2 2 Proportional Reasoning Warm Up Lesson Presentation

2 -2 Proportional Reasoning Warm Up Lesson Presentation Lesson Quiz Holt Algebra 22

2 -2 Proportional Reasoning Warm Up Write as a decimal and a percent. 1. 0. 4; 40% 2. 1. 875; 187. 5% Holt Algebra 2

2 -2 Proportional Reasoning 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

2 -2 Proportional Reasoning 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

2 -2 Proportional Reasoning Objective Apply proportional relationships to rates, similarity, and scale. Holt Algebra 2

2 -2 Proportional Reasoning Vocabulary ratio proportion rate similar indirect measurement Holt Algebra 2

2 -2 Proportional Reasoning 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

2 -2 Proportional Reasoning 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. Holt Algebra 2

2 -2 Proportional Reasoning Example 1: Solving Proportions Solve each proportion. A. 16 = 24 p 12. 9 B. 16 24 = p 12. 9 206. 4 = 24 p 206. 4 24 p = 24 24 8. 6 = p Holt Algebra 2 14 c = 88 132 Set cross products equal. Divide both sides. 88 c = 1848 88 88 c = 21

2 -2 Proportional Reasoning Check It Out! Example 1 Solve each proportion. A. y 77 = 12 84 B. y 77 = 12 84 924 = 84 y 924 84 y = 84 84 11 = y Holt Algebra 2 15 2. 5 = x 7 Set cross products equal. Divide both sides. 2. 5 x =105 2. 5 x = 2. 5 105 2. 5 x = 42

2 -2 Proportional Reasoning 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 Holt Algebra 2

2 -2 Proportional Reasoning 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). Holt Algebra 2

2 -2 Proportional Reasoning 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 22. 5(1800) = 100 x Cross multiply. 405 = x Solve for x. x = 405 So 405 voters are planning to vote for that candidate. 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 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. Reading Math The ratio of the corresponding side lengths of similar figures is often called the scale factor. Holt Algebra 2

2 -2 Proportional Reasoning Example 3: 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. Holt Algebra 2

2 -2 Proportional Reasoning Example 3 Continued Step 1 graph ∆XYZ and vertex B. Z Y A B X Holt Algebra 2

2 -2 Proportional Reasoning 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 3 9 = x 6 9 x = 18, so x = 2 Holt Algebra 2

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

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 9 ft using the shadows are similar, so Luka can use a proportion to find h the height of the house. 6 ft 6 22 = 9 h 6 h = 198 Shadow of tree Shadow of house = Height of tree Height of house h = 33 The house is 33 feet high. Holt Algebra 2 h ft 22 ft

2 -2 Proportional Reasoning Lesson Quiz: Part I Solve each proportion. 1. k=8 2. g = 42 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? 1200 3. 4. Gina earned $68. 75 for 5 hours of tutoring. Approximately how much did she earn per minute? $0. 23 Holt Algebra 2

2 -2 Proportional Reasoning 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. X B Z Holt Algebra 2 A Y

2 -2 Proportional Reasoning 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 Holt Algebra 2