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Solution Manual for Pathways to College Mathematics, 2nd Edition

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    S OLUTIONS M ANUAL D ANIEL S. M ILLER Niagara County Community College P ATHWAYS TO C OLLEGE M ATHEMATICS S ECOND E DITION Robert Blitzer Miami Dade College

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    i Table of Contents Chapter P Prealgebra Pathways ..............................................................................................1 Chapter 1 Numerical Pathways ............................................................................................77 Chapter 2 Algebraic Pathways: Equations and Inequalities .................................................................................143 Chapter 3 Algebraic Pathways: Graphs, Functions, Linear Functions, and Linear Systems ...............................215 Chapter 4 Algebraic Pathways: Polynomials, Quadratic Equations, and Quadratic Functions ...........................295 Chapter 5 Geometric Pathways: Measurement ......................................................................................................373 Chapter 6 Geometric Pathways ..........................................................................................393 Chapter 7 Pathways to Probability: Counting Methods and Probability Theory ........................................................435 Chapter 8 Statistical Pathways ...........................................................................................491

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    Chapter P Prealgebra Pathways 1 Check Points P.1 1. a. millions place b. ones place c. hundreds place d. millions place 2. Work from left to right. Write the name of the three-digit number in each period, followed by the name of the period and a comma. Do not write the name of the last period, “ones.” a. twenty-seven thousand, one hundred forty-three b. five hundred twenty-one million, six hundred thirty thousand, fifty-seven 3. a. Begin by noting how to write the number within each period. 53 406 fifty-three thousand, four hundred six      Write the digits for the number in each period, followed by a comma. The standard form for the number is 53,406. b. Begin by noting how to write the number within each period. 204 932 016 two hundred four million, nine hundred thirty-two thousand, sixteen          Write the digits for the number in each period, followed by a comma. The standard form for the number is 204,932,016.

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    Chapter P Prealgebra Pathways 2 4. a. The place value chart shows that 704,663 contains 7 hundred-thousands, 0 ten-thousands, 4 thousands, 6 hundreds, 6 tens, and 3 ones. Thus 704,663 is written in expanded form as follows: 704,663 700, 000 4000 600 60 3.      b. The place value chart shows that 49,063,400 contains 4 ten-millions, 9 millions, 0 hundred-thousands, 6 ten- thousands, 3 thousands, 4 hundreds, 0 tens, and 0 ones. Thus 49,063,400 is written in expanded form as follows: 49, 063, 400 40, 000, 000 9, 000, 000 60, 000 3000 400.      5. a. 14 5  because 14 is to the right of 5 on the number line. b. 0 16  because 0 is to the left of 16 on the number line. 6. a. The digit to the right of the thousands digit is 4, which is less than 5. This indicates to leave the thousands digit the same. Replace all digits to the right with zeros. 57, 498 57, 000  b. The digit to the right of the hundred-thousands digit is 5. This indicates to add one to the hundred-thousands digit. Replace all digits to the right with zeros. 4,856,902 4,900, 000  c. The digit to the right of the thousands digit is 6, which is greater than 5. This implies to add one to the thousands digit. Replace all digits to the right with zeros. 9602 10, 000  d. The digit to the right of the millions digit is 2, which is less than 5. This implies to leave the millions digit the same. Replace all digits to the right with zeros. 684, 236, 042 684, 000, 000  7. a. The digit to the right of the billions digit is 5. This implies to add one to the billions digit. Replace all digits to the right with zeros. 7,599, 445,183 8, 000,000, 000  b. The digit to the right of the ten-thousands digit is 5. This implies to add one to the ten-thousands digit. Replace all digits to the right with zeros. 7,599, 445,183 7,599, 450, 000  c. The digit to the right of the thousands digit is 1, which is less than 5. This implies to leave the thousands digit the same. Replace all digits to the right with zeros. 7,599, 445,183 7,599, 445, 000  8. a. The cost of a coronary bypass in the United States is $67,583 b. The country with the least amount in the CT scan column is India. The average cost for this procedure in India is $43. c. The charge for an appendectomy in Chile is $5509. The countries in which an appendectomy costs more than in Chile are Canada, Switzerland, and United States. 9. a. We begin with the number of marriages between an African-American husband and a white wife in 2010. Look at the bars labeled with the year 2010. The yellow bar to the right represents the number of marriages between an African-American husband and a white wife. The number above this bar is 390, representing 390 thousand. Thus, in 2010, there were 390,000 marriages between an African-American husband and a white wife. b. Look for the red bar labeled 61 (for 61 thousand, or 61,000). This is the bar to the left for the year labeled 1990. Thus, in 1990, there were 61,000 marriages between a white husband and an African-American wife.

