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Solution Manual for Algebra for College Students, 9th Edition

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    S OLUTIONS M ANUAL E MILY K EATON A LGEBRA FOR C OLLEGE S TUDENTS N INTH E DITION Margaret L. Lial American River College John Hornsby University of New Orleans Terry McGinnis

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    Table of Contents R Review of the Real Number System....................................................................................................... 1 R.1 R.2 R.3 R.4 R.5 Fractions, Decimals, and Percents •................................................................................... Basic Concepts from Algebra • ......................................................................................... Operations on Real Numbers • .......................................................................................... Exponents, Roots, and Order of Operations • .................................................................... Properties of Real Numbers • ............................................................................................ Chapter R Test • ................................................................................................................ 1 19 24 30 38 42 1 Linear Equations, Inequalities, and Applications ............................................................................. 45 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Linear Equations in One Variable • ................................................................................... Formulas and Percent • ...................................................................................................... Applications of Linear Equations • .................................................................................... Further Applications of Linear Equations • ....................................................................... Summary Exercises Applying Problem-Solving Techniques • ........................................... Linear Inequalities in One Variable • ................................................................................ Set Operations and Compound Inequalities • .................................................................... Absolute Value Equations and Inequalities •..................................................................... Summary Exercises Solving Linear and Absolute Value Equations and Inequalities • .................................................................................................... Chapter 1 Review Exercises •............................................................................................ Chapter 1 Mixed Review Exercises • ................................................................................ Chapter 1 Test • ................................................................................................................. Chapters R and 1 Cumulative Review Exercises• ............................................................. 45 55 66 79 88 92 105 113 128 132 138 141 144 2 Linear Equations, Graphs, and Functions ........................................................................................147 2.1 2.2 2.3 2.4 2.5 2.6 Linear Equations in Two Variables • ................................................................................. The Slope of a Line • ......................................................................................................... Writing Equations of Lines • ............................................................................................. Summary Exercises Finding Slopes and Equations of Lines • ........................................... Linear Inequalities in Two Variables • .............................................................................. Introduction to Relations and Functions • ......................................................................... Function Notation and Linear Functions • ......................................................................... Chapter 2 Review Exercises •............................................................................................ Chapter 2 Mixed Review Exercises • ................................................................................ Chapter 2 Test • ................................................................................................................. Chapters R  2 Cumulative Review Exercises • .................................................................. 147 161 175 189 192 202 209 217 223 224 227

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    3 Systems of Linear Equations ..............................................................................................................231 3.1 3.2 3.3 Systems of Linear Equations in Two Variables • .............................................................. Systems of Linear Equations in Three Variables • ............................................................ Applications of Systems of Linear Equations • ................................................................. Chapter 3 Review Exercises •............................................................................................ Chapter 3 Mixed Review Exercises • ................................................................................ Chapter 3 Test • ................................................................................................................. Chapters R  3 Cumulative Review Exercises • .................................................................. 231 252 272 290 297 299 303 4 Exponents, Polynomials, and Polynomial Functions........................................................................307 4.1 4.2 4.3 4.4 4.5 4.6 Integer Exponents • ........................................................................................................... Scientific Notation • .......................................................................................................... Adding and Subtracting Polynomials • .............................................................................. Polynomial Functions, Graphs, and Composition • ........................................................... Multiplying Polynomials • ................................................................................................. Dividing Polynomials •...................................................................................................... Chapter 4 Review Exercises •............................................................................................ Chapter 4 Mixed Review Exercises • ................................................................................ Chapter 4 Test • ................................................................................................................. Chapters R  4 Cumulative Review Exercises • .................................................................. 307 316 321 328 341 352 364 370 371 374 5 Factoring ..............................................................................................................................................379 5.1 5.2 5.3 5.4 5.5 Greatest Common Factors and Factoring by Grouping • .................................................... Factoring Trinomials •......................................................................................................... Special Factoring • .............................................................................................................. A General Approach to Factoring • ..................................................................................... Solving Quadratic Equations Using the Zero-Factor Property • ......................................... Chapter 5 Review Exercises • ............................................................................................. Chapter 5 Mixed Review Exercises • .................................................................................. Chapter 5 Test • ................................................................................................................... Chapters R  5 Cumulative Review Exercises • ................................................................... 379 385 394 402 410 422 427 428 430

