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 MathIn 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 Example 1: Solving Proportions Solve each proportion.c=pA.B.=pc==206.4 = 24pSet cross products equal.88c = 1848p=88c=Divide both sides.8.6 = pc = 21
11 Check It Out! Example 1Solve each proportion.yA.=B.=xyx==Set cross products equal.924 = 84y2.5x =105y==2.5xDivide both sides.11 = yx = 42
12 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!30100
13 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).
14 Method 1 Use a proportion. Method 2 Use a percent equation. Example 2 ContinuedMethod 1 Use a proportion.Method 2 Use a percent equation.Divide the percent by 100.Percent (as decimal) whole = part0.225 1800 = xCross multiply.22.5(1800) = 100x405 = xSolve for x.x = 405So 405 voters are planning to vote for that candidate.
15 Check It Out! Example 2At 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).
16 Check It Out! Example 2 Continued Method 1 Use a proportion.Method 2 Use a percent equation.Divide the percent by 100.35% = 0.35Percent (as decimal) whole = part0.35x = 434Cross multiply.100(434) = 35xx = 1240Solve for x.x = 1240Clay High School has 1240 students.
17 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.
18 Example 3: Fitness Application Ryan ran 600 meters and counted 482 strides. How long is Ryan’s stride in inches? (Hint: 1 m ≈ in.)Use a proportion to find the length of his stride in meters.600 m482 stridesx m1 stride=Write both ratios in the formmetersstrides600 = 482xFind the cross products.x ≈ 1.24 m
19 Example 3: Fitness Application continued Convert the stride length to inches.is the conversion factor.39.37 in.1 m ≈1.24 m1 stride length39.37 in.1 m49 in.Ryan’s stride length is approximately 49 inches.
20 Luis ran 400 meters in 297 strides. Find his stride length in inches. Check It Out! Example 3Luis 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 m297 stridesx m1 stride=Write both ratios in the formmetersstrides400 = 297xFind the cross products.x ≈ 1.35 m
21 Check It Out! Example 3 Continued Convert the stride length to inches.is the conversion factor.39.37 in.1 m ≈1.35 m1 stride length39.37 in.1 m53 in.Luis’s stride length is approximately 53 inches.
22 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
23 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.
24 Example 4 ContinuedStep 2 To find the width of ∆XAB, use a proportion.=height of ∆XAB width of ∆XABheight of ∆XYZ width of ∆XYZ=x9x = 18, so x = 2
25 To graph ∆XAB, first find the coordinate of A. Example 4 ContinuedStep 3To graph ∆XAB, first find the coordinate of A.YZThe width is 2 units, and the height is 3 units, so the coordinates of A are (–2, 3).BAX
26 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.
27 Check It Out! Example 4 Continued Step 2 To find the height of ∆DGH, use a proportion.=width of ∆DGH height of ∆DGHwidth of ∆DEF height of ∆DEF=364x6x = 12, so x = 2
28 Check It Out! Example 4 Continued Step 3To graph ∆DGH, first find the coordinate of H.●E(–6, 0)F(0,–4)G(–3, 0)D(0, 0)H(0, –2)●The width is 3 units, and the height is 2 units, so the coordinates of H are (0, –2).
29 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 ft6 ft=69h22=Shadow of treeHeight of treeShadow of houseHeight of househ ft22 ft6h = 198h = 33The house is 33 feet high.
30 Check It Out! Example 5A 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 ft20 ft=206h90=Shadow of climberHeight of climberShadow of treeHeight of treeh ft90 ft20h = 540h = 27The tree is 27 feet high.
31 Lesson Quiz: Part ISolve 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 = 42k = 81200$0.23
32 Lesson Quiz: Part II5. ∆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.YZABX
33 Lesson Quiz: Part III6. 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