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    Section P.1 Whole Numbers 3 Concept and Vocabulary Check P.1 1. whole; 0 2. standard 3. periods 4. millions ; hundred-thousands; thousands; tens 5. millions; forty-two; nine 6. expanded; 5000; 60; 8 7. number line 8. < 9. > 10. 8; 5; add 1; 9,000,000 11. 2; 3; do not change; 8,542,000 Exercise Set P.1 Note that exercises #1 - 22 use the following table: 1. hundreds 2. hundreds 3. ones 4. ones 5. hundred-thousands 6. hundred-thousands 7. millions 8. millions 9. ten-millions 10. ten-millions 11. hundred-millions 12. hundred-millions 13. two hundred fifty-eight 14. three hundred twenty-four 15. eight thousand, three hundred seventy-six 16. six thousand, two hundred twenty-six 17. thirty-six thousand, eight hundred eighty 18. fifty-two thousand, seven 19. seven million, five hundred sixty-six thousand 20. four million, three hundred two thousand 21. thirty-five million, two hundred sixty thousand, three hundred seventy-five 22. fifty-seven million, forty-four thousand, two hundred eight 23. The standard form is 3468. 24. The standard form is 5283. 25. The standard form is 86,500. 26. The standard form is 58,004. 27. The standard form is 16,402,012. 28. The standard form is 14,204,015. 29. The standard form is 9,000,009. 30. The standard form is 5,000,005. 31. The standard form is 26,034,203. 32. The standard form is 52,028,706. 33. The standard form is 620,595. 34. The standard form is 430,696. 35. The expanded form is 600 40 3.   36. The expanded form is 500 70 2.   37. The expanded form is 5000 40 6.   38. The expanded form is 3000 50 7.  

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    Chapter P Prealgebra Pathways 4 39. The expanded form is 80, 000 1000 300 60 4.     40. The expanded form is 70, 000 2000 500 40 6.     41. The expanded form is 50, 000 5000 30 8.    42. The expanded form is 40, 000 4000 20 9.    43. The expanded form is 20, 000, 000 8, 000, 000 600, 000 40, 000.    44. The expanded form is 50,000, 000 6, 000, 000 300, 000 7000 30 2.      45. 9 3  because 9 is to the right of 3 on the number line. 46. 7 2  because 7 is to the right of 2 on the number line. 47. 0 14  because 0 is to the left of 14 on the number line. 48. 0 16  because 0 is to the left of 16 on the number line. 49. 3600 4500  because 36000 is to the left of 4500 on the number line. 50. 2300 3200  because 2300 is to the left of 3200 on the number line. 51. 200, 000 20, 000  because 200,000 is to the right of 20,000 on the number line. 52. 300, 000 30,000  because 300,000 is to the right of 30,000 on the number line. 53. 624 rounded to the nearest ten is 620. 54. 372 rounded to the nearest ten is 370. 55. 627 rounded to the nearest ten is 630. 56. 378 rounded to the nearest ten is 380. 57. 4891 rounded to the nearest hundred is 4900. 58. 5482 rounded to the nearest hundred is 5500. 59. 4831 rounded to the nearest hundred is 4800. 60. 5432 rounded to the nearest hundred is 5400. 61. 61,529 rounded to the nearest thousand is 62,000. 62. 72,503 rounded to the nearest thousand is 73,000. 63. 61,129 rounded to the nearest thousand is 61,000. 64. 72,103 rounded to the nearest thousand is 72,000. 65. 24,628 rounded to the nearest ten-thousand is 20,000. 66. 34,628 rounded to the nearest ten-thousand is 30,000. 67. 345,207 rounded to the nearest ten-thousand is 350,000.

