Section 2.5 Ratios and Proportions

Transcription

1 Section. Ratios and Proportions Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Apply cross-multiplication to Simplifying fractions (R.) to determine equivalent ratios. Solving equations (.1,., &.) Define and identify proportions. Writing a legend (.) Solve proportions. Use proportions in problem solving. INTRODUCTION: RATIOS A ratio between two numbers compares their relative size, using division. This relative size will not change, no matter how large or small the numbers may get. The ratio of two numbers, A and B, is written as either A : B or A B. In words, this is read, A to B. Consider this: Two brothers, Kenny and Jimi decided to go into business together selling lemonade. Kenny, the older brother, contributed $10 to get the business started and Jimi contributed $6. When the profits were counted at the end of the summer, Kenny got $7 and Jimi got $. Is this a fair sharing of the profits To answer this question, use the ratio Kenny : Jimi to look at the relative size of both (1) the investments each made and () the profits each received: (1) Investments: $10 : $6 or () Profits: $7 : $ or If these two ratios are equivalent, then the share of the profits is fair. One way we can determine whether the ratios are equivalent is to simplify each and see if they simplify to the same ratio or fraction: 10 6 simplifies by a factor of to 7 simplifies by a factor of 1 to Because each fraction simplifies to, it is safe to conclude that the sharing of profits was fair.. Ratios and Proportions 1 Robert H. Prior, 01

2 EQUIVALENT RATIOS: CROSS-MULTIPLICATION When two ratios have the same value, such as 10 6 and 7, we call them equivalent ratios: A technique for verifying whether two ratios, a b and c d, are equivalent is called cross-multiplication. Cross-multiplication has the effect of multiplying the denominators to the opposite side numerators, as shown at right: a b a b a d c d c d b c For eample, we can verify that 10 6 and 7 are equivalent using cross-multiplication: Multiply each denominator across the equal sign Evaluate. 0 0 True Because both products are 0, we have verified that the ratios 10 6 Here is why cross-multiplication works: and 7 are equivalent. a b c d a b bd 1 bd 1 c d bd Multiply each side by the common denominator bd, or 1. On the left side, b divides out, and on the right side, d divides out. a 1 1 d 1 b 1 1 c 1 Simplify. a d b c Cross-multiplication! As mentioned, cross-multiplication provides a way to verify whether two ratios are equivalent. After cross-multiplying, if the products are not the same, then the two ratios are not equivalent. Eample 1: Determine whether or not the ratios are equivalent by cross-multiplying b) 1 0. Ratios and Proportions 1 Robert H. Prior, 01

3 Procedure: Multiply each denominator across the equal sign to the numerator on the opposite side. Answer: Cross-multiply. b) 0 1 Cross-multiply Multiply. 0 1 Multiply Yes, 6 and 1 10 are equivalent. No, 0 and 1 are not equivalent. You Try It 1 Determine whether the ratios are equivalent by cross-multiplying. Use Eample 1 as a guide. 6 1 b) c) 10 PROPORTIONS An equation that shows the equivalence of two ratios is called a proportion. Based on the results in Eample 1, is a proportion because 6 1 and 10 are equivalent ratios. However, we cannot set up a proportion between 0 and 1 because they are not equivalent ratios. When a proportion contains a variable either in one or both of the numerators or in one or both of the denominators we can solve for that variable by cross-multiplying. Eamples of such proportions are w 7 1, 0 y 1 1, and Ratios and Proportions 1 Robert H. Prior, 01

4 Caution: If the equation involves more than just two equivalent fractions, then it is not a proportion and we cannot use cross-multiplication to clear the fractions. Here are some eamples of equations with fractions and some eamples of proportions. Equations with fractions To solve each of these, we first clear the fractions by multiplying each side by the LCD. Proportions To solve each proportion, we can use cross-multiplication Eample : Solve for in each proportion. Verify the solution. b) Procedure: Use cross-multiplication to get the equation into a more familiar form. Then solve the resulting equation. We can verify the solution by replacing in the proportion and simplifying each fraction. Answer: Cross-multiply. Simplify each side. Verify 6: Divide each side by. 6 Simplify. 6 Verify 6. b) Simplify each side. Cross-multiply. You finish it: Verify that is the solution Divide each side by Simplify. Verify.. Ratios and Proportions 16 Robert H. Prior, 01

