264 CHAPTER 4. FRACTIONS cm in cm cm ft pounds

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1 6 CHAPTER. FRACTIONS 9. 7cm 61. cm 6. 6ft 6. 0in cm 69. pounds

2 .. DIVIDING FRACTIONS 6. Dividing Fractions Suppose that you have four pizzas and each of the pizzas has been sliced into eight equal slices. Therefore, each slice of pizza represents 1/ of a whole pizza. Figure.: One slice of pizza is 1/ of one whole pizza. Now for the question: How many one-eighths are there in four? This is a division statement. To find how many one-eighths there are in, divide by 1/. That is, Number of one-eighths in four = 1. On the other hand, to find the number of one-eights in four, Figure. clearly demonstrates that this is equivalent to asking how many slices of pizza are there in four pizzas. Since there are slices per pizza and four pizzas, Number of pizza slices =. The conclusion is the fact that (1/) is equivalent to. That is, 1 = =. Therefore, we conclude that there are one-eighths in. Reciprocals The number 1 is still the multiplicative identity for fractions.

3 66 CHAPTER. FRACTIONS Multiplicative Identity Property. Let a/b be any fraction. Then, a b 1 = a b and 1 a b = a b. The number 1 is called the multiplicative identity because the identical number is returned when you multiply by 1. Next, if we invert /, that is, if we turn / upside down, we get /. Note what happens when we multiply / by /. = Multiply numerators; multiply denominators. = Simplify numerators and denominators. = 1 Divide. The number / is called the multiplicative inverse or reciprocal of /. The product of reciprocals is always 1. Multiplicative Inverse Property. Let a/b be any fraction. The number b/a is called the multiplicative inverse or reciprocal of a/b. The product of reciprocals is 1. a b b a = 1 Note: To find the multiplicative inverse (reciprocal) of a number, simply invert the number (turn it upside down). For example, the number 1/ is the multiplicative inverse (reciprocal) of because 1 = 1. Note that can be thought of as /1. Invert this number (turn it upside down) to find its multiplicative inverse (reciprocal) 1/. You Try It! Find the reciprocals of: EXAMPLE 1. Find the multiplicative inverses (reciprocals) of: (a) /, (a) /7 and (b) 1 (b) /, and (c). Solution. a) Because = 1, the multiplicative inverse (reciprocal) of / is /.

4 .. DIVIDING FRACTIONS 67 b) Because ( ) = 1, the multiplicative inverse (reciprocal) of / is /. Again, note that we simply inverted the number / to get its reciprocal /. c) Because ( 1 ) = 1, the multiplicative inverse (reciprocal) of is 1/. Again, note that we simply inverted the number (understood to equal /1) to get its reciprocal 1/. Answer: (a) 7/, (b) 1/1 Division Recall that we computed the number of one-eighths in four by doing this calculation: 1 = =. Note how we inverted the divisor (second number), then changed the division to multiplication. This motivates the following definition of division. Division Definition. If a/b and c/d are any fractions, then a b c d = a b d c. That is, we invert the divisor (second number) and change the division to multiplication. Note: We like to use the phrase invert and multiply as a memory aid for this definition. EXAMPLE. Divide 1/ by /. Solution. To divide 1/ by /, invert the divisor (second number), then multiply. 1 = 1 = 6 Invert the divisor (second number). Multiply. Divide: You Try It! 10 Answer: 1/

5 6 CHAPTER. FRACTIONS You Try It! Divide: EXAMPLE. Simplify the following expressions: (a) and (b). 1 7 Solution. In each case, invert the divisor (second number), then multiply. a) Note that is understood to be /1. = 1 = 9 Invert the divisor (second number). Multiply numerators; multiply denominators. b) Note that is understood to be /1. Answer: 7 = 1 = Invert the divisor (second number). Multiply numerators; multiply denominators. After inverting, you may need to factor and cancel, as we learned to do in Section.. You Try It! Divide: ( 6 1 ) EXAMPLE. Divide 6/ by /. Solution. Invert, multiply, factor, and cancel common factors. 6 = 6 = 6 = ( ) ( ) ( 7) ( ) = 7 = 7 Invert the divisor (second number). Multiply numerators; multiply denominators. Factor numerators and denominators. Cancel common factors. Remaining factors. Answer: 1/ Note that unlike signs produce a negative answer. Of course, you can also choose to factor numerators and denominators in place, then cancel common factors.

6 .. DIVIDING FRACTIONS 69 You Try It! EXAMPLE. Divide 6/x by /x. Solution. Invert, factor numerators and denominators, cancel common factors, then multiply. 6 ( x ) x = 6 ) ( x x Invert second number. = x x Factor numerators and denominators. x = x x Cancel common factors. x = x Multiply. Note that like signs produce a positive answer. Divide: a Answer: a ( 1 ) a

7 70 CHAPTER. FRACTIONS Exercises In Exercises 1-16, find the reciprocal of the given number /. / / / /19. /7 1. /17 1. / In Exercises 17-, determine which property of addition is depicted by the given identity = = = 19 1 = ( 1 ) = 1 6 ( ) = = 16 1 = 7 6. ( 1 1 ) = ( ) = = = = = = = 1 1

8 .. DIVIDING FRACTIONS 71 In Exercises -6, divide the fractions, and simplify your result In Exercises 7-6, divide the fractions, and simplify your result ( 6) ( 9) 6. 9 ( )

9 7 CHAPTER. FRACTIONS In Exercises 69-0, divide the fractions, and simplify your result. 69. x x 70. x x y 9 10y6 7. y y 7. x y6 x y 1 7. x 10 x 76. 1y y y 1 7. x 0 x 79. 1x y6 7 10y 1 0x 19 1y 9 In Exercises 1-96, divide the fractions, and simplify your result. 1. y 1x 9y 7x. x y x 1y. 10x y 7x y. 0x y x 6y. y 1x y 6x 6. 7y x 6 1y x 7. x 1y 17x y. 7y x 1y x 9. 16y x y6 x 90. 0x 1y x y x y x 16y 9. 0x 17y x 1y 9. y x 9y x 9. 10y 1x y6 6x 9. 1x6 1y x y 96. 0x 9y 6 1x 17y

10 .. DIVIDING FRACTIONS 7 Answers multiplicative inverse property 19. multiplicative identity property 1. multiplicative inverse property. multiplicative identity property. multiplicative inverse property 7. multiplicative identity property 9. multiplicative inverse property 1. multiplicative identity property

11 7 CHAPTER. FRACTIONS x 17 0y 7. 1x 7 7. x y 79. 7x 1. y x xy. y x 7. xy 9 9. x y x y 9. x 9y 9. 6x 1y