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Solution Manual for Beginning Algebra, 13th Edition

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    S OLUTIONS M ANUAL E MILY K EATON B EGINNING A LGEBRA T HIRTEENTH E DITION Margaret L. Lial American River College John Hornsby University of New Orleans Terry McGinnis

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    Table of Contents R Prealgebra Review ................................................................................................................... 1 R.1 R.2 Fractions •................................................................................................................. Decimals and Percents • ........................................................................................... 1 16 1 The Real Number System ....................................................................................................... 28 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Exponents, Order of Operations, and Inequality • .................................................... Variables, Expressions, and Equations • ................................................................... Real Numbers and the Number Line • ...................................................................... Adding and Subtracting Real Numbers • .................................................................. Multiplying and Dividing Real Numbers • ............................................................... Summary Exercises: Performing Operations with Real Numbers • ......................... Properties of Real Numbers • .................................................................................... Simplifying Expressions • ......................................................................................... Chapter 1 Review Exercises • ................................................................................... Chapter 1 Mixed Review Exercises • ........................................................................ Chapter 1 Test • ......................................................................................................... 28 34 42 47 57 67 69 77 83 88 90 Chapters R–1 Cumulative Review Exercises • ....................................................... 92 2 Linear Equations and Inequalities in One Variable ............................................................ 94 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 The Addition Property of Equality • ....................................................................... The Multiplication Property of Equality • ............................................................... Solving Linear Equations Using Both Properties of Equality• ............................... Clearing Fractions and Decimals When Solving Linear Equations • ...................... Summary Exercises: Applying Methods for Solving Linear Equations • ............... Applications of Linear Equations • ......................................................................... Formulas and Additional Applications from Geometry • ....................................... Ratio, Proportion, and Percent • .............................................................................. Further Applications of Linear Equations •............................................................. Solving Linear Inequalities • ................................................................................... Chapter 2 Review Exercises • ................................................................................. Chapter 2 Mixed Review Exercises • ...................................................................... Chapter 2 Test • ....................................................................................................... Chapters R–2 Cumulative Review Exercises • ....................................................... 94 104 113 121 135 139 155 168 181 193 210 219 221 224 3 Linear Equations and Inequalities in Two Variables; Functions ......................................... 228 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Linear Equations and Rectangular Coordinates • .................................................... Graphing Linear Equations in Two Variables • ...................................................... The Slope of a Line • .............................................................................................. Slope-Intercept Form of a Linear Equation • .......................................................... Point-Slope Form of a Linear Equation and Modeling • ......................................... Summary Exercises: Applying Graphing and Equation-Writing Techniques for Lines • ........................................................................................................ Graphing Linear Inequalities in Two Variables •.................................................... Introduction to Functions •...................................................................................... Chapter 3 Review Exercises • ................................................................................. Chapter 3 Mixed Review Exercises • ...................................................................... 228 240 258 268 282 293 298 307 314 320

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    Chapter 3 Test • ....................................................................................................... Chapters R–3 Cumulative Review Exercises • ....................................................... 323 325 4 Systems of Linear Equations and Inequalities ....................................................................... 328 4.1 4.2 4.3 4.4 4.5 Solving Systems of Linear Equations by Graphing • .............................................. Solving Systems of Linear Equations by Substitution • .......................................... Solving Systems of Linear Equations by Elimination • .......................................... Summary Exercises: Applying Techniques for Solving Systems of Linear Equations •........................................................................................... Applications of Linear Systems • ............................................................................ Solving Systems of Linear Inequalities • ................................................................ Chapter 4 Review Exercises • ................................................................................. Chapter 4 Mixed Review Exercises • ...................................................................... Chapter 4 Test • ....................................................................................................... Chapters R–4 Cumulative Review Exercises • ....................................................... 328 341 352 365 371 389 398 405 408 412 5 Exponents and Polynomials .................................................................................................... 416 5.1 5.2 5.3 5.4 5.5 5.6 5.7 The Product Rule and Power Rules for Exponents • ................................................ Integer Exponents and the Quotient Rule • .............................................................. Summary Exercises: Applying the Rules for Exponents • ....................................... Scientific Notation • ................................................................................................. Adding, Subtracting, and Graphing Polynomials • .................................................. Multiplying Polynomials • ....................................................................................... Special Products • .................................................................................................... Dividing Polynomials • ............................................................................................ Chapter 5 Review Exercises • .................................................................................. Chapter 5 Mixed Review Exercises • ....................................................................... Chapter 5 Test • ........................................................................................................ Chapters R–5 Cumulative Review Exercises • ........................................................ 416 424 433 437 446 459 471 478 493 499 501 504 6 Factoring and Applications.......................................................................................................508 6.1 6.2 6.3 6.4 6.5 6.6 Greatest Common Factors; Factoring by Grouping • ............................................. Factoring Trinomials • ............................................................................................ More on Factoring Trinomials • ............................................................................. Special Factoring Techniques • .............................................................................. Summary Exercises: Recognizing and Applying Factoring Strategies • ............... Solving Quadratic Equations Using the Zero-Factor Property • ............................. Applications of Quadratic Equations •.................................................................... Chapter 6 Review Exercises • ................................................................................. Chapter 6 Mixed Review Exercises • ....................................................................... Chapter 6 Test • ...................................................................................................... Chapters R–6 Cumulative Review Exercises • ....................................................... 508 517 527 539 552 556 567 579 583 585 587 iv

