CHAPTER 2 MULTIPLYING AND DIVIDING FRACTIONS

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1 2.1 Basics of Fractions 69 CHAPTER 2 MULTIPLYING AND DIVIDING FRACTIONS 2.1 Basics of Fractions 2.1 Margin Exercises 1. a The figure has equal parts. Three parts are shaded: One part is unshaded: b The figure has equal parts. One part is shaded: Five parts are unshaded: c The figure has equal parts. Seven parts are shaded: One part is unshaded: 2. a An area equal to of the parts is shaded. 3. b An area equal to of the parts is shaded. Numerator a à à Denominator 4. a Proper fractions: numerator smaller than denominator. ß ß b Improper fractions: numerator greater than or equal to denominator. 2.1 Section Exercises ß ß 1. The figure has equal parts. Three parts are shaded: One part is unshaded: 2. The figure has equal parts. Five parts are shaded: Three parts are unshaded: 3. The figure has equal parts. One part is shaded: Two parts are unshaded: 4. An area equal to of the parts is shaded: One part is unshaded: 5. Each of the two figures is divided into parts and are shaded: Three are unshaded: 6. An area equal to of the parts is shaded: b c d à Numerator à Denominator à Numerator à Denominator à Numerator à Denominator One part is unshaded: 7. Five of the bills have a lifespan of years or greater: Four of the bills have a lifespan of years or less: Two of the bills have a lifespan of years: 8. Two of the pets are dogs: Three of the pets are cats: One of the pets is tan: 9. There are students, and are hearing impaired. 10. There are bicycles of which are mountain bikes œ are not mountain bikes.

2 70 Chapter 2 Multiplying and Dividing Fractions 11. There are! rooms. are for nonsmokers, and! œ! are for smokers.!! 12. There are employees. œ are parttime à Numerator à Denominator à Numerator à Denominator à Numerator à Denominator à Numerator à Denominator 17. Proper fractions: numerator smaller than denominator. ß ß Improper fractions: numerator greater than or equal to denominator. ß ß 18. Proper fractions: numerator smaller than denominator. ß ß Improper fractions: numerator greater than or equal to denominator.! ß ß 19. Proper fractions: numerator smaller than denominator. ß ß Improper fractions: numerator greater than or equal to denominator. ß ß 20. Proper fractions: numerator smaller than denominator. none Improper fractions: numerator greater than or equal to denominator. ß ß ß ß ß 21. Answers will vary. One possibility is à Numerator à Denominator The denominator shows the number of equal parts in the whole and the numerator shows how many of the parts are being considered. 22. An example is as a proper fraction and as an improper fraction. A proper fraction has a numerator smaller than the denominator. An improper fraction has a numerator that is greater than or equal to the denominator. Proper fraction Improper fraction 23. The fraction represents of the equal parts into which a whole is divided. 24. The fraction represents of the equal parts into which a whole is divided. 25. The fraction represents of the equal parts into which a whole is divided. 26. The fraction represents of the equal parts into which a whole is divided. 2.2 Mixed Numbers 2.2 Margin Exercises 1. a The figure shows whole object with equal parts, all shaded, and a second whole with parts shaded, so parts are shaded in all. œ b The figure shows wholes with equal parts, all shaded, and a third whole with part shaded, so parts are shaded in all. œ 2. a œ Multiply and. œ Add. œ

3 2.2 Mixed Numbers 71 b œ Multiply and. œ Add. œ c œ Multiply and. œ Add. œ d œ Multiply and. œ Add. œ 3. a à Whole number part à Remainder œ b à Whole number part à Remainder œ c à Whole number part! à Remainder d œ ÃWhole number part à Remainder œ 2.2 Section Exercises 1. œ Multiply and. œ Add. œ 2. œ Multiply and. œ Add. œ 3. œ! Multiply and.! œ Add. œ 4. œ Multiply and. œ Add. œ 5. œ Multiply and. œ Add. œ 6. œ Multiply and. œ Add. œ 7. œ Multiply and. œ Add. œ 8. œ Multiply and. œ Add. œ 9. œ Multiply and. œ Add. œ 10. œ Multiply and. œ Add. œ

4 72 Chapter 2 Multiplying and Dividing Fractions 11. œ Multiply and. œ Add. œ 12. œ Multiply and. œ Add. œ 13.!! œ! Multiply! and.! œ Add.! œ 14. œ Multiply and. œ Add. œ 15.!! œ! Multiply! and.! œ Add.! œ 16. œ Multiply and. œ Add. œ 17. œ Multiply and. œ Add. œ 18. œ Multiply and. œ Add. œ 19. œ! Multiply and.! œ Add. œ 20. œ Multiply and. œ Add. œ 21.! œ Multiply and.! œ Add!.! œ 22. œ Multiply and. œ Add. œ 23. œ Multiply and. œ Add. œ 24.!! œ! Multiply and!.! œ Add. œ!! 25. œ Multiply and. œ Add. œ 26. œ! Multiply and.! œ Add. œ 27. œ Multiply and. œ Add. œ 28. œ Multiply and. œ Add. œ 29. œ Multiply and. œ Add. œ 30. œ! Multiply and.! œ Add. œ

