242 CHAPTER 5 Number Theory and the Real Number System To divide fractions we make use of the reciprocal of a number. The reciprocal of any number is 1 divided by that number. The product of a number and its reciprocal must equal 1. Examples of some numbers and their reciprocals follow. Multiplication of Fractions ⋅ = ⋅ ⋅ = ≠ ≠ a b c d a c b d ac bd b d , 0, 0 Example 11 Multiplying Fractions Evaluate. a) ⋅ 2 5 9 11 b) ⎛ − ⎝ ⎞ ⎠ ⎛ − ⎝ ⎞ ⎠ 2 3 4 9 c) ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ 1 7 8 2 1 4 Solution a) ⋅ = ⋅ ⋅ = 2 5 9 11 2 9 5 11 18 55 b) 2 3 4 9 ( 2)( 4) (3)(9) 8 27 ⎛ − ⎝⎜ ⎞ ⎠⎟ ⎛ − ⎝⎜ ⎞ ⎠⎟ = − − = c) ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ = ⋅ = ⋅ ⋅ = = 1 7 8 2 1 4 15 8 9 4 15 9 8 4 135 32 4 7 32 7 Now try Exercise 57 Number Reciprocal Product 3 ⋅ 1 3 = 1 3 5 ⋅ 5 3 = 1 6− ⋅ − 1 6 = 1 To determine the quotient of two fractions, multiply the first fraction by the reciprocal of the second fraction. Division of Fractions ÷ = ⋅ = ≠ ≠ ≠ a b c d a b d c ad bc b d c , 0, 0, 0 Example 12 Dividing Fractions Evaluate. a) ÷ 2 7 3 5 b) ⎛ − ⎝ ⎞ ⎠ ÷ 2 9 3 13 Solution a) ÷ = ⋅ = ⋅ ⋅ = 2 7 3 5 2 7 5 3 2 5 7 3 10 21 b) ⎛ − ⎝ ⎞ ⎠ ÷ = − ⋅ = − ⋅ ⋅ = − = − 2 9 3 13 2 9 13 3 2 13 9 3 26 27 26 27 7 Now try Exercise 61
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