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    7 Rational Expressions and Applications ...................................................................................591 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 The Fundamental Property of Rational Expressions • ........................................... Multiplying and Dividing Rational Expressions • ................................................. Least Common Denominators • ............................................................................. Adding and Subtracting Rational Expressions • .................................................... Complex Fractions •............................................................................................... Solving Equations with Rational Expressions • ..................................................... Summary Exercises: Simplifying Rational Expressions vs. Solving Rational Equations •............................................................................ Applications of Rational Expressions • .................................................................. Variation • .............................................................................................................. Chapter 7 Review Exercises • ................................................................................ Chapter 7 Mixed Review Exercises • ..................................................................... Chapter 7 Test •...................................................................................................... Chapters R–7 Cumulative Review Exercises • ...................................................... 591 605 615 625 643 657 679 682 695 702 712 715 719 8 Roots and Radicals ....................................................................................................................724 8.1 8.2 8.3 8.4 8.5 8.6 Evaluating Roots • ................................................................................................. Multiplying, Dividing, and Simplifying Radicals • ............................................... Adding and Subtracting Radicals • ........................................................................ Rationalizing the Denominator • ............................................................................ More Simplifying and Operations with Radicals •................................................. Summary Exercises: Applying Operations with Radicals • .................................. Solving Equations with Radicals • ......................................................................... Chapter 8 Review Exercises • ................................................................................ Chapter 8 Mixed Review Exercises • ..................................................................... Chapter 8 Test •...................................................................................................... Chapters R–8 Cumulative Review Exercises • ...................................................... 724 735 743 751 759 770 773 789 795 797 799 9 Quadratic Equations .................................................................................................................805 9.1 9.2 9.3 9.4 Solving Quadratic Equations by the Square Root Property • ................................. Solving Quadratic Equations by Completing the Square • .................................... Solving Quadratic Equations by the Quadratic Formula • ..................................... Summary Exercises: Applying Methods for Solving Quadratic Equations • ........ Graphing Quadratic Equations • ............................................................................ Chapter 9 Review Exercises • ................................................................................ Chapter 9 Mixed Review Exercises • ..................................................................... Chapter 9 Test •...................................................................................................... Chapters R–9 Cumulative Review Exercises • ...................................................... 805 814 831 847 853 868 874 876 880 v

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    R.1 Fractions 1 Chapter R Prealgebra Review R.1 Fractions Classroom Examples, Now Try Exercises 1. 90 is composite and can be written as Writing 90 as the product of primes gives us 90 2 3 3 5     . N1. 60 is composite and can be written as Writing 60 as the product of primes gives us 60 2 2 3 5.     2. (a) 12 3 4 3 4 3 3 1 20 5 4 5 4 5 5         (b) 8 8 1 1 48 6 8 6 1 6      (c) 90 5 18 5 5 1 162 9 18 9 9       N2. (a) 30 5 6 5 6 5 5 1 42 7 6 7 6 7 7         (b) 10 10 1 1 70 7 10 7 1 7       (c) 72 3 24 3 3 1 120 5 24 5 5       3. The fraction bar represents division. Divide the numerator of the improper fraction by the denominator. 3 10 37 30 7 Thus, 37 7 3 . 10 10  N3. The fraction bar represents division. Divide the numerator of the improper fraction by the denominator. 18 5 92 5 42 40 2 Thus, 92 2 18 . 5 5  4. Multiply the denominator of the fraction by the natural number and then add the numerator to obtain the numerator of the improper fraction. 5 3 15   and 15 4 19   The denominator of the improper fraction is the same as the denominator in the mixed number. Thus, 4 19 3 . 5 5  N4. Multiply the denominator of the fraction by the natural number and then add the numerator to obtain the numerator of the improper fraction. 3 11 33   and 33 2 35   The denominator of the improper fraction is the same as the denominator in the mixed number. Thus, 2 35 11 . 3 3  5. (a) To multiply two fractions, multiply their numerators and then multiply their denominators. Then simplify and write the answer in lowest terms. 5 18 5 18 9 25 90 22 9 25 2 5 5 2 45 5 45         

