3.2 Equivalent Fractions: Simplifying and Building

Transcription

1 3.2 Equivalent Fractions: Simplifying and Building Two fractions are said to be equivalent if they have the same value. Naturally, one approach we could use to determine if two fractions are equivalent is to convert each fraction to a decimal. For example, since 3 = 0.6 and 2 = 0.6, the fractions 3 and 2 write 3 =. Alternatively, consider the following forms of the number 1: 2 are equivalent, and we could 1 = 2 2 = 3 3 = 4 4 =... = = n n Clearly 2 parts out of 2 is equal to 1, as is 100 parts out of 100, or n parts out of n. Now consider the following property (the Fundamental Property of Fractions): If a, b, and c are nonzero: a b = a c b c This statement is saying if both the numerator and denominator of a fraction have the same factor (called a common factor), then that factor can be eliminated resulting in an equivalent fraction. It is true because a b = a b c c = a c b c, so we are multiplying the fraction a b by c c (a form of 1) to result in the fraction a c. Recall that multiplying a number by 1 does not change its value (the b c Identity Property of Multiplication). Using our fractions 3 2 = 3 = 3 and, note that: 2 Thus 2 and 3 are equivalent fractions, or 2 = 3. 14

2 Example 1 Determine whether the two fractions are equivalent by using the Fundamental Property of Fractions. a. 9 10, 4 0 b.! 60,! 7 84 c. 7 12, d. 4 9, 4xy 9xy Solution a. For the two fractions to be equivalent, there must be a form of 1 (or a common factor) which can be multiplied by one fraction to create the other. Note that: 9 10 = 4 0 Since is a form of 1, the two fractions are equivalent. b. We must find a form of 1 (or common factor) which can be multiplied by one fraction to create the other. Note that:! =! Since 12 is a form of 1, the two fractions are equivalent. 12 c. We must find a form of 1 (or common factor) which can be multiplied by one fraction to create the other. Note that: = Since 7 is not a form of 1, the two fractions are not equivalent. An 8 alternate way to verify this is to convert each fraction to decimal: 7 12 = = Note that these two decimal forms are not the same. 146

3 d. We must find a form of 1 (or common factor) which can be multiplied by one fraction to create the other. Note that: 4 9 xy xy = 4xy 9xy Since xy is a form of 1, the two fractions are equivalent. xy Given a fraction, could you find other fractions which are equivalent to it. For example, given the fraction 3, what would be some other fractions equivalent to it? We could multiply by 7 different forms of 1: = = a a = 3a 7a Note that we could list as many equivalent fractions as we can list forms of 1, which is infinite. Example 2 For each fraction, list three equivalent fractions. Use variables in at least one of your fractions. a. 8 b.! 7 c. 2x

4 Solution a. Using the fractions 2 2, different): = = 2 40, and ab ab 8 ab ab = ab 8ab Three equivalent fractions are 10 16, 2 ab, and 40 8ab. b. Using the fractions 9 9, , and x2 y x 2 y different):! =! ! =! ! 7 12 x2 y x 2 y =! 7x2 y 12x 2 y as forms of 1 (yours will probably be as forms of 1 (yours will probably be Three equivalent fractions are! ,! , and! 7x2 y 12x 2 y. c. Using the fractions 8 8, probably be different): 2x 8 8 = 16x 40 2x = 30x 7 2x xy xy = 2x2 y xy, and xy xy as forms of 1 (again yours will Three equivalent fractions are 16x 40, 30x 7, and 2x2 y. Note how we xy multiplied the numbers, and how the exponent was used to represent x x in the third fraction. 148

5 In addition and subtraction of fractions, it will be necessary to convert a fraction to a specified denominator. For example, given the fraction, how could this fraction be converted to one 6 with a denominator of 72? That is, what numerator x would result in 6 = x being equivalent? 72 Since 6 12 = 72 (we can find 12 by dividing 6 into 72), the form of 1 to use is Thus: = The missing numerator is x = 60. This idea is often referred to as building fractions. Example 3 Find the variable such that the two given fractions are equivalent. 9 a. 14 = x 70 b.! 3 4 =! 33 y a c. = d.! b =! Solution a. Since =, the form of 1 to use is. Therefore: 9 14 = 4 70 The missing numerator is x = 4. b. Since 33 3 = 11, the form of 1 to use is Therefore:! =! The missing denominator is y =