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    6 Rational Expressions and Functions .................................................................................................434 6.1 6.2 6.3 6.4 6.5 6.6 Rational Expressions and Functions; Multiplying and Dividing • ...................................... Adding and Subtracting Rational Expressions •.................................................................. Complex Fractions • ............................................................................................................ Equations with Rational Expressions and Graphs • ............................................................ Summary Exercises Simplifying Rational Expressions vs. Solving Rational Equations • ................................................................................................ Applications of Rational Expressions • ............................................................................... Variation • ........................................................................................................................... Chapter 6 Review Exercises • ............................................................................................. Chapter 6 Mixed Review Exercises • .................................................................................. Chapter 6 Test • ................................................................................................................... Chapters R  6 Cumulative Review Exercises • ................................................................... 434 443 457 468 483 487 504 514 520 523 527 7 Roots, Radicals, and Root Functions .................................................................................................532 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Radical Expressions and Graphs •....................................................................................... Rational Exponents • ........................................................................................................... Simplifying Radicals, the Distance Formula, and Circles • ................................................ Adding and Subtracting Radical Expressions • ................................................................... Multiplying and Dividing Radical Expressions • ................................................................ Summary Exercises Performing Operations with Radicals and Rational Exponents • ................................................................................................ Solving Equations with Radicals • ...................................................................................... Complex Numbers • ............................................................................................................ Chapter 7 Review Exercises • ............................................................................................. Chapter 7 Mixed Review Exercises • .................................................................................. Chapter 7 Test • ................................................................................................................... Chapters R  7 Cumulative Review Exercises • ................................................................... 532 539 547 559 567 579 581 594 602 609 611 614

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    8 Quadratic Equations and Inequalities...............................................................................................619 8.1 8.2 8.3 8.4 8.5 The Square Root Property and Completing the Square • ..................................................... The Quadratic Formula • ...................................................................................................... Equations That Lead to Quadratic Methods • ...................................................................... Summary Exercises Applying Methods for Solving Quadratic Equations • ......................... Formulas and Further Applications • ................................................................................... Polynomial and Rational Inequalities • ................................................................................ Chapter 8 Review Exercises • .............................................................................................. Chapter 8 Mixed Review Exercises • ................................................................................... Chapter 8 Test • .................................................................................................................... Chapters R  8 Cumulative Review Exercises • .................................................................... 619 634 649 674 678 692 713 721 724 728 9 Additional Graphs of Functions and Relations ................................................................................735 9.1 9.2 9.3 9.4 9.5 Review of Operations and Composition • ......................................................................... Graphs of Quadratic Functions • ....................................................................................... More About Parabolas and Their Appications • ................................................................ Symmetry; Increasing and Decreasing Functions • ........................................................... Piecewise Linear Functions • ............................................................................................ Chapter 9 Review Exercises • ........................................................................................... Chapter 9 Mixed Review Exercises • ................................................................................ Chapter 9 Test • ................................................................................................................. Chapters R  9 Cumulative Review Exercises • ................................................................. 735 748 758 773 783 795 801 803 806 10 Inverse, Exponential, and Logarithmic Functions ...........................................................................812 10.1 10.2 10.3 10.4 10.5 10.6 Inverse Functions • ............................................................................................................ Exponential Functions • .................................................................................................... Logarithmic Functions • .................................................................................................... Properties of Logarithms • ................................................................................................ Common and Natural Logarithms • .................................................................................. Exponential and Logarithmic Equations; Further Applications • ...................................... Chapter 10 Review Exercises • ......................................................................................... Chapter 10 Mixed Review Exercises • .............................................................................. Chapter 10 Test • ............................................................................................................... Chapters R  10 Cumulative Review Exercises • ............................................................... 812 823 833 844 850 856 867 874 876 879

