Proportions PRE-ACTIVITY PREPARATION

Transcription

1 Section 4.2 PRE-ACTIVITY PREPARATION Proportions In the photo at right, you can see that the top teapot is a smaller version of the lower one. In fact, if you were to compare the ratios of each teapot s width to its height you would find that they are equal. For this reason, you can say that the two are in proportion to each other. Building upon your knowledge of equivalent fractions, you will learn in this section how to determine if two ratios are equal (in proportion) to each other and how to determine the value of the missing numerator or denominator in a pair of equal ratios. These are the important skills to master before you explore, in the next section, the many and varied contexts in which you can use a proportion. LEARNING OBJECTIVES Validate proportions to see if they are true statements. Solve for an unknown quantity in a proportion. Test the validity of the solution to a proportion. TERMINOLOGY PREVIOUSLY USED cross-multiply cross-product denominator multiplier numerator ratio reduce NEW TERMS TO LEARN approximately equals (represented by the symbol ) equate equation expression in proportion proportion solution solve an equation solve for the unknown solve for the variable unknown variable 38

2 382 Chapter 4 Ratios and Proportions BUILDING MATHEMATICAL LANGUAGE Equations An expression is a mathematical symbol or combination of symbols that represents a value. An equation is a mathematical statement of equality of two expressions. When you write an equation, you equate two values. You have already written many equations in this course. Whenever you wrote a statement with an equal sign () and the statement was true, you presented an equation. Up until now, the equations you presented contained only numbers and operation signs. For example: However, an equation might be written with one or both expressions containing unknown quantities. For example, an equation might ask, 4 what number 42 In mathematics, the common way to express what number is with a letter to represent the unknown number (any letter of the alphabet you choose) that is, to represent the unknown number with a variable. When you substitute a variable (say, n) for the unknown quantity (the unknown), the same equation becomes 4 n 42 You might assign any value to n, but only one value for n will make this equation a true statement. Only if n 3, will the equation be true (4 3 42). Any other value for n (for example, n 2.8) will result in a false statement ( ) To solve an equation containing one variable, is to determine the value(s) for the variable that will make the equation a true statement that is, to determine its solution(s). This process is referred to as solving for the unknown or solving for the variable. To maintain the equality of the values represented on both sides of an equation, whenever you make a change to the expression on one side of the statement, you must do the same to the other side. For example,

3 Section 4.2 Proportions 383 Proportions A proportion is a special type of equation that states the equality of two ratios. For example, The proportion may be read Three to five equals thirty-nine to sixty-five. However, it is more commonly read, Three is to five as thirty-nine is to sixty-five. The parts of a proportion are not always whole numbers. When two ratios are equal, we say they are in proportion to each other. Determining if a Given Proportion is or False A true (valid) proportion is a proportion whose ratios are equal to each other. If the ratios are not equal, the proportion is false. When unit labels are included, the first thing to verify is that the comparison of units is in the same order for both ratios. For example, miles gallons and miles gallons, or gallons miles and gallons miles ; not miles gallons gallons and miles If the units do not match numerator to numerator and denominator to denominator, the proportion is false, no matter what the numbers are. Once you have verified that the units match in their comparisons, you can test the validity of the proportion (determine whether it is true or false) by using any one of the following three techniques presented on the next several pages. They will be familiar to you from your knowledge of fractions.

4 384 Chapter 4 Ratios and Proportions TECHNIQUES Testing a Proportion by Applying the Equality Test for Fractions Technique Apply the Equality Test for Fractions and cross-multiply to determine if the proportion is true. MODELS A Is this proportion true or false Cross-multiply to determine the equality of the ratios The proportion is true. B 7 miles Is this proportion true or false gallons 2. miles 9. gallons Verify that the units are compared in the same order. THINK miles per gallon in both ratios of the proportion Cross-multiply The proportion is false. 3 C Is this proportion true or false Cross-multiply The proportion is true.