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    Chapter R Prealgebra Review 2 (b) To multiply two mixed numbers, first write them as improper fractions. Multiply their numerators and then multiply their denominators. Then simplify and write the answer as a mixed number in lowest terms. 3 10 7 3 1 10 7 3 4 2 5 7 3 2 2 5 , or 5 1 3 4 3 4 35 6 6             N5. (a) To multiply two fractions, multiply their numerators and then multiply their denominators. Then simplify and write the answer in lowest terms. 4 5 4 5 7 8 7 8 20 56 5 4 14 4 5 14          (b) To multiply two mixed numbers, first write them as improper fractions. Multiply their numerators and then multiply their denominators. Then simplify and write the answer as a mixed number in lowest terms. 2 2 17 20 3 6 5 3 5 3 17 20 5 3 17 5 4 5 3 68 2 , or 22 3 3            6. (a) To divide fractions, multiply by the reciprocal of the divisor. 9 5 3 3 5 2 5 3 3 1 , or 9 3 10 5 10 1 2 3 2          (b) Change both mixed numbers to improper fractions. Then multiply by the reciprocal of the second fraction. 3 1 11 10 2 3 4 4 3 11 3 4 0 4 3 33 1 0       N6. (a) To divide fractions, multiply by the reciprocal of the divisor. 2 8 7 9 7 8 2 9 2 3 3 7 2 4 28 9          (b) To divide fractions, multiply by the reciprocal of the divisor. 3 2 15 30 3 4 4 7 4 7 15 7 4 30 15 7 4 2 15 7 8           7. To find the sum of two fractions having the same denominator, add the numerators and keep the same denominator. 9 9 6 1 5 1 5 9 3 3 3 9 2 3 2        

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    R.1 Fractions 3 N7. To find the sum of two fractions having the same denominator, add the numerators and keep the same denominator. 1 3 1 3 8 8 8 4 8 1 4 2 4 1 2         8. (a) Since 30 2 3 5    and 45 3 3 5,    the least common denominator must have one factor of 2 (from 30), two factors of 3 (from 45), and one factor of 5 (from either 30 or 45), so it is 2 3 3 5 90.     Write each fraction with a denominator of 90. 7 3 21 30 7 3 9 3 0 0    and 2 2 2 4 45 9 45 2 0    Now add. 7 2 21 4 21 4 25 30 45 90 90 90 90       Write 25 90 in lowest terms. 25 5 5 5 90 18 5 18     (b) Write each mixed number as an improper fraction. 5 6 3 1 29 7 4 2 6 3    The least common denominator is 6, so write each fraction with a denominator of 6. 29 6 and 7 7 2 14 3 3 2 6    Now add. 7 29 14 29 29 6 3 6 6 6 43 6 14 1 , or 7 6       N8. (a) Since 12 2 2 3    and 8 2 2 2,    the least common denominator must have three factors of 2 (from 8) and one factor of 3 (from 12), so it is 2 2 2 3 24.     Write each fraction with a denominator of 24. 5 5 2 10 12 12 2 24    and 3 3 9 8 3 2 3 8 4    Now add. 5 3 10 9 10 9 19 12 8 24 24 24 24       (b) Write each mixed number as an improper fraction. 1 4 8 5 13 45 3 5 4 8    The least common denominator is 8, so write each fraction with a denominator of 8. 45 8 and 13 2 26 4 13 4 2 8    Now add. 13 45 26 45 26 45 4 8 8 8 8 71 7 , or 8 8 8       9. (a) Since 10 2 5   and 4 2 2,   the least common denominator is 2 2 5 20.    Write each fraction with a denominator of 20. 3 3 2 6 10 10 2 20    and 1 1 5 5 4 5 4 20    Now subtract. 3 1 6 5 1 10 20 20 4 20     (b) Write each mixed number as an improper fraction. 8 2 3 1 27 3 3 1 8 2    The least common denominator is 8. Write each fraction with a denominator of 8. 27 8 remains unchanged, and 3 3 4 12 . 2 2 4 8    Now subtract. 3 27 12 27 12 15 2 8 , 8 8 8 8 27       or 1 7 8 N9. (a) Since 11 11  and 9 3 3,   the least common denominator is 3 3 11 99.    Write each fraction with a denominator of 99. 5 5 9 45 11 11 9 99    and 2 2 11 22 9 9 11 99    Now subtract. 5 2 45 22 23 11 9 99 99 99    