5 2.2 Mixed Numbers œ à Whole number part à Remainder à à Whole number part Remainder œ à Whole number part à Remainder œ à Whole number part à Remainder œ à Whole number part! à Remainder œ à Whole number part! à Remainder œ œ à Whole number part à Remainder à Whole number part à Remainder œ à Whole number part à Remainder œ! à Whole number part! à Remainder! œ à Whole number part! à Remainder œ à Whole number part! à Remainder œ

6 74 Chapter 2 Multiplying and Dividing Fractions ÃWhole number part! à Remainder œ à Whole number part à Remainder œ à Whole number part à Remainder œ à Whole number part à Remainder œ à Whole number part à Remainder œ à Whole number part à Remainder œ ÃWhole number part! à Remainder œ! ÃWhole number part! à Remainder œ! ÃWhole number part! à Remainder œ ÃWhole number part! à Remainder œ ÃWhole number part à Remainder œ

7 2.2 Mixed Numbers ÃWhole number part! à Remainder œ 55. Multiply the denominator by the whole number and add the numerator. The result becomes the new numerator, which is placed over the original denominator. 56. Divide the numerator by the denominator. The quotient is the whole number of the mixed number and the remainder is the numerator of the fraction part. The denominator is unchanged. R 57.!! œ!!!! œ!!! œ 58. œ!! œ œ 59. œ œ!!!!!! œ 60. œ!! œ œ 61. œ œ œ 62. œ œ œ 63. ÃWhole number part! à Remainder 64. œ! ÃWhole number part!!!! à Remainder! œ 65. The commands used will vary. The following is from a TI-83 Plus: œ 66. The commands used will vary. The following is from a TI-83 Plus: œ

8 76 Chapter 2 Multiplying and Dividing Fractions 67. The commands used will vary. The following is from a TI-83 Plus: 72. a The improper fractions in Exercise 71 are the ones where the numerator is equal to or greater than the denominator. b ; œ Note: You can use the following procedure on any calculator. Divide by to get Þ!. Subtract. Multiply by to get. The mixed number is. 68. The commands used will vary. The following is from a TI-83 Plus: œ 69. The following fractions are proper fractions. ß ß ß! 70. a The proper fractions in Exercise 69 are the ones where the numerator is less than the denominator. b ; ;! ; ; c The proper fractions in Exercise 69 are all less than. 71. The following fractions are improper fractions.! ß ß! ; ; c The improper fractions in Exercise 71 are all equal to or greater than. 73. The following fractions can be written as whole or mixed numbers. à Whole number part à Remainder œ à Whole number part! à Remainder œ à à Whole number part Remainder œ 74. a The fractions that can be written as whole or mixed numbers in Exercise 73 are improper fractions, and their value is always greater than or equal to.

9 b ; œ œ ; œ ; c Divide the numerator by the denominator. Use the quotient as the whole number and place the remainder over the denominator. 2.3 Factors 2.3 Margin Exercises 1. a Factorizations of : œ œ œ The factors of are,,,,, and. b Factorizations of : œ œ œ The factors of are,,,, and. c Factorizations of : œ œ œ œ œ The factors of are,,,,,,,, and. d Factorizations of!:! œ!! œ!! œ! œ!! œ! The factors of! are,,,,,!,,!,!, and!. 2.,,,, and each have no factor other than themselves or ;,,,!,,, and each have a factor of ;,, and have a factor of. So,,,!,,,,,, and are composite. 3.,,,, and are prime because they are divisible only by themselves and. 4. a ƒœ ƒœ prime œ b ƒ œ ƒ œ œ prime 5. c ƒ œ ƒœ œ d! ƒ œ!! ƒ œ!! ƒ œ! œ prime prime a Divide by. Divide by. Divide by. Divide by. œ œ b Divide by. Divide by. Divide by. Divide by. œ œ c!! Divide! by.! Divide! by. Divide by. Divide by.! œ œ d Divide by. Divide by. Divide by. Divide by. œ œ 2.3 Factors 77

10 78 Chapter 2 Multiplying and Dividing Fractions 6. a Divide by. Divide by. Divide by. Divide by. Divide by. œ œ b Divide by. Divide by. Divide by. œ œ c d! Divide! by. Divide by. Divide by. Divide by.! œ œ!! Divide! by.!! Divide! by.! Divide! by. Divide by. Divide by.! œ œ e!! Divide! by.! Divide! by. Divide by. Divide by. Divide by.! œ œ 7. a œ œ b œ c œ 2.3 Section Exercises 1. Factorizations of : œ œ The factors of are,,, and. 2. Factorizations of : œ œ œ The factors of are,,,,, and. 3. Factorizations of : œ œ The factors of are,,, and. 4. Factorizations of : œ œ œ The factors of are,,,,, and. 5. Factorizations of : œ œ œ œ œ