5 Section 4.2 Proportions 38 Testing a Proportion by Building Up One of the Ratios Technique If it is obvious by inspection that you can multiply the numerator and denominator of one of the ratios by the same multiplier to make it equal to the other ratio, then the proportion is true. MODELS A Is this proportion true or false It is apparent that multiplying the numerator by 3 will equal. Is the denominator 36 multiplied by 3 equal to 08 Yes The proportion is true. B Is this proportion true or false 84 males 42 males 60 females 30 females Verify that the units are compared in the same order. THINK males to females on both sides of the proportion It is apparent that multiplying both 42 and 30 by the multiplier 2 in the ratio on the right will yield the ratio on the left The proportion is true C Is this proportion true or false You can see that the denominator of the first ratio () times 0 will yield 0. Is the numerator 2 times the same multiplier (0) equal to 23 No. ( ) The proportion is false.

6 386 Chapter 4 Ratios and Proportions Testing a Proportion by Comparing the Reduced Forms of the Ratios Technique If easily done, reduce both ratios to their simplest forms. If the reduced ratios are equal, then the proportions are true. MODELS A Is this proportion true or false cats dogs cats dogs Verify that the units are compared in the same order. THINK cats to dogs in both ratios Reduce both sides: and The proportion is true. B Is this proportion true or false Reduce both sides: and The proportion is false. Solving for a Variable in a Proportion A variable can take the place of any one of the four components of the proportion. When the other three components are known quantities, you can determine the value for the unknown that will make the proportion true. For example, in the proportion the proportion true n, the task is to determine the value for n (solve for n) that will make Occasionally, when the solution for the variable is a decimal number and the answer is not exact, you will find it necessary to round your answer to a specified decimal place. When this is the case, use the symbol for is approximately equal to or approximately equals, as in n for the example proportion above. That is, n is approximately equal to Also, when the solution has been rounded and you cross-multiply to test the equality of the two ratios, use the symbol to indicate that the cross-products are close but not exactly equal.

7 Section 4.2 Proportions 387 TECHNIQUE If the relationship between the two ratios of a proportion is easily recognizable, that is, if one ratio is a multiple of the other, you can use the following technique to solve for the unknown quantity. Solving a Proportion when One Ratio is a Multiple of the Other Technique Step : Step 2: If one numerator is a multiple of the other numerator (or one denominator is a multiple of the other denominator), use the multiplier to determine the value of the variable. Validate that the solution is correct by substituting the answer into the original proportion. Apply the Equality Test for Fractions by cross-multiplying. MODELS A Solve for n in the proportion 9 n 26 2 Step THINK Then n Answer : n 8 Step 2 Validate: B Solve for n in the proportion n Step THINK Then 4 20 n Answer : n 20 Step 2 Validate:

8 388 Chapter 4 Ratios and Proportions METHODOLOGY If the relationship between the two ratios of a proportion is not easily recognizable, use the following methodology to solve for the unknown quantity. Solving a Proportion Example : is to as what number is to 37. Example 2: n 39 6 Try It! Steps in the Methodology Example Example 2 Step Set up the proportion with a variable. Write the given proportion. If it is not already set up in proportion form, set up the proportion using a variable for the unknown component. 37 n. Step 2 Equate the cross products. Cross multiply and equate the crossproducts. Why can you do this Shortcut # Reduce the known ratio fi rst (see page 390, Model ) 37. n Step 3 Divide by the multiplier of the variable. Divide both sides of the equation by the multiplier of the variable. Why do you do this 37. n Step 4 Solve for the variable. Calculate the value for the variable. Compute the numerator and divide the product by the denominator. Round, if necessary to the specified place value. Shortcut #2 Cancel common factors before dividing (see page 39, Model 2) )

9 Section 4.2 Proportions 389 Steps in the Methodology Example Example 2 Step Present the answer. State your answer, the value for the unknown. n 27. Step 6 Validate your answer. Validate your answer. In the original proportion, replace the variable with your answer and use one of the Techniques for Testing a Proportion. Note: When the answer is rounded, the cross-products will be close but not exactly equal. (See Model on page 390.) Why can you do Step 2 You are determining the value for the variable that will make a true proportion the value that will make the two ratios equal. The Equality Test for Fractions says that, if the ratios are equal, then their cross-products will have to be equal. Why do you do Step 3 You want your end result to be an equation that either states, n the solution number or the solution number n, as that statement will provide the missing quantity of your proportion. Because the Special Property of Division Involving One tells you that any number divided by itself, and the Identity Property of Multiplication tells you that times any number that same number, you divide by the multiplier of the variable n to yield as its multiplier: n n n Recall that to maintain the equality of the statement, however, you must also do the same to the expression on the other side of the equal sign.