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    Chapter R Prealgebra Review 4 (b) Write each mixed number as an improper fraction. 1 5 13 17 4 2 3 6 3 6    The least common denominator is 6. Write each fraction with a denominator of 6. 17 6 remains unchanged, and 13 13 2 26 . 3 3 2 6    Now subtract. 13 17 26 17 26 17 9 3 6 6 6 6 6       Now reduce. 9 3 3 3 6 2 2 , 3     or 1 1 2 10. To find out how many yards of fabric Jen should buy, add the lengths needed for each piece to obtain the total length. The common denominator is 12. 1 2 1 3 8 6 17 1 1 2 1 1 2 4 4 3 2 12 12 12 12       Because 17 5 1 12 12  , we have 17 5 5 4 4 1 5 12 12 12    . Jen should buy 5 5 12 yd of fabric. N10. To find out how long each piece must be, divide the total length by the number of pieces. 1 21 4 21 1 21 10 4 2 2 1 2 4 8 ,       or 5 2 8 Each piece should be 5 2 8 feet long. 11. (a) In the circle graph, the sector for Other is the second largest, so Other had the second largest share of Internet users, 23 . 100 (b) T he total number of Internet users, 3900 million, can be rounded to 4000 million (or 4 billion). Multiply 1 10 by 4000. 1 4000 400 million 10   (c) Multiply the fraction from the graph for Africa by the actual number of users. 1 3900 390 million 10   N11. (a) In the circle graph, the sector for Africa is the smallest, so Africa had the least number of Internet users. (b) T he total number of Internet users, 3900 million, can be rounded to 4000 million (or 4 billion). Multiply 1 2 by 4000. 1 4000 2000 million, 2   or 2 billion (c) Multiply the fraction from the graph for Asia by the actual number of users. 1 3900 1950 million, 2   or 1.95 billion Exercises 1. True; the number above the fraction bar is called the numerator and the number below the fraction bar is called the denominator. 2. True; 5 divides the 31 six times with a remainder of one, so 31 1 6 . 5 5  3. False; this is an improper fraction. Its value is 1. 4. False; the number 1 is neither prime nor composite. 5. False; the fraction 13 39 can be written in lowest terms as 1 3 since 13 13 1 1 . 39 13 3 3     6. False; the reciprocal of 6 3 2  is 2 1 . 6 3  7. False; product refers to multiplication, so the product of 10 and 2 is 20. The sum of 10 and 2 is 12. 8. False; difference refers to subtraction, so the difference between 10 and 2 is 8. The quotient of 10 and 2 is 5. 9. 16 2 8 2 24 3 8 3     Therefore, C is correct.

