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Chapter 2 Section 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.6 1 2 3 Ratios and Proportions Write ratios. Solve proportions. Solve applied problems by using proportions. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Write ratios. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2.6 - 3 Write ratios. A ratio is a comparison of two quantities using a quotient. The ratio of the number a to the number b (b ≠ 0) is written atob, a : b, or a . b The last way of writing a ratio is most common in algebra. Percents are ratios in which the second number is always 100. For example, 50% represents the ratio 50 to 100, 27% represents the ratio 27 to 100, and so on. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2.6 - 4 EXAMPLE 1 Writing Word Phrases as Ratios Write a ratio for each word phrase. 3 days to 2 weeks Solution: 2weeks 7days 14days 3days 3days 14days weeks 3 14 12 hr to 4 days Solution: 4days 24hours 96hours 12hours 1 hours 96hours 8 4days Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2.6 - 5 EXAMPLE 2 Finding Price per Unit The local supermarket charges the following prices for a popular brand of pancake syrup. Which size is the best buy? What is the unit cost for that size? Solution: The 36 oz. size is the best buy. The unit price is $0.108 per oz. $3.89 $0.108 36 $2.79 $0.116 24 $1.89 $0.158 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2.6 - 6 Objective 2 Solve proportions. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2.6 - 7 Solve proportions. A ratio is used to compare two numbers or amounts. A proportion says that two ratios are equal, so it is a special type of equation. For example, 3 15 4 20 is a proportion which says that the ratios In the proportion 3 4 and 15 20 are equal. a c b, d 0 , b d a, b, c, and d are the terms of the proportion. The terms a and d are called the extremes, and the terms b and c are called the a c means. We read the proportions b d as “a is to b as c is to d.” Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2.6 - 8 Solve proportions. (cont’d) Beginning with this proportion and multiplying each side by the common denominator, bd, gives a c b d a c bd bd b d ad bc. We can also find the products ad and bc by multiplying diagonally. bc a c b d ad For this reason, ad and bc are called cross products. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2.6 - 9 Solve proportions. (cont’d) If a c , b d then the cross products ad and bc are equal. a c b, d 0 . b d a c From this rule, if then ad = bc; that is, the product of b d Also, if ad bc, then the extremes equals the product of the means. a b If , then ad = cb, or ad = bc. This means that the two c d proportions are equivalent, and the proportion a c can b d a b also be written as c, d 0 . c d Sometimes one form is more convenient to work with than the other. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2.6 - 10 EXAMPLE 3 Deciding whether Proportions Are True Decide whether the proportion is true or false. Solution: False 15 62 930 21 62 15 45 13 91 17 119 21 45 945 17 91 1547 Solution: True 13119 1547 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2.6 - 11 EXAMPLE 4 Finding an Unknown in a Proportion x 35 Solve the proportion . 6 42 Solution: x 42 6 35 42 x 210 42 42 x 5 The solution set is {5}. The cross-product method cannot be used directly if there is more than one term on either side of the equals symbol. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2.6 - 12 EXAMPLE 5 Solving an Equation by Using Cross Products a6 2 . Solve 2 5 Solution: a 6 5 2 2 5a 30 30 4 30 5a 26 5 5 26 a 5 The solution set is 26 . 5 When you set cross products equal to each other, you are really multiplying each ratio in the proportion by a common denominator. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2.6 - 13 Objective 3 Solve applied problems with proportions. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2.6 - 14 EXAMPLE 6 Applying Proportions Twelve gal of diesel fuel costs $37.68. How much would 16.5 gal of the same fuel cost? Solution: Let x = the price of 16.5 gal of fuel. $37.68 x 12 gal 16.5 gal 12 x 621.72 12 12 x 51.81 16.5 gal of diesel fuel costs $51.81. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2.6 - 15