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Ratios and Proportions 1 ALGEBRA 7-1 7-1 7-1 Ratios and Proportions 1. Plan Objectives What You’ll Learn Check Skills You’ll Need GO for Help Skills Handbook p. 756 and Lesson 5-1 1 To write ratios and solve • To write ratios and solve Simplify each ratio. proportions proportions 2 1 8 2 6 3 10 1.4 2 2.12 3 3.8 4 4. 10 1 Examples . And Why 1 Real-World Connection Ϻ 2 1 Ϻ 4 5. 20 30 3 6. 8 to 2 4 7. 2 to 8 4 8. 12 9 3 2 Properties of Proportions To find dimensions from a scale drawing, as in Example 4 3 Solving for a Variable 9. Draw a triangle. Then draw its three midsegments to form a smaller 4 Real-World Connection triangle. How do the lengths of the sides of the smaller triangle compare to the lengths of the sides of the larger triangle? k 1 k Each side of the smaller is 2 the length of a side of the larger . New Vocabulary • ratio • proportion • extended proportion Math Background • Cross-Product Property • scale drawing • scale The properties of proportions are variations of applying the Multiplication and Addition Properties of Equality. Thus the 1 Using Ratios and Proportions same units must be within each ratio or in comparable positions Aratio is a comparison of two quantities. You can write the ratio of a to b or a Ϻ b in a proportion. Vocabulary Tip a as the quotient b when b 2 0. Unless otherwise stated, the terms and expressions Ϻ appearing in ratios in this book are assumed to be nonzero. More Math Background: p. 364C You can read a b as the ratio a to b. 1 EXAMPLE Real-World Connection Lesson Planning and 1 Photography A photo that is 8 in. wide and 53 in. high is enlarged to a poster that Resources 1 is 2 ft wide and 1 3 ft high. What is the ratio of the width of the photo to the width See p. 364E for a list of the of the poster? resources that support this lesson. PowerPoint Bell Ringer Practice Check Skills You’ll Need For intervention, direct students to: Simplifying Expressions Skills Handbook, p. 756 Midsegments of Triangles Lesson 5-1: Example 1 Extra Skills, Word Problems, Proof Practice, Ch. 5 width of photo ==8 in. 8 in. ==8 1 width of poster 2 ft 24 in. 24 3 1 The ratio of the width of the photo to the width of the poster is 1 Ϻ 3 or 3. Quick Check 1 What is the ratio of the height of the photo to the height of the poster? 1:3 366 Chapter 7 Similarity Special Needs L1 Below Level L2 Have each student make a scale drawing of their Have students make a list of equivalent fractions and bedroom as shown in Example 4. Students need to test them for equivalence by applying the properties measure their room and use graph paper to make of proportions. their drawings. Each drawing must include a scale. 366 learning style: tactile learning style: verbal A proportion is a statement that two ratios are equal. You can write a proportion Vocabulary Tip in these forms: 2. Teach You can read both a 5 c a c b d = and a Ϻ b = c Ϻ d and a b = c d b d as a is to b as c is to d. When three or more ratios are equal, you can write an extended proportion. For Guided Instruction example, you could write the following: 6 4 1 24 ==16 4 1 EXAMPLE Math Tip Students should understand that Two equations are equivalent when either can be deduced from the other using the units of measurement must be Properties of Equality. Several equations are equivalent to a proportion. Some of the same. them are important enough to be called Properties of Proportions. 