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DA2SE_763_02.qxd 10/17/2005 18:53 LESSON 2.1 Page 96 Proportions When you say, “I got 21 out of 24 questions correct on the last quiz,” you are comparing two numbers. The ratio of your correct questions to the total number of questions is 21 to 24. You can write the ratio as 21:24, or as a fraction, 2214 , or as a decimal, 0.875. The fraction bar means division, so these expressions are equivalent: Mathematics is not a way of hanging numbers on things so that quantitative answers to ordinary questions can be obtained. It is a language that allows one to think of extraordinary questions. 21 24 21 24 0.875 JAMES BULLOCK EXAMPLE A 7 8 Write the ratio 210 : 330 in several ways. 7 210 or 11 330 Solution 210 330 or 7 11 To change a common fraction into a decimal fraction, divide the numerator by the denominator. When you see the same difference that you’ve seen before, you know the decimal repeats. .636363 33021 0. 00 00 00 198 0 12 00 9 90 2 100 1 980 1200 990 ... When you divide 21 by 24, the decimal form of the quotient ends, or terminates. The ratio 2214 equals 0.875 exactly. But when you divide 210 by 330 or 7 by 11, you see a repeating decimal pattern, 0.636363 . . . . You can use a bar over the numerals that repeat to show a repeating decimal pattern, 7 0.6 3. [ See Calculator Note 0A for more 11 about converting fractions to decimals. ] 96 CHAPTER 2 Proportional Reasoning and Variation DA2SE_763_02.qxd 10/17/2005 18:53 Page 97 A proportion is an equation stating that two ratios are equal. For example, 32 182 is a proportion. You can use the numbers 2, 3, 8, and 12 to write these true proportions: 12 8 8 3 2 3 12 2 3 2 12 12 3 2 8 8 Do you agree that these are all true equations? One way to check that a proportion is true is by finding the decimal equivalent of each side. The statement 38 122 is not true; 0.375 is not equal to 0.16. In algebra, a variable can stand for an unknown number or for a set of numbers. In the proportion 23 M 6 , you can replace the letter M with any number, but only one number will make the proportion true. That number is unknown until the proportion is solved. Investigation Multiply and Conquer You can easily guess the value of M in the proportion 23 M 6 . In this investigation you’ll examine ways to solve a proportion for an unknown number when guessing 56 is not easy. It’s hard to guess the value of M in the proportion 1M9 133 . Step 1 56 Multiply both sides of the proportion 1M9 133 by 19. Why can you do this? What does M equal? Step 2 For each equation, choose a number to multiply both ratios by to solve the proportion for the unknown number. Then multiply and divide to find the missing value. p 132 21 Q a. 1 b. 3 2 176 5 20 L 30 130 n c. 3 d. 7 0 200 8 15 Step 3 Check that each proportion in Step 2 is true by replacing the variable with your answer. Step 4 In each equation in Step 2, the variables are in the numerator. Write a brief explanation of one way to solve a proportion when one of the numerators is a variable. Step 5 The proportions you solved in Step 2 have been changed by switching the numerators and denominators. That is, the ratio on each side has been inverted. p (You may recall that inverted fractions, like 12 and 1p2 are called reciprocals.) Do the solutions from Step 2 also make these new proportions true? 12 176 a. p 132 Step 6 35 20 b. 2 1 Q 30 200 c. L 3 0 78 15 d. 130 n How can you use what you just discovered to help you solve a proportion that 20 12 has the variable in the denominator, such as 135 k ? Why does this work? Solve the equation. LESSON 2.1 Proportions 97 DA2SE_763_02.qxd 10/17/2005 18:53 Page 98 There are many ways to solve proportions. Here are three student papers each answering the question “13 is 65% of what number?” What are the steps each student followed? What other methods can you use to solve proportions? Step 7 a. b. 65 13 ___ __ 100 = x 100 x ___ = __ 65 13 13 x 13 __ 100 ___ = __ __ 1 65 13 1 c. 65 13 ___ __ 100 = x 13 65 = 13 ___ __ 100 x 20 20 = x 20 = x 65 13 ___ __ 100 = x 100 x ___ 65 = 13 x ___ __ __ 100 ___ __ 1 1 100 x 1 1 65x _____ 1300 ____ 65 = 65 x = 20 