10 390 Chapter 4 Ratios and Proportions MODELS Model Shortcut #: Reduce the Known Ratio First A Solve for the unknown. Round to the nearest hundredth place: Step n n Shortcut (optional): To simplify the next computations, reduce the known ratio before equating the cross-products. Step 2 08 n 2 4 Step n n 08 n 2 4 Step 3 Step n Step 4 n Step 4 n ) Step Step 6 n 0.42 Validate: )

11 Section 4.2 Proportions 39 B 7 Solve for n: 00 Step n 9 n 7 2 Step 2 Use the shortcut and reduce: 00 2 At this point, there is no need to cross-multiply. Use the Technique and skip Steps n THINK n Step n 2 Step 6 Validate: Model 2 Shortcut #2: Cancel Common Factors before Dividing 64 is to.2 as 4 is to what number Step n Step 2 Step 3 64 n n Step 4 n Step ) Step Step 6 n 0.34 Validate: Shortcut: Cancel common factors before dividing to simplify the computation. n )

12 392 Chapter 4 Ratios and Proportions Model Solve for the unknown. Round to the nearest hundredth place: 20 n Step Use shortcut #. 20 n Reduce: 20 4 n Step 2 2 n 64 Step 3 Step 4 Step n Step 6 Validate: 6 2 n n Use shortcut #2. Cancel common factors ) Model 4 Solve: What number is to 3 as ½ is to 2 Step Step 2 n n Step 3 Step 4 Step n Compute the numerator: Divide by the denominator: n Step 6 Validate:

13 Section 4.2 Proportions 393 How Estimation Can Help When solving for the unknown quantity in a proportion, an effective practice is to estimate the size of the solution relative to its position in the proportion. This can go a long way in preventing decimal placement errors in your answers. Example : n Before solving, think about the fact that the two ratios must be in proportion to one another. This means that, minimally, you can predict that your solution for n will be greater than because 37 is greater than 2.8 in the ratio on the right. You can also predict the value for n that will make this a true proportion. THINK On the right side, estimate that 37 is approximately three times 2.8, so n will be approximately three times. Estimate: n 4 Actual answer: n 43.4, reasonably close to the estimate Example 2: 2. n Here is a case in which you might estimate in either of two ways: THINK Since numerator 8.7 is about two times numerator 9, denominator n will be about two times denominator.2. Estimate: n 2.4 OR THINK Since numerator 9 is about nine times its denominator.2, numerator 8.7 will be about nine times its denominator n. Estimate: n 2 Actual answer: n 2.49, reasonably close to the estimates Example 3: THINK n n will be considerably less than 6 because 2.8 is considerably less than is about one hundred times as large as 2.8, so 6 will be about one hundred times as large as n. Estimate: n 0.06 Actual answer: n 0.3, reasonably close to the estimate Go back and estimate the answers for the previous Models. Were the answers reasonable as compared to your estimates

14 394 Chapter 4 Ratios and Proportions ADDRESSING COMMON ERRORS There are no additional new common errors to address. However, the errors that do most frequently occur are those addressed in previous sections of this book. Issue Resolution Incorrectly rounding the answer when solving for the unknown in a proportion Review the Methodology for Rounding a Decimal Number on page 36. Incorrect placement of the decimal point when multiplying decimal numbers Review the Methodology for Multiplying Decimal Numbers on page 67. Errors in the quotient when dividing decimal numbers Review the Methodology for Dividing Decimal Numbers on page 86. PREPARATION INVENTORY Before proceeding, you should have an understanding of each of the following: the terminology and notation associated with proportions how to determine if a proportion is true how to solve a proportion for the unknown component how to validate the solution to a proportion

15 Section 4.2 ACTIVITY Proportions PERFORMANCE CRITERIA Validating a proportion Solving a proportion answer rounded to the specified place validation of the answer CRITICAL THINKING QUESTIONS. What are two different ways the following proportion can be read Answers include: 8 is to as 2.8 is to fi fteenths equals 2.8 halves 8 to equals 2.8 to 2 2. Why must the comparison order of the units be verified before determining if a proportion is true The numerators should both refl ect the same unit and likewise the denominators of both ratios should refl ect the same unit. Therefore the comparison will both be the same, like dollars per share or miles per hour. 3. What are the ways you can validate that a proportion is true You can accomplish this by cross multiplication. If the answers are the same, then the proportion is true. You could also reduce or build both ratios to show that the fractions are equivalent. Corresponding parts can be reduced: numerator with denominator on the same side of the equal sign or numerator to numerator or denominator to denominator on opposite sides of the equal signs. 39