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    11 Polynomial and Rational Functions...................................................................................................886 11.1 11.2 11.3 11.4 Zeros of Polynomial Functions (I) • .................................................................................. Zeros of Polynomial Functions (II) •................................................................................. Graphs and Applications of Polynomial Functions • ........................................................ Summary Exercises Examining Polynomial Functions and Graphs • ............................... Graphs and Applications of Rational Functions • ............................................................. Chapter 11 Review Exercises • ......................................................................................... Chapter 11 Mixed Review Exercises • .............................................................................. Chapter 11 Test • ............................................................................................................... Chapters R  11 Cumulative Review Exercises • ............................................................... 886 892 904 925 931 949 957 958 961 12 Conic Sections and Nonlinear Systems..............................................................................................968 12.1 12.2 12.3 12.4 Circles Revisited and Ellipses • ........................................................................................ Hyperbolas and Functions Defined by Radicals •............................................................. Nonlinear Systems of Equations • .................................................................................... Second-Degree Inequalities, Systems of Inequalities, and Linear Programming • ........... Chapter 12 Review Exercises • ......................................................................................... Chapter 12 Mixed Review Exercises • ............................................................................. Chapter 12 Test • .............................................................................................................. Chapters R  12 Cumulative Review Exercises • ............................................................... 968 977 986 1000 1021 1028 1030 1034 13 Further Topics in Algebra ................................................................................................................1041 13.1 13.2 13.3 13.4 13.5 13.6 13.7 Sequences and Series • ..................................................................................................... Arithmetic Sequences • .................................................................................................... Geometric Sequences •..................................................................................................... The Binomial Theorem • .................................................................................................. Mathematical Induction • ................................................................................................. Counting Theory • ............................................................................................................ Basics of Probability • ...................................................................................................... Chapter 13 Review Exercises • ........................................................................................ Chapter 13 Mixed Review Exercises • ............................................................................. Chapter 13 Test • .............................................................................................................. Chapters R  13 Cumulative Review Exercises • .............................................................. 1041 1047 1057 1069 1075 1086 1093 1100 1106 1108 1112

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    R.1 Fractions, Decimals, and Percents 1 Chapter R Review of the Real Number System R.1 Fractions, Decimals, and Percents Classroom Examples, Now Try Exercises 1. (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       N1. (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       2. 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  N2. 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  3. 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  N3. 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  4. 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          N4. 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          5. (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         

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    Chapter R Review of the Real Number System 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       N5. (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           6. (a) 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         (b) 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     (c) 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     (d) 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 N6. (a) 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         (b) 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      

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    R.1 Fractions, Decimals, and Percents 3 (c) 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     (d) 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 7. (a) 15 0.15 100  (b) 9 0.009 1000  (c) 5 25 2.5 2 10 10   N7. (a) 8 0.8 10  (b) 431 0.431 1000  (c) 58 258 2.58 2 100 100   8. (a) 42.830 71.000 3.074 116.904  (b) 32.50 21.72 10.78  N8. (a) 68.900 42.720 8.973 120.593  (b) 351.800 2.706 349.094  9. (a) 30.2 1 decimal place 0.052 3 decimal places 604 1510 1 3 4 1.5704 4 decimal places     (b) 0.06 2 decimal places 0.12 2 decimal places 12 6 2 2 4 0.0072 4 decimal places     (c) To change the divisor 0.37 into a whole number, move each decimal point two places to the right. Move the decimal point straight up and divide as with whole numbers. 14.8 37 547.6 37 177 148 296 296 0 Therefore, 5.476 0.37 14.8.   (d) To change the divisor 3.1 into a whole number, move each decimal point one place to the right. Move the decimal point straight up and divide as with whole numbers. 1.21 31 37.60 31 66 62 40 31 9

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    Chapter R Review of the Real Number System 4 We carried out the division to 2 decimal places so that we could round to 1 decimal place. Therefore, 3.76 3.1 1.2.   N9. (a) 9.32 2 decimal places 1.4 1 decimal place 3728 932 2 1 3 13.048 3 decimal places     (b) 0.6 1 decimal place 0.004 3 decimal places 24 1 3 4 0.0024 4 decimal places    (c) To change the divisor 1.3 into a whole number, move each decimal point one place to the right. Move the decimal point straight up and divide as with whole numbers. 5.641 13 73.340 65 83 78 54 52 20 13 7 We carried out the division to 3 decimal places so that we could round to 2 decimal places. Therefore, 7.334 1.3 5.64.   10. (a) Move the decimal point three places to the right. 19.5 1000 19,500   (b) Move the decimal point one place to the left. 960.1 10 96.01   N10. (a) Move the decimal point one place to the right. 294.72 10 2947.2   (b) Move the decimal point two places to the left. Insert a 0 in front of the 4 to do this. 4.793 100 0.04793   11. (a) Divide 3 by 50. Add a decimal point and as many 0s as necessary. 0.06 50 3.00 3 00 0 Therefore, 3 0.06. 50  (b) Divide 11 by 1. Add a decimal point and as many 0s as necessary. 0.090909... 11 1.000000... 99 100 99 100 99 1 Note that the pattern repeats. Therefore, 1 0.09, 11  or about 0.091. N11. (a) Divide 20 by 17. Add a decimal point and as many 0s as necessary. 0.85 20 17.00 160 100 100 0 Therefore, 17 0.85. 20  (b) Divide 2 by 9. Add a decimal point and as many 0s as necessary. 0.222... 9 2.000... 18 20 18 20 18 2 Note that the pattern repeats. Therefore, 2 0.2, 9  or 0.222.