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    Section P.1 Whole Numbers 5 68. 645,308 rounded to the nearest ten-thousand is 650,000. 69. 86,609,100 rounded to the nearest million is 87,000,000. 70. 75,809,100 rounded to the nearest million is 76,000,000. 71. 86,409,100 rounded to the nearest million is 86,000,000. 72. 75,309,100 rounded to the nearest million is 75,000,000. 73. 86,609,100 rounded to the nearest ten-million is 90,000,000. 74. 75,809,100 rounded to the nearest million is 80,000,000. 75. ninety-two quadrillion, two hundred thirty-three trillion, seven hundred twenty billion, three hundred sixty-eight million, five hundred forty-seven thousand, eight hundred. 76. ten-quadrillions 77. 700, 000, 000, 000 20, 000, 000, 000  78. 90,000, 000, 000, 000, 000 2, 000, 000,000, 000, 000  79. 92,233,720,368,547,800 rounded to the nearest ten-quadrillion is 90,000,000,000,000,000. The word name is ninety quadrillion. 80. 92,233,720,368,547,800 rounded to the nearest quadrillion is 92,000,000,000,000,000. The word name is ninety-two quadrillion. 81. The greatest yearly earnings are for males with a Doctorate which is $131,569. In word form this is one hundred thirty- one thousand, five hundred sixty-nine dollars. 82. The least yearly earnings are for females with a 9 th to 12 th grade, non-graduate which is $16,812. In word form this is sixteen thousand, eight hundred twelve dollars. 83. Women with bachelor’s degree earns $50,856. In word form this is fifty thousand, eight hundred fifty-six dollars. 84. Men with associate degree earns $51,865. In word form this is fifty-one thousand, eight hundred sixty-five dollars. 85. Men with bachelor’s degree will earns $79,927 which would round to $80,000. In word form this is seventy-nine thousand, nine hundred twenty-seven dollars. 86. Women with master’s degree will earns $64,834 which would round to $60,000. In word form this is sixty-four thousand, eight hundred thirty-four dollars. 87. In 1960 there were only 9,700,000 immigrants. 88. In 2016 there were 43,700,000 immigrants. 89. In 1900 there were only 10,300,000 immigrants. In 1940 there were only 11,600,000 immigrants. In 1960 there were only 9,700,000 immigrants. 90. In 1980 there were 14,100,000 immigrants. In 2000 there were 31,100,000 immigrants. In 2016 there were 43,700,000 immigrants. 91. In 1980 there were 14,100,000 immigrants.

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    Chapter P Prealgebra Pathways 6 92. In 2000 there were 31,100,000 immigrants. 93. 2,376,206; two million, three hundred seventy-six thousand, two hundred six 94. 1,857,160; one million; eight hundred fifty-seven thousand, one hundred sixty 95. Williams 96. Brown and Jones 97. 1,857,160 rounded to the nearest hundred-thousand is 1,900,000. 98. 2,376,206 rounded to the nearest hundred-thousand is 2,400,000. 99. The 3 is in the ten-thousands place. 100. The 8 is in the ten-thousands place. 101. 1,380,145 < 1,534,042 102. 1,857,160 > 1,380,145 103. two thousand, four hundred fifty-three 104. two hundred two thousand, twenty-two 105. 102,063 106. 12,042 107. – 117. Answers will vary. 118. does not make sense; Explanations will vary. Sample explanation: Adding one to this number would create a bigger number. 119. makes sense 120. makes sense 121. makes sense 122. true 123. false; Changes to make the statement true will vary. A sample change is: The number 32,864 is written in standard form. 124. false; Changes to make the statement true will vary. A sample change is: When rounding whole numbers, the digit to be rounded either stays the same or increases by 1. 125. false; Changes to make the statement true will vary. A sample change is: When comparing numbers of various items, tables are just as effective as bar graphs. 126. The whole numbers from 10 to 40 would be rounded to 10 or 20 or 30 or 40. So there are four different rounded numbers.