6 c. Since 120 = 8, the form of 1 to use is 8. Instead of multiplying the 8 first fraction by 8, we can alternatively divide the second fraction: = 12 The missing numerator is a = 12. Note how we used the idea that division is the inverse of multiplication to do this problem. d. Since 200 = 40, the form of 1 to use is 40. Again, we do this problem 40 backwards by dividing the second fraction:! =! 8 The missing denominator is b = 8. This last example leads to the idea of simplifying (or reducing) fractions. That is, given a fraction such as 32, can we apply the Fundamental Property of Fractions to reduce the numbers 40 to a simpler form? Using the form of 1 as 8, we can write: 8 We say that reduces to 4. Note that = = 4 does not reduce further, since there is no other form of 1 we can use in the Fundamental Property of Fractions. But where did 8 come from? Recall 8 from Chapter 1 that the greatest common factor (GCF) of 32 and 40 is the largest number that will divide into both 32 and 40, which is precisely the number 8. In other words, using the GCF of the numerator and denominator as the common factor will always result in the form of 1 to use. In the past, you may have learned to reduce fractions by dividing the numerator and denominator by the same number (this is the same as our form of 1). The big problem, however, is knowing when to stop. 0

7 For example, we can attempt to reduce as: = = However, the result can be reduced further. Thus the GCF becomes the quickest (and safest, in terms of errors) approach to simplify fractions. Example 4 Use the greatest common factor to simplify each fraction. 6 a. 80 b.! 2 0 c.! 48 d. 132 xy 10x Solution a. The GCF of 6 and 80 is 8, so the form of 1 to use is 8 8. Therefore: 6 80 = = 7 10 b. The GCF of 2 and 0 is 2, so the form of 1 to use is 2 2. Therefore:! =! =! 1 6 Note how our invisible factor of 1 is used in this fraction. 1

8 c. The GCF of 48 and 132 is 12, so the form of 1 to use is Therefore:! =! =! 4 11 d. The GCF of xy and 10x is x, so the form of 1 to use is x x. Therefore: xy 10x = y x 2 x = y 2 This illustrates how we can simplify fractions with symbols also. Thus far, we have found the GCF by guessing at it, but recall our alternate approach using primes, which works particularly well for larger numbers. For example, to reduce the fraction 168, it would be difficult to guess at the GCF of 168 and 180. We first factor each number into 180 primes: 168 = 8 21 = = = 2 ( ) ( 3 7) = ( 2 2 2) ( 3 7) = ( ) ( 3 6) = ( 2 ) ( 3 2 3) = Instead of finding the GCF, we will use the primes in our fraction, remembering that common factors of the numerator and denominator will cancel: = = /2 /2 2 /3 7 /2 /2 /3 3 = = 14 prime factorizations cancelling common factors writing the remaining factors multiplying For fractions with larger numbers, this is usually the most efficient, and more importantly the most accurate, approach. 2

9 Example Use prime numbers to simplify each fraction. 21 a. 112 b.! c.! x 2 y 3 d. 4x 4 y Solution a. First find the prime factorizations of 21 and 112: 21 = 3 7 ( ) ( 2 2 7) = = 4 28 = 2 2 Now rewrite the fraction using prime numbers, and simplify: = 3 7 prime factorizations /7 = cancelling common factors /7 3 = writing the remaining factors = 3 16 multiplying b. First find the prime factorizations of 90 and 198: 90 = 9 10 = 3 3 ( ) ( 2 ) = ( ) = = 2 99 = Now rewrite the fraction using prime numbers, and simplify:! =! prime factorizations =! =! 11 /2 /3 /3 /2 /3 /3 11 cancelling common factors writing the remaining factors 3

10 c. First find the prime factorizations of 168 and 224: 168 = 4 42 = 2 2 ( ) ( 2 21) = ( 2 2) ( 2 3 7) = ( ) ( 4 14) = ( 2 2) ( ) = = 4 6 = 2 2 Now rewrite the fraction using prime numbers, and simplify:! =! prime factorizations =! /2 /2 /2 3 /7 /2 /2 /2 2 2 /7 =! cancelling common factors writing the remaining factors =! 3 4 multiplying d. This may not seem to fit, but look carefully at the fraction. Treating the variables as prime numbers, we simplify: x 2 y 3 4x 4 y = x x y y y prime factorizations 2 2 x x x x y = /x /x y y /y 2 2 /x /x x x /y = y y 2 2 x x cancelling common factors writing the remaining factors = y2 multiplying 4x 2 This example is a typical algebra problem, and it is included here to illustrate that the ideas we are developing extend directly to algebra. In the last section, we found that some fractions result in a terminating decimal, while others result in a repeating decimal. Is there a way to tell in advance which will occur? Consider the two fractions and their decimal forms: = =