16 396 Chapter 4 Ratios and Proportions 4. When should you use the Methodology versus the Technique for solving a proportion If the relationship between the two ratios is easily recognizable, then use a Technique, otherwise use the Methodology.. Why can you cross-multiply and equate the cross-products in Step 2 of the Methodology for Solving a Proportion This can be used because of the Equality Test for Fractions. 6. Why do you divide each side of the equation by the multiplier of the variable in Step 3 of the Methodology for Solving a Proportion Because the Special Properties of Division Involving One tell us that any number divided by itself equals one, and the Identity Property of Multiplication tells us that one times any number equals that same number. 7. How do you assure that your answer for the unknown value in a proportion is correct Replace the unknown with the answer that you found then cross-multiply to show that the cross products are equivalent. This will validate your answer.

17 Section 4.2 Proportions 397 TIPS FOR SUCCESS To assure a correct answer when solving a proportion, do not skip steps. Write the equation for the crossmultiplication followed by the equation showing the division by the multiplier of the variable. In this way, the numbers you must multiply and divide will be clearly presented. Use your skills of reducing and building up fractions to shortcut the process of validating proportions and the process of solving a proportion. When validating a proportion with decimal components, be attentive to the placement of the decimal point in the cross-products. To check for exact computational accuracy, validate the decimal division (see Methodology for Dividing Decimal Numbers in Section 2.4). DEMONSTRATE YOUR UNDERSTANDING In problems -, determine if the given proportions are true or false. ) Proportion Worked Solution or False False 2) 6.4 gallons 4 gallons. 2 acres 7 acres 3) 28 SUVs 6 SUVs 9 total vehicles 2 total vehicles

18 398 Chapter 4 Ratios and Proportions Proportion Worked Solution or False 4) ) ) The directions on a well-known brand of parboiled rice say to combine the rice and water in the ratio of /2 cup rice to 2 /4 cups water for four servings, and in the ratio of /2 cups rice to 3 /3 cups water for six servings. Are the ratios in proportion to each other For the following proportions, solve for the unknown. Round to the nearest tenth, if necessary. 7) Proportion n Worked Solution or False

19 Section 4.2 Proportions 399 8) Proportion 2 n 7 0 Worked Solution or False 9) n. 0) 39 3 n 72 ) n

20 400 Chapter 4 Ratios and Proportions Proportion 2) is to 8, as 42 is to what number Worked Solution or False IDENTIFY AND CORRECT THE ERRORS In the second column, identify the error(s) you find in each of the following worked solutions. If the answer appears to be correct, validate it in the second column and label it Correct. If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer in the last column. Worked Solution What is Wrong Here Identify Errors or Validate Correct Process Validation ) Is this a true proportion The decimal point was placed improperly in the product of 0.72 and 90. (Should have been 3 decimal places.) TRUE Validation is not necessary for this problem. or TRUE

21 Section 4.2 Proportions 40 Worked Solution What is Wrong Here Identify Errors or Validate Correct Process Validation 2) Solve for the unknown. 44 n You cannot cancel across the equal () sign. 3) is to 9 as 33 is to what number The problem is set up incorrectly. The correct set up is: 9 33 n

22 402 Chapter 4 Ratios and Proportions Worked Solution What is Wrong Here Identify Errors or Validate Correct Process Validation 4) Solve for n. Round to the nearest tenth. 2 n 9. Did not need to carry out the answer to the thousandths place. Rounding to tenths, you only need to carry out the division to the hundredths place. ) Solve for n. Round to the nearest tenth n Carried out division to the wrong place. Did not round to the correct place.

23 Section 4.2 Proportions 403 ADDITIONAL EXERCISES Determine if the following proportions are true or false.. 6 feet 24 feet 0 seconds seconds False False False. 7 cups sauce 39 servings 9 cups sauce 3 2 servings Solve for the unknown in each of the following proportions. Round to the nearest tenth if necessary. Validate your solutions n ṇ n or ṅ. 0. n n is to 0 as what number is to