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    R.1 Fractions 5 10. Simplify each fraction to find which are equal to 5 9 . 3 5 5 3 15 2 9 7 9     6 5 5 6 30 5 9 4 9     2 20 20 2 37 40 74 37     11 5 5 11 55 99 9 9     Therefore, C is correct. 11. A common denominator for p q and r s must be a multiple of both denominators, q and s . Such a number is . q s  Therefore, A is correct. 12. We need to multiply 8 by 3 to get 24 in the denominator, so we must multiply 5 by 3 as well. 5 5 3 15 8 8 3 24     Therefore, B is correct. 13. Since 19 has only itself and 1 as factors, it is a prime number. 14. Since 31 has only itself and 1 as factors, it is a prime number. 15. 2 15 2 3 30 5      Since 30 has factors other than itself and 1, it is a composite number. 16. 50 2 25 2 5 5,      so 50 is a composite number. 17. 2 32 2 2 16 2 2 2 8 2 2 2 2 4 2 2 4 2 2 2 6 2                     Since 64 has factors other than itself and 1, it is a composite number. 18. 27 3 3 9 8 3 3 1 3 3 3          Since 81 has factors other than itself and 1, it is a composite number. 19. As stated in the text, the number 1 is neither prime nor composite, by agreement. 20. The number 0 is not a natural number, so it is neither prime nor composite. 21. 57 3 19,   so 57 is a composite number. 22. 51 3 17,   so 51 is a composite number. 23. Since 79 has only itself and 1 as factors, it is a prime number. 24. Since 83 has only itself and 1 as factors, it is a prime number. 25. 2 62 124 2 2 31,      so 124 is a composite number. 26. 2 69 138 2 3 23,      so 138 is a composite number. 27. 2 250 2 2 125 2 2 5 25 2 2 5 5 0 5 5, 0               so 500 is a composite number. 28. 2 350 2 2 175 2 2 5 35 2 2 5 7 0 5 7, 0               so 700 is a composite number. 29. 2 1729 2 7 247 2 7 13 19 3458          Since 3458 has factors other than itself and 1, it is a composite number. 30. 205 1 5 5 4 0 1 25 5      Since 1025 has factors other than itself and 1, it is a composite number. 31. 8 1 8 1 8 1 1 1 16 2 8 2 8 2 2         32. 1 4 1 4 1 1 1 12 3 4 3 4 3 3 4         33. 15 3 5 3 5 5 5 1 18 3 6 3 6 6 6        

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    Chapter R Prealgebra Review 6 34. 16 4 4 4 4 2 4 4 1 5 4 5 0 5 4 5         35. 90 3 30 3 30 3 1 150 5 30 5 30 5 5 3         36. 100 5 140 7 5 20 5 20 5 1 7 20 7 20 7         37. 18 1 18 1 18 1 1 1 90 5 18 5 18 5 5         38. 16 1 16 1 16 1 1 1 64 4 16 4 4 4 16         39. 144 6 24 6 24 6 6 1 120 5 24 5 24 5 5         40. 132 12 11 12 11 12 12 1 7 11 7 11 7 77 7         41. 1 7 12 7 5 Therefore, 12 5 1 . 7 7  42. 1 9 16 9 7 Therefore, 16 7 1 . 9 9  43. 6 12 77 72 5 Therefore, 77 5 6 . 12 12  44. 6 15 101 90 11 Therefore, 101 11 6 . 15 15  45. 7 11 83 77 6 Therefore, 83 6 7 . 11 11  46. 5 13 67 65 2 Therefore, 67 2 5 . 13 13  47. Multiply the denominator of the fraction by the natural number and then add the numerator to obtain the numerator of the improper fraction. 5 2 10   and 10 3 13   The denominator of the improper fraction is the same as the denominator in the mixed number. Thus, 3 13 2 . 5 5  48. Multiply the denominator of the fraction by the natural number and then add the numerator to obtain the numerator of the improper fraction. 7 5 35   and 35 6 41   The denominator of the improper fraction is the same as the denominator in the mixed number. Thus, 6 41 5 . 7 7  49. Multiply the denominator of the fraction by the natural number and then add the numerator to obtain the numerator of the improper fraction. 8 10 80   and 80 3 83   The denominator of the improper fraction is the same as the denominator in the mixed number. Thus, 3 83 10 . 8 8  50. Multiply the denominator of the fraction by the natural number and then add the numerator to obtain the numerator of the improper fraction. 3 12 36   and 36 2 38   The denominator of the improper fraction is the same as the denominator in the mixed number. Thus, 2 38 12 . 3 3 