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    Section P.2 Fractions and Mixed Numbers 7 Check Points P.2 1. a. 3 10 Numerator is 3. Denominator is 10. b. 10 3 Numerator is 10. Denominator is 3. c. 180   Numerator is .  Denominator is 180 .  2. a. There are 7 parts shaded out of a total 10 equal parts. Thus, the fraction 7 10 (seven-tenths) represents the shaded portion of the figure. b. There are 2 parts shaded out of a total 5 equal parts. Thus, the fraction 2 5 (two-fifths) represents the shaded portion of the figure. c. There are 13 parts shaded out of a total 16 equal parts. Thus, the fraction 13 16 (thirteen-sixteenths) represents the shaded portion of the figure. 3. Fraction of fatal gun accidents with children between 15 and 34 is number of fatal gun accidents with children between 15 and 34 . total number of fatal gun accidents 145 107 606 252 606    4. a. 3 10 Because the numerator is less than the denominator, 3 10 is a proper fraction. b. 10 10 Because the numerator and denominator are equal, 10 10 is an improper fraction. c. 10 3 Because the numerator is greater than the denominator, 10 3 is an improper fraction. 5. a. Improper Fraction: Each whole object is divided into 3 equal parts, or thirds. A total of 5 of the thirds are shaded. The improper fraction 5 3 represents the shaded portion of the group of figures. Mixed Number: The shaded portions include 1 whole object and 2 3 of a second object. The mixed number 2 1 3 represents the shaded portion of the group of figures.

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    Chapter P Prealgebra Pathways 8 b. Improper Fraction: Each whole object is divided into 4 equal parts, or fourths. A total of 9 of the fourths are shaded. The improper fraction 9 4 represents the shaded portion of the group of figures. Mixed Number: The shaded portions include 2 whole object and 1 4 of a second object. The mixed number 1 2 4 represents the shaded portion of the group of figures. 6. a. 5 8 2 5 16 5 21 2 8 8 8 8       b. 7 16 12 7 192 7 199 12 16 16 16 16       7. a. 5 3 1 3 5 3 2 Write the mixed number using remainder quotient . original denominator Thus, 5 2 1 . 3 3  b. 86 7 12 7 86 7 16 1 2 2 4 Write the mixed number using remainder quotient . original denominator Thus, 86 2 12 . 7 7  c. 513 19 27 19 513 38 133 3 20 13 0 Write the mixed number using remainder quotient . original denominator Thus, 513 27. 19  Concept and Vocabulary Check P.2 1. numerator; denominator 2. 3 3; 8; 8 3. proper; improper 4. mixed 5. 5; 3; 2; 5

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    Section P.2 Fractions and Mixed Numbers 9 6. 2; 1 Exercise Set P.2 1. 5 13 Numerator is 5. Denominator is 13. 2. 7 19 Numerator is 7. Denominator is 19. 3. 13 5 Numerator is 13. Denominator is 5. 4. 19 7 Numerator is 19. Denominator is 7. 5. 60  Numerator is  . Denominator is 60. 6. 180  Numerator is  . Denominator is 180. 7. There are 3 parts shaded out of a total 5 equal parts. Thus, the fraction 3 5 represents the shaded portion of the figure. 8. There are 2 parts shaded out of a total 5 equal parts. Thus, the fraction 2 5 represents the shaded portion of the figure. 9. There is 1 part shaded out of a total 6 equal parts. Thus, the fraction 1 6 represents the shaded portion of the figure. 10. There are 2 parts shaded out of a total 6 equal parts. Thus, the fraction 2 6 represents the shaded portion of the figure. 11. There are 7 parts shaded out of a total 16 equal parts. Thus, the fraction 7 16 represents the shaded portion of the figure. 12. There are 11 parts shaded out of a total 16 equal parts. Thus, the fraction 11 16 represents the shaded portion of the figure. 13. a. Probability of landing on a red region red region 3 . total regions 10   b. Probability of landing on a blue region b lue region 2 . total regions 10   c. Red is more likely.

