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

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

  2. Warm Up Write as a decimal and a percent. 1. 2. 0.4; 40% 1.875; 187.5%

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

  4. 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. Objective Apply proportional relationships to rates, similarity, and scale.

  6. Vocabulary ratio proportion rate similar indirect measurement

  7. 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. 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. 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.

  10. y77 y 77 152.5 = = = = = 1284 1284 x7 924 84y 2.5x 105 = 2.5 2.5 8484 Check It Out! Example 1 Solve each proportion. 15 2.5 A. B. x7 Set cross products equal. 924 = 84y 2.5x =105 Divide both sides. 11 = y x = 42

  11. 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

  12. 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).

  13. 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.

  14. 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.

  15. Write both ratios in the form . meters strides 400 m 297 strides x m 1 stride = 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 = 297x Find the cross products. x ≈1.35 m

  16. 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  ≈ Check It Out! Example 3 Continued Convert the stride length to inches. Luis’s stride length is approximately 53 inches.

  17. Reading Math The ratio of the corresponding side lengths of similar figures is often called the scale factor. 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.

  18. Step 1 Graph ∆DEF. Then draw DG. 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.

  19. 3 x = = 6 4 width of ∆DGH height of ∆DGH width of ∆DEF height of ∆DEF Check It Out! Example 4 Continued Step 2To find the height of ∆DGH, use a proportion. 6x = 12, so x = 2

  20. G(–3, 0) ● ● ● ● D(0, 0) E(–6, 0) ● H(0, –2) ● F(0,–4) 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).

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

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