In the investigation you discovered that you can solve for an unknown numerator in a proportion by multiplying both sides of the proportion by the denominator 56 under the unknown value. You can also think of a proportion such as 1M9 133 56 .” To find the original like this: “When a number is divided by 19, the result is 133 number, you need to undo the division. Multiplying by 19 undoes the division. EXAMPLE B Solution Jennifer estimates that two out of every three students will attend the class party. She knows there are 750 students in her class. Set up and solve a proportion to help her estimate how many people will attend. To set up the proportion, be sure both ratios make the same comparison. Use a to represent the number of students who will attend. Students who will attend Students who will attend a 2 3 750 Students who are invited 98 CHAPTER 2 Proportional Reasoning and Variation Students who are invited DA2SE_763_02.qxd 10/17/2005 18:53 Page 99 In the proportion, when a is divided by 750, the answer is 23. 2 750 3 a Multiply by 750 to undo the division. 500 a Multiply and divide. Jennifer can estimate that 500 students will attend the party. EXAMPLE C After the party, Jennifer found out that 70% of the class attended. How many students attended? Solution 70% is 70 out of 100. So write and solve a proportion to answer the question “If 70 students out of 100 attended the party, how many students out of 750 attended?” History Let s represent the number of students who attended. The Pythagoreans, a group of philosophers begun by Pythagoras in about 520 B.C.E., realized that not all numbers are rational. For example, for a square one unit on a side, the diagonal 2 is irrational. Another irrational number is pi, or , the ratio of the circumference of a circle to its diameter. 70 s 100 750 70 750 100 s 525 s Write the proportion. Multiply by 750 to undo the division. Multiply and divide. 525 out of 750 students attended the party. You have worked with ratios and proportions in this lesson. Numbers that can be written as the ratio of two integers are called rational numbers. EXERCISES You will need your graphing calculator for Exercise 14. Practice Your Skills 1. List these fractions in increasing order by estimating their values. Then use your calculator to find the decimal value of each fraction. 7 a. 8 13 b. 2 0 13 c. 5 52 d. 2 5 2. Ms. Lenz collected information about the students in her class. Eye Color Brown eyes Blue eyes Hazel eyes 9th graders 9 3 2 8th graders 11 4 1 Write these ratios as fractions. a. ninth graders with brown eyes to ninth graders a b. eighth graders with brown eyes to students with brown eyes c. eighth graders with blue eyes to ninth graders with blue eyes a d. all students with hazel eyes to students in both grades LESSON 2.1 Proportions 99 DA2SE_763_02.qxd 10/17/2005 18:54 Page 100 3. Phrases such as miles per gallon, parts per million (ppm), and accidents per 1000 people indicate ratios. Write each ratio named below as a fraction. Use a number and a unit in both the numerator and the denominator. a. In 2000, the McLaren was the fastest car produced. Its top speed was recorded at 240 miles per hour. a b. Pure capsaicin, a substance that makes hot peppers taste hot, is so strong that 10 ppm in water can make your tongue blister. a c. In 2000, women owned approximately 350 of every thousand firms in the United States. a d. The 2000 average income in Philadelphia, Pennsylvania, was approximately $35,500 per person. 4. What number should you multiply by to solve for the unknown in each proportion? T 24 a a. 4 0 30 49 R b. 5 6 32 5. Find the value of the unknown number in each proportion. T 24 49 R 52 42 b. 5 c. 91 S a a. 4 0 30 6 32 M 87 6 62 c 36 e. 1 f. n g. 1 6 232 217 5 13 M 87 a c. 1 6 232 100 7 d. 3 0 x 220 60 h. 3 3 W Reason and Apply 6. APPLICATION Write a proportion for each problem, and solve for the unknown number. a. Leaf-cutter ants that live in Central and South America weigh about 1.5 grams (g). One ant can carry a 4 g piece of leaf that is about the size of a dime. If a person could carry proportionally as much as the leaf-cutter ant, how much could a 55 kg algebra student carry? b. The leaf-cutter ant is about 1.27 cm long and takes strides of 0.84 cm. If a person could take proportionally equivalent strides, what size strides would a 1.65 m tall algebra student take? c. The 1.27 cm long ants travel up to 0.4 km from home each day. If a person could travel a proportional distance, how far would a 1.65 m tall person travel? 