1. Holt Algebra 2 2-2 Proportional Reasoning 2-2 Proportional Reasoning Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

2. Holt Algebra 2 2-2 Proportional Reasoning Warm Up Write as a decimal and a percent. 1. 2. 0.4; 40% 1.875; 187.5%

3. 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)

4. 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

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

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

7. 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.

8. 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.

9. 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

10. Holt Algebra 2 2-2 Proportional Reasoning 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 1848 88 c = 21

11. 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 105 2.5 x = 42 Check It Out! Example 1

12. Holt Algebra 2 2-2 Proportional Reasoning Percent is a ratio that means per hundred. For example: 30% = 0.30 = Remember! 30 100 Because percents can be expressed as ratios, you can use the proportion to solve percent problems.

13. Holt Algebra 2 2-2 Proportional Reasoning EXAMPLE 1 Finding the Part A. Find 20% of 60. B. Find 210% of 8. C. Find 4% of 36.

14. Holt Algebra 2 2-2 Proportional Reasoning EXAMPLE 2 Finding the Percent A. What percent of 60 is 15? B. What percent of 35 is 7? C. 27 is what percent of 9?

15. Holt Algebra 2 2-2 Proportional Reasoning EXAMPLE 3 Finding the Whole A. 40% of what number is 14? B. 120% of what number is 90? C. 48 is 15% of what number?

16. Holt Algebra 2 2-2 Proportional Reasoning 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).

17. Holt Algebra 2 2-2 Proportional Reasoning 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. Divide the percent by 100. Percent (as decimal)  whole = part 0.225  1800 = x 405 = x x = 405 22.5(1800) = 100x

18. 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 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

19. 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

20. 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.

21. 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. meters strides

22. 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 m 1 stride length 39.37 in. 1 m 49 in. 1 stride length

23. Holt Algebra 2 2-2 Proportional Reasoning Use a proportion to find the length of his stride in meters. Check It Out! Example 3 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. meters strides

24. Holt Algebra 2 2-2 Proportional Reasoning Convert the stride length to inches. Luis’s stride length is approximately 53 inches. Check It Out! Example 3 Continued is the conversion factor. 39.37 in. 1 m  ≈ 1.35 m 1 stride length 39.37 in. 1 m 53 in. 1 stride length

25. 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

26. 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.

27. Holt Algebra 2 2-2 Proportional Reasoning 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.

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

29. Holt Algebra 2 2-2 Proportional Reasoning ∆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. Check It Out! Example 4 Step 1 Graph ∆DEF. Then draw DG.

30. Holt Algebra 2 2-2 Proportional Reasoning = width of ∆DGH height of ∆DGH width of ∆DEF height of ∆DEF Check It Out! Example 4 Continued Step 2 To find the height of ∆DGH, use a proportion. 6x = 12, so x = 2 = 3 64 x

31. Holt Algebra 2 2-2 Proportional Reasoning The width is 3 units, and the height is 2 units, so the coordinates of H are (0, –2). Check It Out! Example 4 Continued 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.

32. 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 h 22 = 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

33. 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 tree Height 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

34. Holt Algebra 2 2-2 Proportional Reasoning Lesson Quiz: Part I Solve 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

35. 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 A B X Lesson Quiz: Part II

36. 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