2 EXAMPLE Teaching Tip Key Concepts Property Properties of Proportions Replace the variables with numbers to verify that the a = c = b = d b d is equivalent to (1) ad bc (2) a c proportions are equivalent. a b a 1 b c 1 d (3) c = d (4) b = d 4 EXAMPLE means Point out that the scale compares TT inches to feet, so each ratio a c = Multiplying both sides of b = d by bd results in the first a b c d has inches in the numerator property, called the Cross-Product Property. You may state extremes and feet in the denominator. this property as “The product of the extremes is equal to the a = c b d product of the means.” ad = bc PowerPoint Additional Examples 2 EXAMPLE Properties of Proportions x 5 1 A scale model of a car is 4 in. Algebra If = , complete each statement. y 6 long. The actual car is 15 ft long. y 7 x 7 x 1 y 7 What is the ratio of the length of a. 6x = 7 b. x = 7 c. 5 = 7 d. y = 7 the model to the length of the car? y 6 x y x 1 y 11 6x = 5y x ==5 5 6 y =6 1:45 2 Complete: If a = 12, then Quick Check 2 m = n 4 b Critical Thinking Write two proportions that are equivalent to 4 11. b = ? 4 4 ≠ 11 m 1 4 ≠ n 1 11 12 ?. a Answers may vary. Sample: m n , 4 11 3 Solve each proportion. You solve a proportion by finding the value of the variable. 2 = n a. 5 35 14 b. x 1 1 = x 2 3 EXAMPLE Solving for a Variable 3 2 4 1 Two cities are 32 in. apart on a Algebra Solve each proportion. = x 12 y 1 3 y map with the scale 1 in. 50 mi. a. 5 = 7 b. 8 = 4 Find the actual distance. 175 mi 7x = 5(12) Cross-Product Property 4(y + 3) = 8y Resources 7x = 60 4y + 12 = 8y • Daily Notetaking Guide 7-1 L3 60 x = 7 12 = 4y • Daily Notetaking Guide 7-1— Adapted Instruction L1 y = 3 Quick Check 3 Solve each proportion. 5 = 20 18 = 6 a. z 3 0.75 b. n 1 6 n 3 Closure A baseball batting average is the ratio of hits to at-bats, expressed Lesson 7-1 Ratios and Proportions 367 as a decimal. If a player with 540 at-bats has a batting average of 0.350, how many hits did the Advanced Learners L4 English Language Learners ELL player make? 189 hits 2 2 2 Point out that a proportion involves two ratios that Ask: If a = c , does a equal c ? Does a equal a ? b d b2 d2 b b2 are equal. Sometimes the term proportion is used ≠ ≠ incorrectly for a ratio, such as “the proportion of Yes; yes, only if a b or a 0 students who have cell phones is 3 out of 5.” learning style: verbal learning style: verbal 367 In a scale drawing, thescale compares each length in the drawing to the actual 3. Practice length. The lengths used in a scale can be in different units. A scale might be written as 1 in. to 100 mi, 1 in. = 12 ft, or 1 mm i 1 m. You can use proportions to Assignment Guide find the actual dimensions represented in a scale drawing. 1 AB1-55 4 EXAMPLE Real-World Connection C Challenge 56-61 D E B C 1 A D E B C 2 A D E Multiple Choice Use a ruler to measure the length B C 3 A D E B C 4 A D E B C D E O 5 A Test Prep 62-66 B C Test-Taking Tip of the bedroom in the scale drawing. What is the Mixed Review 67-77 length of the actual bedroom? w If you use a ruler and 2 2 your answer does not 10 ft 14 ft 187 ft 245 ft Homework Quick Check match any answer ᐉ / 7 To check students’ understanding choice, first check that An inch ruler shows that 5 8 in. 7 of key skills and concepts, go over you measured correctly. 1 = 8 drawing length (in.) Exercises 16, 22, 26, 34, 44. 16 / actual length (ft) 7 Scale: 1 in. ϭ 16 ft Error Prevention! O = 16 Q8R Cross-Product Property O = 14 Exercise 2 Ask: Why is writing the 6 The actual bedroom is 14 ft long. The correct answer is B. ratio as 185 incorrect? 185 ft first must be converted to inches, or Quick Check 4 6 in. to feet. Find the width of the actual bedroom. 10 ft Auditory Learners Exercises 3–11 Have students explain their reasoning aloud EXERCISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. to the class, citing the Property Practice and Problem Solving of Proportions. A Practice by Example 1. The base of the pyramid at the right is a square whose sides Example 1 measure 0.675 m. The intent was (page 366) for for the sides to measure 675 m. GO Help What is the ratio of the length of a base side in the small pyramid to the length of a base side in the intended pyramid? 1 : 1000 GPS Guided Problem Solving L3 2. Models The Leaning Tower of Enrichment L4 Pisa in Italy is about 185 ft tall. A model of the Leaning Tower Reteaching L2 is 6 in.
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