11 Now look at the prime factorizations of the denominators: = = = = 0.63 Recall that the decimal system has denominators which are powers of 10 = 2. Suppose we want to build the first fraction 17 up to a power of 10 in the denominator. Since each 10 = 2, 40 we will need to multiply by two additional factors of : = = = = 0.42 However, with the second fraction 19, the prime factor of 3 will always be in the prime 30 factorization of the denominator. Thus we can never build its denominator to a power of 10, and thus it can t be represented as a terminating decimal. In summary, only fractions whose denominators have prime factors of 2 and can be converted to terminating decimals. If a denominator of a fraction has prime factors other that 2 or, it will result in a repeating decimal (assuming the fraction is simplified). Thus the vast majority of fractions have repeating, rather than terminating, decimal forms. Example 6 Determine whether the decimal form of each rational number will be terminating or repeating. Do not actually convert the fraction to decimal! a b.! c. 320 d.!

12 Solution a. Finding the prime factorization of 8: 8 = 17 Since the denominator has a prime factor other than 2 or (17), the decimal form is a repeating decimal. Terminology b. Finding the prime factorization of 400: 400 = ( ) ( 10 10) ( ) ( 2 2 ) = 2 2 = 2 2 = Since the denominator has only 2 and prime factors, the decimal form is a terminating decimal. c. Finding the prime factorization of 320: 320 = ( ) ( 4 8) ( ) ( ) = 2 = 2 = Since the denominator has only 2 and prime factors, the decimal form is a terminating decimal. d. Finding the prime factorization of 440: 440 = ( ) ( 4 11) ( ) ( ) = 2 = 2 = Since the denominator has a prime factor other than 2 or (11), the decimal form is a repeating decimal. equivalent fractions common factor simplifying (or reducing) fractions Fundamental Property of Fractions building fractions greatest common factor (GCF) 6

13 Exercise Set 3.2 Determine whether the two fractions are equivalent by using the Fundamental Property of Fractions. 1. 7, ! 13 2,! 7.! ,! x 7. 4y, 2ax 20ay 9.! 3a 6ab,! 7b 14ab , ! 20,! ! 2,! y 8. 9a, 3y2 4ay 10.! 9w 27awz,! 10z 30az For each fraction, list three equivalent fractions. Use variables in at least one of your fractions ! 12.! 3x 17. 6a 7b ! 8 16.! 4a 18. s 9t 7 Find the variable such that the two given fractions are equivalent = x 6 21.! 14 2 =! x = 19 y 00 2.! 7 =! 112 y = x ! 9 =! x = 216 y ! 9 16 =! 162 y 7

14 a = ! a =! b = ! 12 b =! a = ! a 13 =! b = ! 16 b =! Use the greatest common factor to simplify each fraction ! ! ! xy 47. ax 49.! 16abx 30axy ! ! ! ab 40ax 0.! 4axy 7aby 8

15 Use prime numbers to simplify each fraction ! ! ! ! ! ! ! ! x 2 y x 3 y x 3 y 7 3 x 6 y 4 8a 7 b 6 6a b a 6 b 2 18a 4 b 3 71.! 16x8 y 12 20x 9 y 72.! 36x6 y x 9 y 8 9

16 Determine whether the decimal form of each rational number will be terminating or repeating. Do not actually convert the fraction to decimal! ! ! ! ! (Hint: Simplify first) (Hint: Simplify first) ! ! Answer each question as true or false. If it is false, give a specific example to show that it is false. If it is true, explain why. 83. Every fraction has either a terminating or repeating decimal form. 84. Every repeating decimal can be written as a fraction. 8. If the numbers a and b are relatively prime, then the fraction a b is already simplified. 86. If the numbers a and b are not relatively prime, then the fraction a b is not simplified. 87. There is only one decimal form for every fraction. (Hint: Refer back to Exercises 9 and 96 of the previous section.) 88. There is only one fraction for every terminating decimal. 160