportional Reasoning
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%

Graph on a coordinate plane.
Warm Up Continued Graph on a coordinate plane. 3. A(–1, 2) 4. B(0, –3) A(–1, 2) B(0, –3)

Warm Up Continued 5. The distance from Max’s house to the park is 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 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.

Example 1: Solving Proportions
Solve each proportion. c = p A. B. = p c = = 206.4 = 24p Set cross products equal. 88c = 1848 p = 88c = Divide both sides. 8.6 = p c = 21

Check It Out! Example 1 Solve each proportion. y A. = B. = x y x = = Set cross products equal. 924 = 84y 2.5x =105 y = = 2.5x Divide both sides. 11 = y x = 42

Because percents can be expressed as ratios, you can use the proportion to solve percent problems.
Percent is a ratio that means per hundred. For example: 30% = 0.30 = Remember! 30 100

Example 2: Solving 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).

Method 1 Use a proportion. Method 2 Use a percent equation.
Example 2 Continued 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! 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! 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 rate is a ratio that involves two different units
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.

Example 3: Fitness Application
Ryan ran 600 meters and counted 482 strides. How long is Ryan’s stride in inches? (Hint: 1 m ≈ in.) Use a proportion to find the length of his stride in meters. 600 m 482 strides x m 1 stride = Write both ratios in the form meters strides 600 = 482x Find the cross products. x ≈ 1.24 m

Example 3: Fitness Application continued
Convert the stride length to inches. is the conversion factor. 39.37 in. 1 m  ≈ 1.24 m 1 stride length 39.37 in. 1 m 49 in. Ryan’s stride length is approximately 49 inches.

Luis ran 400 meters in 297 strides. Find his stride length in inches.
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 m 297 strides x m 1 stride = Write both ratios in the form meters strides 400 = 297x Find the cross products. x ≈ 1.35 m

Check It Out! Example 3 Continued
Convert the stride length to inches. is the conversion factor. 39.37 in. 1 m  ≈ 1.35 m 1 stride length 39.37 in. 1 m 53 in. Luis’s stride length is approximately 53 inches.

Similar figures have the same shape but not necessarily the same size
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

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.

Example 4 Continued Step 2 To find the width of ∆XAB, use a proportion. = height of ∆XAB width of ∆XAB height of ∆XYZ width of ∆XYZ = x 9x = 18, so x = 2

To graph ∆XAB, first find the coordinate of A.
Example 4 Continued Step 3 To graph ∆XAB, first find the coordinate of A. Y Z The width is 2 units, and the height is 3 units, so the coordinates of A are (–2, 3). B A X

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. Step 1 Graph ∆DEF. Then draw DG.

Check It Out! Example 4 Continued
Step 2 To find the height of ∆DGH, use a proportion. = width of ∆DGH height of ∆DGH width of ∆DEF height of ∆DEF = 3 6 4 x 6x = 12, so x = 2

Check It Out! Example 4 Continued
Step 3 To graph ∆DGH, first find the coordinate of H. E(–6, 0) F(0,–4) G(–3, 0) D(0, 0) H(0, –2) The width is 3 units, and the height is 2 units, so the coordinates of H are (0, –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. 9 ft 6 ft = 6 9 h 22 = Shadow of tree Height of tree Shadow of house Height of house h ft 22 ft 6h = 198 h = 33 The house is 33 feet high.

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. 6 ft 20 ft = 20 6 h 90 = Shadow of climber Height of climber Shadow of tree Height of tree h ft 90 ft 20h = 540 h = 27 The tree is 27 feet high.

Lesson Quiz: Part I 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

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. Y Z A B X

Lesson Quiz: Part III 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