7. Write three other true proportions using the four values in each proportion. 2 10 a a. 5 2 5 100 CHAPTER 2 Proportional Reasoning and Variation a 12 b. 9 2 7 j l c. k m DA2SE_763_02.qxd 10/17/2005 18:54 Page 101 8. APPLICATION Jeremy has a job at the movie theater. His hourly wage is $7.38. Suppose 15% of his income is withheld for taxes and Social Security. a. What percent does Jeremy get to keep? b. What is his hourly take-home wage? 9. In a resort area during the summer months, only one out of eight people is a year-round resident. The others are there on vacation. If the year-round population of the area is 3000, how many people are there in the summer? a 10. APPLICATION To make three servings of Irish porridge, you need 4 cups of water and 1 cup of steel-cut oatmeal. How much of each ingredient will you need for two servings? For five servings? 11. APPLICATION When chemists write formulas for chemical compounds, they indicate how many atoms of each element combine to form a molecule of that compound. For instance, they write H2O for water, which means there are two hydrogen atoms and one oxygen atom in each molecule of water. Acetone (or nail polish remover) has the formula C3H6O. The C stands for carbon. carbon hydrogen oxygen hydrogen oxygen Model of acetone molecule Model of water molecule a. How many of each atom are in one molecule of acetone? a b. How many atoms of carbon must combine with 470 atoms of oxygen to form acetone molecules? How many atoms of hydrogen are required? a c. How many acetone molecules can be formed from 3000 atoms of carbon, 3000 atoms of hydrogen, and 1000 atoms of oxygen? a Review 12. In the dot plot below, circle the points that represent values for the five-number summary. If a value is actually the mean of two data points, draw a circle around the two points. 0 4 8 12 16 20 LESSON 2.1 Proportions 101 DA2SE_763_02.qxd 10/17/2005 18:54 Page 102 13. The Forbes Celebrity 100 (www.forbes.com) listed these ten people (and their incomes for 2004 in millions) among the celebrities that got the most media attention. Find the three measures of center for their incomes, and explain why they are so different. 210) Mel Gibson ($ ($210) Oprah Winfrey 147) J.K. Rowling ($ acher ($80) Michael Schum 80) Tiger Woods ($ g ($75) Steven Spielber ) Jim Carrey ($66 en ($64) Bruce Springste 60) Nora Roberts ($ eld ($57) David Copperfi [Data set: CELEB] Tiger Woods 14. Use the order of operations to evaluate these expressions. Check your results on your calculator. a. 5 4 8 b. 12 (7 4) d. 18(3) 81 c. 3 6 25 30 THE GOLDEN RATIO In this project you’ll research the amazing number mathematicians call the golden ratio. (There are plenty of books and web sites on the topic. Find links at www.keymath.com/DA .) Your project should include 102 Basic information on the golden ratio, such as its exact value, why it represents a mathematically “ideal” ratio, and how to construct a golden rectangle. (Its length-to-width ratio is the golden ratio.) Some history of the golden ratio, including its role in ancient Greek architecture. At least one other interesting mathematical fact about the golden ratio, such as its relationship to the Fibonacci sequence or its own reciprocal. A report on where to find the golden ratio in the environment, architecture, or art. List items and their measurements or include prints from photographs, art, or architecture on which you have drawn the golden rectangle. CHAPTER 2 Proportional Reasoning and Variation Once you’ve learned how to construct the golden ratio, The Geometer’s Sketchpad is an ideal tool for further exploration. You can create a Custom Tool for dividing segments into the golden ratio, and then use this tool to help you construct the golden rectangle or even the golden spiral.