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    R.1 Fractions 7 51. Multiply the denominator of the fraction by the natural number and then add the numerator to obtain the numerator of the improper fraction. 5 10 50   and 50 1 51   The denominator of the improper fraction is the same as the denominator in the mixed number. Thus, 1 51 10 . 5 5  52. Multiply the denominator of the fraction by the natural number and then add the numerator to obtain the numerator of the improper fraction. 6 18 108   and 108 1 109   The denominator of the improper fraction is the same as the denominator in the mixed number. Thus, 1 109 18 . 6 6  53. 4 6 4 6 24 5 7 5 7 35      54. 2 5 2 10 7 9 7 5 9 63      55. 3 6 1 15 8 120 2 2 2 3 1 6 15 8 20 6 0          56. 3 5 15 1 20 21 42 3 5 1 15 20 21 28 0 8 15 2          57. 1 12 1 12 1 2 6 6 10 5 10 5 2 5 5 25           58. 1 10 1 10 1 2 5 5 8 7 2 4 7 28 8 7           59. 15 8 15 8 4 25 4 25 3 5 4 2 4 5 5 3 2 5 6 1 , or 1 5 5              60. 21 4 21 4 7 8 7 3 7 4 4 2 7 3 1 , or 8 1 2 2           61. 3 21 3 21 7 1 7 21 3 1 7 3 7 3 1 7 3 3 9 1              62. 4 36 4 36 1 36 4 1 9 4 9 4 1 9 4 4 16 9 9 1              63. Change both mixed numbers to improper fractions. 1 2 13 5 3 1 4 3 4 3 13 5 4 3 65 5 , or 5 12 12        64. Change both mixed numbers to improper fractions. 2 3 3 3 3 8 8 2 1 5 5 8 8 5 64 4 , or 4 15 15        65. Change both mixed numbers to improper fractions. 3 1 19 16 2 3 8 5 8 5 19 16 8 5 19 2 8 8 5 38 3 , or 7 5 5           

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    Chapter R Prealgebra Review 8 66. Change both mixed numbers to improper fractions. 3 1 18 43 3 7 5 6 5 6 18 43 5 6 3 6 43 5 6 3 43 5 129 4 , or 25 5 5              67. Change both numbers to improper fractions. 1 21 5 2 10 10 5 21 1 10 5 2 5 1 21 1 1 2 5 21 , or 10 1 2 1 2 2              68. Change both numbers to improper fractions. 2 38 3 4 9 9 3 38 1 9 3 38 1 3 1 38 1 3 3 3 , o 38 2 3 3 r 12              69. To divide fractions, multiply by the reciprocal of the divisor. 3 3 7 7 2 9 2 9 14 27 7 2 9 3        70. To divide fractions, multiply by the reciprocal of the divisor. 6 5 6 4 1 6 1 4 11 5 2 4 1 5 4 55 1        71. To divide fractions, multiply by the reciprocal of the divisor. 5 3 5 8 4 8 4 3 5 8 4 3 5 4 2 4 3 5 2 3 10 1 , or 3 3 3              72. To divide fractions, multiply by the reciprocal of the divisor. 3 7 10 10 3 7 10 5 3 7 2 5 5 3 7 5 5 14 2 , or 4 3 3            73. To divide fractions, multiply by the reciprocal of the divisor. 32 8 32 15 5 15 5 8 32 15 5 8 8 4 3 5 1 5 8 4 3 12 1               

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    R.1 Fractions 9 74. To divide fractions, multiply by the reciprocal of the divisor. 24 24 21 24 21 7 6 6 7 21 7 6 4 6 3 7 1 7 6 4 3 12 1                75. To divide fractions, multiply by the reciprocal of the divisor. 3 3 1 12 4 4 12 3 1 4 12 3 1 4 3 4 1 1 4 4 16              76. To divide fractions, multiply by the reciprocal of the divisor. 2 2 1 30 30 2 1 5 30 2 1 5 2 15 1 1 5 15 5 5 75              77. To divide fractions, multiply by the reciprocal of the divisor. 3 6 5 6 5 1 3 6 5 1 3 2 3 5 1 3 2 5 10 1              78. To divide fractions, multiply by the reciprocal of the divisor. 4 8 9 8 1 8 9 1 4 2 4 9 1 4 2 9 9 18 1 4              79. Change the first number to an improper fraction, and then multiply by the reciprocal of the divisor. 3 3 27 3 6 4 8 4 8 27 8 4 3 27 8 4 3 3 9 2 4 4 3 9 2 18 1                 80. Change the first number to an improper fraction, and then multiply by the reciprocal of the divisor. 3 7 28 7 5 5 28 10 28 10 5 7 4 10 5 10 7 2 5 5 7 4 2 8 1 5 7                 81. Change both mixed numbers to improper fractions, and then multiply by the reciprocal of the divisor. 1 5 5 12 2 1 2 7 2 7 5 7 2 12 5 7 2 12 35 11 , or 1 24 24         