5 Note: In Eample b), we could have simplified the left side fraction first. That way, the proportion is easier to work with: Simplify the left side by a factor of. 10 Cross-multiply. 0 Divide each side by. This is the same solution as in Eample b). You Try It Solve for in each proportion. Use Eample as a guide. 9 7 b) 10 1 c) 0 Sometimes proportions have variables in more than one place, such as + 6. This proportion can still be solved by cross-multiplying. First, though, we must recognize that because the division bar is a grouping symbol the left side numerator is a quantity: ( + 6) Cross-multiply. ( + 6) As we cross-multiply, the parentheses remain until we distribute. ( + 6) The result of cross-multiplying is a linear equation. + 1 Now we solve as we would any linear equation by isolating the variable. This process remains true even if both variable terms are in the denominator.. Ratios and Proportions 17 Robert H. Prior, 01

6 Eample : Solve for the variable in each proportion. Verify the solution. + 6 b) w 1 6 w + Procedure: Use parentheses to group all quantities; then use cross-multiplication to create a linear equation. Solve the linear equation by isolating the variable. Answer: + 6 Show parentheses in the first numerator. ( + 6) Cross-multiply. ( + 6) Distribute on the right side. You finish it: + 1 Add - to each side. Verify that is the solution Divide each side by. 1 Simplify. Verify. b) w 1 6 w + Show parentheses in each denominator. (w 1) 6 (w + ) Cross-multiply. (w + ) (w 1) 6 Distribute on each side. w + 6w 6 Add -6w to each side. You finish it: -6w -6w Verify that w 7 is the solution. -w + -6 Add - to each side w -1 Divide each side by -. -w Simplify. w 7 Verify w 7.. Ratios and Proportions 1 Robert H. Prior, 01

7 You Try It Solve each proportion. Verify the solution. Use Eample as a guide. + b) y y 9 c) w + w To this point, we ve worked only with integer values. However, many times a ratio will include fractional values in the numerator or denominator. For eample, if it takes cup of milk for pancakes, then we can double the recipe to make pancakes by doubling the amount of milk needed:, or 1 1, cups of milk. (For our purposes here, we leave fractions in their improper form.) This can be seen in the proportion with ratios, milk : pancakes. Also, it is possible that a solution is a fraction. If the solution is an improper fraction, we can leave it as a simplified improper fraction.. Ratios and Proportions 19 Robert H. Prior, 01

8 Eample : Solve for the variable in each proportion. y b) 6 Procedure: Use cross-multiplication to get the equation into a linear form. Answer: y Cross-multiply. Verify y : y 1 b) y Divide each side by. (Treat the right side as.) y y 6 Show parentheses in the left side denominator. 1 0 ( 6) Cross-multiply. ( 6) Distribute Add -1 to each side Divide each side by Simplify. 9 Verify 9.. Ratios and Proportions 0 Robert H. Prior, 01

9 Here is the work involved in verifying the solution for Eample b): Verify 9 : The left side denominator simplifies to ( ) To evaluate the right side, think of the fraction as 9. This can be evaluated as Note: In a proportion, when the solution is a fraction, it is often more efficient to verify that the work was done correctly than to verify the solution. You Try It Solve each proportion. Use Eample as a guide. p b) 1 m c) + 0 d) y 6 6y +. Ratios and Proportions 1 Robert H. Prior, 01

10 APPLICATIONS WITH PROPORTIONS: THE PROPORTION TABLE A rate is a ratio in which the two parts, numerator and denominator, have a different unit of measure, 0 miles such as a car s gas mileage, 1 gallon (0 miles per gallon) or the cost of renting a bowling lane, $1 hours ($1 for every two hours of bowling). If a rate is consistent, then we can use a proportion to answer questions about scale, or size. For eample, on a road map you might find that every inch on the map is equivalent to miles of actual road. This could be set up as a rate, 1 inch for every miles: 1 inch miles This is one set of information. We might be curious about a certain length of highway that, on the map, is inches long. We would find that the actual road would be 10 miles long; this is a second set of information. Let s look at the two sets of information as a proportion: first set of information second set of information 1 inch miles inches 10 miles Numerically, we get: 1 10 Cross-multiplying, we can see that these are equivalent ratios: In working with rates (ratios) and proportions, consistency is the key. This means that: 1. the numerator of each fraction should be of the same units of measure (or of the same count);. the denominator of each fraction should be of the same units of measure (or of the same count); and. the smaller units stay together and the larger units stay together. In comparing the inches on the map to the actual miles of the road, we can set up each ratio according to a scheme, such as inches miles. (A scheme is a plan of approach; it gives guidelines on how to approach a problem.). Ratios and Proportions Robert H. Prior, 01

