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

Warm Up Write as a decimal and a percent. 1. 2. 0.4; 40% 1.875; 187.5% 3. The distance from Max’s house to the park is 3.5 mi. What is the distance in feet? (1 mi = 5280 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. 14 c = 16 24 p 12.9 A. B. = 88 132 y 77 12 84 15 2.5 C. = D. = x 7

4 = r + 3 9 + m = 15 9 45 5 4

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?

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?

Opener-SAME SHEET-4/20 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

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 ≈ 39.37 in.) . meters strides Ryan’s stride length is approximately 49 inches.

Check It Out! Example 3 Luis ran 400 meters in 297 strides. Find his stride length in inches. 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 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 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