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    R.1 Fractions, Decimals, and Percents 5 12. (a) 85% 0.85  (b) 110% 1.10,  or 1.1 (c) 0.30 30%  (d) 0.165 16.5%  N12. (a) 52% 0.52  (b) 2% 02% 0.02   (c) 0.45 45%  (d) 3.5 3.50 350%   13. (a) 65 65% 100  In lowest terms, 65 13 5 13 100 20 5 20     (b) 1.5 1.5 10 15 3 1.5% 100 100 10 1000 200      N13. (a) 20 20% 100  In lowest terms, 20 1 20 1 100 5 20 5     (b) 160 160% 100  In lowest terms, 160 8 20 8 3 , or 1 100 5 20 5 5     14. (a) 3 3 100% 50 50 3 100 % 50 1 3 50 2 % 50 6%         (b) 1 1 100% 3 3 1 100 % 3 1 100 % 3 1 33 %, or 33.3% 3       N14. (a) 6 6 100% 25 25 6 100 % 25 1 6 25 4 % 25 24%         (b) 7 7 100% 9 9 7 100 % 9 1 700 % 9 7 77 %, or 77.7% 9       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 reciprocal of 6 3 2  is 2 1 . 6 3  5. 16 2 8 2 24 3 8 3     Therefore, C is correct. 6. 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. 7. 8 1 8 1 8 1 1 1 16 2 8 2 8 2 2        

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    Chapter R Review of the Real Number System 6 8. 1 4 1 4 1 1 1 12 3 4 3 4 3 3 4         9. 15 3 5 3 5 5 5 1 18 3 6 3 6 6 6         10. 16 4 4 4 4 2 4 4 1 5 4 5 0 5 4 5         11. 90 3 30 3 30 3 1 150 5 30 5 30 5 5 3         12. 100 5 140 7 5 20 5 20 5 1 7 20 7 20 7         13. 18 1 18 1 18 1 1 1 90 5 18 5 18 5 5         14. 16 1 16 1 16 1 1 1 64 4 16 4 4 4 16         15. 144 6 24 6 24 6 6 1 120 5 24 5 24 5 5         16. 132 12 11 12 11 12 12 1 7 11 7 11 7 77 7         17. 1 7 12 7 5 Therefore, 12 5 1 . 7 7  18. 1 9 16 9 7 Therefore, 16 7 1 . 9 9  19. 6 12 77 72 5 Therefore, 77 5 6 . 12 12  20. 5 13 67 65 2 Therefore, 67 2 5 . 13 13  21. 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  22. 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  23. 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  24. 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  25. 4 6 4 6 24 5 7 5 7 35      26. 2 5 2 10 7 9 7 5 9 63      27. 3 6 1 15 8 120 2 2 2 3 1 6 15 8 20 6 0          28. 3 5 15 1 20 21 42 3 5 1 15 20 21 28 0 8 15 2         

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    R.1 Fractions, Decimals, and Percents 7 29. 1 12 1 12 1 2 6 6 10 5 10 5 2 5 5 25           30. 1 10 1 10 1 2 5 5 8 7 2 4 7 28 8 7           31. 15 8 15 8 4 25 4 25 3 5 4 2 4 5 5 3 2 5 6 1 , or 1 5 5              32. 21 4 21 4 7 8 7 3 7 4 4 2 7 3 1 , or 8 1 2 2           33. 3 21 3 21 7 1 7 21 3 1 7 3 7 3 1 7 3 3 9 1              34. 4 36 4 36 1 36 4 1 9 4 9 4 1 9 4 4 16 9 9 1              35. 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        36. 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        37. To divide fractions, multiply by the reciprocal of the divisor. 3 3 7 7 2 9 2 9 14 27 7 2 9 3        38. 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        39. 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              40. 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           

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    Chapter R Review of the Real Number System 8 41. 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                42. 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                43. 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              44. 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              45. 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              46. 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              47. 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                 48. 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                

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    R.1 Fractions, Decimals, and Percents 9 49. 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          50. 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          51. 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                52. 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           53. 7 4 7 4 11 15 15 15 15     54. 2 5 2 5 7 9 9 9 9     55. 7 1 7 1 12 12 12 8 12 2 4 3 4 2 3         56. 5 3 5 1 16 16 1 3 8 6 16 2      57. 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     58. 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          59. 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 60. 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   

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