11 We can be sure that we are consistent by setting up a table to outline our scheme, as shown here: This table uses the scheme we ve established, inches miles : we put inches in the numerator, and the corresponding miles in the denominator. Scheme: inches miles smaller units larger units (first set) (second set) 1 10 The table indicates that 1 inch will be paired with miles and inches will be paired with 10 miles, so we can get the proportion directly from the table: 1 inch miles inches 10 miles or just 1 10 In general, to solve a proportion problem: 1. Set up a scheme based on the information given; write a legend identifying the unknown value.. Set up a proportion table based on that scheme.. Set up a proportion based on the proportion table.. Solve the proportion and write a concluding sentence. Eample : trucks are needed to haul away 10 tons of trash in a day. How many trucks are needed to haul away 1 tons of trash in a day Procedure: The first set of information is trucks and 10 tons; the second set is trucks and 1 tons. Let s use a scheme of trucks tons. Set up the legend and the proportion table. Legend: Let the number of trucks needed for 1 tons of trash. Scheme: smaller units larger units trucks tons 10 1 In the table, place a fraction bar between each numerator and denominator, and place an equal sign between the fractions. This is the proportion.. Ratios and Proportions Robert H. Prior, 01

12 Answer: 10 1 Cross-multiply. An optional first step is to simplify the fraction on the left Multiply Divide each side by Simplify. 6 Concluding sentence: 6 trucks are needed to haul away 1 tons of trash. For the following You Try It eercises, use Eample as a guide. Note that the variable, the unknown value, could be in the numerator or the denominator. You Try It Miguel earns $0 for hours of work. How much does he earn for 6 hours of work Legend Sentence: You Try It 6 adults are needed to supervise 1 children on a field trip. How many adults are needed to supervise children on a field trip Legend Sentence:. Ratios and Proportions Robert H. Prior, 01

13 You Try It 7 On a map of Germany, 6 inches represents 0 km. How many inches represent 00 kilometers Legend Sentence: You Try It Tunde s Jeep can travel 10 miles on 9 gallons of gas. How many gallons of gas are required for his Jeep to travel 0 miles Legend Sentence:. Ratios and Proportions Robert H. Prior, 01

14 You Try It Answers You Try It 1: b) 7 7 c) not equivalent equivalent equivalent You Try It : 6 b) 1 c) 1 You Try It : b) y c) w 9 You Try It : p b) m 6 c) d) y 6 You Try It : You Try It 6: You Try It 7: You Try It : Miguel earns $7 for 6 hours of work. 10 adults are needed to supervise children. 1 inches represent 00 kilometers 1 gallons of gas are required for his Jeep to travel 0 miles Section. Eercises Think Again. 1. Can we use cross-multiplication to solve the equation Why or why not. If two ratios are not equivalent, is it still possible to set up a proportion between them Eplain your answer or show an eample that supports your answer. Focus Eercises. Use cross-multiplication to determine whether the pair of ratios is equivalent or not Ratios and Proportions 6 Robert H. Prior, 01

15 Solve each proportion. Write any improper fraction answer as a mied number w v m w y y y 1. p + 7 p 19. y y m 7 m y 1 6 y r r m p y 6 6. n For each of the following, set up a proportion table. Write a legend and solve the corresponding proportion. Write a sentence answering the question. 7. Melissa is traveling in London. She can echange $ (American Dollars) for (British Pounds). How many pounds will she get for $6. At a candy store, the price of 1 inches of licorice is. What is the price for 0 inches of licorice 9. Tom received 1 votes for every 0 people who voted. If a total of 160 people voted, how many votes did Tom receive. Ratios and Proportions 7 Robert H. Prior, 01

16 0. Cheyenne was able to drive 7 miles on 1 gallons of gasoline. At that rate, how many miles will he be able to drive on gallons 1. A flooring company sells flooring material for $1 per square yard. How many square yards of flooring material can be purchased for $70. At a hardware store, the price of feet of chain is $.0. What length of chain will $.0 buy. Delon can ride his bike an average of 1 miles in minutes. At that rate, how many minutes will it take him to ride 0 miles. Sandy s car can travel 1 miles on gallons of gas. How far can her car travel on 1 gallons of gas. Bryan can run an average of 6 miles in minutes. How far can he run in minutes 6. At Charles County Community College, there are 10 math classes for every English classes. If the college offers math classes, how many English classes does it offer 7. At the gym, Chandra s favorite eercise is a workout on the cross-trainer. He has found that he burns 6 calories in a 0-minute workout. How many minutes will Chandra need to work out on the cross-trainer to burn 1,060 calories. At the gym, Luana s favorite eercise is a workout on the elliptical machine. She has found that she burns calories in a 0-minute workout. How many calories will Luana burn in 70 minutes on the elliptical machine Think Outside the Bo: Solve each proportion for Ratios and Proportions Robert H. Prior, 01