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    Chapter R Prealgebra Review 10 82. Change both mixed numbers to improper fractions, and then multiply by the reciprocal of the divisor. 2 2 20 7 2 1 5 20 5 9 7 9 5 100 3 9 5 20 9 7 63 7 , or 1 63          83. Change both mixed numbers to improper fractions, and then multiply by the reciprocal of the divisor. 5 15 21 47 2 1 21 32 21 32 8 47 21 8 4 8 47 21 4 47 84 37 , or 8 32 8 32 8 1 7 47 47 4                84. Change both mixed numbers to improper fractions, and then multiply by the reciprocal of the divisor. 3 4 23 9 2 1 23 5 10 9 23 5 2 5 9 23 5 , or 1 8 10 5 10 5 18           85. 7 4 7 4 11 15 15 15 15     86. 2 5 2 5 7 9 9 9 9     87. 7 1 7 1 12 12 12 8 12 2 4 3 4 2 3         88. 5 3 5 1 16 16 1 3 8 6 16 2      89. Since 9 3 3,   and 3 is prime, the LCD (least common denominator) is 3 3 9.   1 1 3 3 3 3 3 9    Now add the two fractions with the same denominator. 5 1 5 3 8 9 3 9 9 9     90. To add 4 15 and 1 , 5 first find the LCD. Since 15 3 5   and 5 is prime, the LCD is 15. 4 1 4 1 3 5 5 3 4 3 15 15 15 15 4 3 7 15 15          91. Since 8 2 2 2    and 6 2 3,   the LCD is 2 2 2 24. 3     3 3 3 9 8 8 3 24    and 5 5 4 20 6 6 4 24    Now add fractions with the same denominator. 3 5 9 20 29 8 6 24 24 4 , 2     or 5 1 24 92. Since 6 2 3   and 9 3 3,   the LCD is 2 3 3 18.    5 5 3 3 15 6 6 18    and 2 2 2 9 9 2 4 18    Now add fractions with the same denominator. 5 15 , 6 9 18 18 2 4 19 18     or 1 1 18

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    R.1 Fractions 11 93. Since 9 3 3   and 16 4 4,   the LCD is 3 3 4 4 144.     16 80 9 9 1 5 5 6 144    and 3 3 9 27 16 16 9 144    Now add fractions with the same denominator. 5 3 80 27 9 16 144 144 107 144     94. Since 4 2 2   and 25 5 5,   the LCD is 2 2 5 5 100.     3 3 25 75 4 4 25 100    and 6 6 4 24 25 25 4 100    Now add fractions with the same denominator. 3 6 75 24 99 4 25 100 0 100 1 0     95. 1 1 24 1 25 3 3 8 8 8 8 8 1 1 8 1 9 2 2 4 4 4 4 4 1 1 25 9 3 2 8 4 8 4              Since 8 2 2 2    and 4 2 2,   the LCD is 2 2 2   or 8. 1 1 25 9 2 3 2 8 4 8 4 2 25 18 8 8 43 3 , or 5 8 8        96. 2 12 2 14 4 4 1 1 12 1 13 2 2 2 3 3 3 3 3 6 6 6 6 6           Since 6 2 3,   the LCD is 6. 3 6 2 1 14 2 13 4 6 2 6 3 2 6 28 13 41 5 , 6 6 or 6        97. 1 1 12 1 13 3 3 4 4 4 4 4 4 4 5 4 9 1 1 5 5 5 5 5           Since 4 2 2,   and 5 is prime, the LCD is 2 2 5 20.    1 4 13 5 9 4 3 1 4 5 4 5 5 4 65 36 20 20 101 1 , or 5 20 20         98. To add 5 3 4 and 1 1 , 3 first change to improper fractions then find the LCD, which is 12. 3 1 23 5 1 3 3 23 3 4 4 4 3 3 4 69 16 12 1 , or 7 12 4 4 4 12 85 12           99. 7 2 7 2 5 9 9 9 9     100. 8 3 8 3 1 11 11 5 1 11     101. 13 3 13 3 15 15 15 10 15 2 5 2 3 5 3         102. 11 3 11 3 12 12 12 12 2 4 4 8 2 3 3         103. Since 12 4 3   (12 is a multiple of 3), the LCD is 12. 1 4 4 3 4 12   Now subtract fractions with the same denominator. 7 1 7 4 3 1 3 1 12 3 12 12 12 4 3 4        

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