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

2 -2 Proportional Reasoning * Warm Up Write as a decimal and a percent. 1. 2. Holt Algebra 2

2 -2 Proportional Reasoning * Warm Up Continued Graph on a coordinate plane. 3. A(– 1, 2) 4. 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) 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 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 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 Holt Algebra 2 B. 15 2. 5 = x 7

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 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). Holt Algebra 2

2 -2 Proportional Reasoning * Check It Out! Example 2 Continued Method 1 Use a proportion. Method 2 Use a percent equation. 35% = 0. 35 Holt Algebra 2 Divide the percent by 100.

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 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 m xm = 482 strides 1 stride 600 = 482 x x ≈ 1. 24 m Holt Algebra 2 meters Write both ratios in the form strides. Find the cross products.

2 -2 Proportional Reasoning Example 3: Fitness Application continued Convert the stride length to inches. 39. 37 in. 1 m 1. 24 m 1 stride length is the conversion factor. 39. 37 in. ≈ 49 in. 1 m 1 stride length Ryan’s stride length is approximately 49 inches. Holt Algebra 2

2 -2 Proportional Reasoning * 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. Holt Algebra 2

2 -2 Proportional Reasoning * Check It Out! Example 3 Continued Convert the stride length to inches. 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 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. 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, and the height is 3 units, so the coordinates of A are (– 2, 3). Holt Algebra 2 Z Y A B X

2 -2 Proportional Reasoning 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. Holt Algebra 2

2 -2 Proportional Reasoning * Check It Out! Example 4 Continued Step 2 To find the height of ∆DGH, use a proportion. width of ∆DGH width of ∆DEF Holt Algebra 2 = height of ∆DGH height of ∆DEF

2 -2 Proportional Reasoning * 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). 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 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 * 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. Shadow of climber Shadow of tree = Height of climber Height of tree Holt Algebra 2