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    Chapter P Prealgebra Pathways 10 14. a. Probability of landing on a brown region brown region 3 . total regions 10   b. Probability of landing on a yellow region yellow region 2 . total regions 10   c. Brown is more likely. 15. 9 11 Because the numerator is less than the denominator, 9 11 is a proper fraction. 16. 8 13 Because the numerator is less than the denominator, 8 13 is a proper fraction. 17. c. 11 9 Because the numerator is greater than the denominator, 11 9 is an improper fraction. 18. c. 13 8 Because the numerator is greater than the denominator, 13 8 is an improper fraction. 19. 2 1 9 is a mixed number. 20. 5 1 8 is a mixed number. 21. 6 6 Because the numerator and denominator are equal, 6 6 is an improper fraction. 22. 8 8 Because the numerator and denominator are equal, 8 8 is an improper fraction. 23. Improper Fraction: 4 . 3 Mixed Number: 1 1 . 3 24. Improper Fraction: 5 . 3 Mixed Number: 2 1 . 3 25. Improper Fraction: 11 . 4 Mixed Number: 3 2 . 4

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    Section P.2 Fractions and Mixed Numbers 11 26. Improper Fraction: 5 . 2 Mixed Number: 1 2 . 2 27. Improper Fraction: 17 . 5 Mixed Number: 2 3 . 5 28. Improper Fraction: 18 . 5 Mixed Number: 3 3 . 5 29. Improper Fraction: 25 . 7 Mixed Number: 4 3 . 7 30. Improper Fraction: 26 . 7 Mixed Number: 5 3 . 7 31. 3 8 2 3 19 2 8 8 8     32. 7 9 2 7 25 2 9 9 9     33. 3 5 7 3 38 7 5 5 5     34. 2 5 6 2 32 6 5 5 5     35. 7 16 8 7 135 8 16 16 16     36. 5 16 9 5 149 9 16 16 16     37. 18 25 12 18 318 12 25 25 25     38. 21 25 15 21 396 15 25 25 25    

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    Chapter P Prealgebra Pathways 12 39. 23 3 4 5 5  40. 47 7 5 8 8  41. 76 4 8 9 9  42. 59 5 6 9 9  43. 711 11 35 20 20  44. 788 13 31 25 25  45. 1247 29 43  46. 1665 45 37  47. 48. 49. 50. 51. a. Shark attacks 24 14 3 2 1 1 1 46         b. Florida had the greatest number of attacks, 24. Thus, the fraction of attacks that occurred in Florida is 24 . 46 c. Alabama, North Carolina, and Texas had one shark attack each. Thus, the fraction of attacks that occurred in these three states is 3 . 46 52. a. Shark attacks 24 14 3 2 1 1 1 46        

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    Section P.3 Simplifying Fractions; Operations with Fractions 13 b. Hawaii had the second greatest number of attacks, 14. Thus, the fraction of attacks that occurred in Hawaii is 14 . 46 c. South Carolina and California had a combined 5 shark attacks. Thus, the fraction of attacks that occurred in these two states is 5 . 46 53. – 58. Answers will vary. 59. does not make sense; Explanations will vary. Sample explanation: Since 3 2 is an improper fraction the price would be higher than the original price. 60. makes sense 61. does not make sense; Explanations will vary. Sample explanation: There are 8 total people, so the denominator would be 8. 62. true 63. true 64. false; Changes to make the statement true will vary. A sample change is: A mixed number has a whole number and a proper fraction. 5 4 is not a proper fraction. 65. false; Changes to make the statement true will vary. A sample change is: 17 2 3 . 5 5  Check Points P.3 1. a. 300 3 100 3 10 10 3 2 5 2 5 2 2 3 5 5                b. 36 6 6 2 3 2 3 2 2 3 3           2. a. 10 2 5 15   3 5  2 3 

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