11 2.3 Factors 79 The factors of are,,,,,,,,, and. 6. Factorizations of! :! œ! œ!! œ! œ! The factors of! are,,,,,!,, and!. 7. Factorizations of : œ œ œ œ œ The factors of are,,,,,,,, and. 8. Factorizations of!:! œ!! œ! œ! The factors of! are,,,,!, and!. 9. Factorizations of!:! œ!! œ!! œ! œ! The factors of! are,,,,,!,!, and!. 10. Factorizations of!:! œ!! œ!! œ! œ! œ!! œ! The factors of! are,,,,,, 1!,,,!,!, and!. 11. Factorizations of : œ œ œ œ The factors of are,,,,,, and. 12. Factorizations of : œ œ œ œ œ œ The factors of are,,,,,,,,,,, and. 13. is divisible by and, so is composite. 14. is divisible by, so is composite. 15. is only divisible by itself and, so it is prime. 16. is only divisible by itself and, so it is prime. 17. is divisible by, so is composite. 18.! is divisible by and, so! is composite. 19. is only divisible by itself and, so it is prime. 20. is divisible by and, so is composite. 21. is only divisible by itself and, so it is prime. 22. is only divisible by itself and, so it is prime. 23. is divisible by, so is composite. 24. is divisible by and, so is composite. 25. is divisible by and, so is composite. 26. is only divisible by itself and, so it is prime. 27. is divisible by and, so is composite. 28. is only divisible by itself and, so it is prime. 29. Divide by. Divide by. Divide by. œ œ 30. Divide by. Divide by. œ 31. We can also use a factor tree.!!! œ œ 32.!!! Divide! by.!! Divide! by.! Divide! by. Divide by.! œ œ 33. œ œ

12 80 Chapter 2 Multiplying and Dividing Fractions 34. Divide by. Divide by. Divide by. œ œ 35. œ œ 36. Divide by. Divide by. Divide by. Divide by. œ œ ! œ œ! Divide! by. Divide by. Divide by.! œ 39. Divide by. Divide by. Divide by. Divide by. Divide by. œ œ œ œ Divide by. Divide by. Divide by. œ œ 42.! á à! œ œ

13 2.3 Factors !!!!! Divide!! by.! Divide! by. Divide by. Divide by.!! œ œ 44. œ œ 45. Divide by. Divide by. Divide by. œ œ ! œ œ 48.!!! 49.!!! œ œ!! Divide! by.!! Divide! by.!! Divide! by.!! Divide! by.!! Divide! by.! Divide! by. Divide by.! œ œ 50.!!!!!!! Divide! by.! Divide! by. Divide by. Divide by. Divide by.! œ œ! œ œ

14 82 Chapter 2 Multiplying and Dividing Fractions 51.!!! Divide! by.!! Divide! by.! Divide! by. Divide by. Divide by. Divide by.! œ œ 52.!!!!!!!!! œ œ 53. Answers will vary. A sample answer follows. A composite number has a factors other than itself or. Examples include,,,, and!. A prime number is a whole number that has exactly two different factors, itself and. Examples include,,,, and Þ The numbers! and are neither prime nor composite. 54. No even number other than is prime because all even numbers have as a factor. Many odd numbers are multiples of prime numbers and are not prime. For example,,,, and are all multiples of. 55. All the possible factors of are,,,,,,, and. This list includes both prime numbers and composite numbers. The prime factors of include only prime numbers. The prime factorization of is œ œ Þ 56. Yes, you can divide by s before you divide by. No, the order of division does not matter. As long as you use only prime numbers, your answers will be correct. However, it does seem easier to always start with and then use progressively greater prime numbers. The prime factorization of is œ œ. 57.! 58.!! Divide! by. Divide by. Divide by. Divide by.! œ œ!! Divide! by.!! Divide! by.!! Divide! by.!! Divide! by.!! Divide! by.!! Divide! by.! Divide! by. Divide by.! œ œ

15 59.!!! Divide! by.!! Divide! by.!! Divide! by.!! Divide! by.!! Divide! by.! Divide! by. Divide by. Divide by.! œ œ 60.!!!!!! 61.!!!! œ œ!! Divide! by.!! Divide! by Þ! Divide! by Þ Divide by. Divide by. Divide by.! œ œ 2.3 Factors !!!!!!!!! Divide!!! by.!!!!! Divide!!! by.!!! Divide!! by.! Divide! by. Divide by. Divide by. Divide by.!!! œ œ 63.!!!! œ œ 64.!!!!!!! œ œ 65. The prime numbers less than! are,,,,,,,,,,,,,, and. 66. A prime number is a whole number that is evenly divisible by itself and only. 67. No. Every other even number is divisible by in addition to being divisible by itself and. 68. No. A multiple of a prime number can never be prime because it will always be divisible by the prime number.

16 84 Chapter 2 Multiplying and Dividing Fractions 69.!!!!!! Divide!! by.!! Divide!! by. Divide by. Divide by. Divide by. Divide by.!! œ 70.!! œ œ 2.4 Writing a Fraction in Lowest Terms 2.4 Margin Exercises 1. a, ; œ œ Yes, is a common factor of and. b, ; œ œ Yes, is a common factor of and. c, ; œ, but is not a factor of. No. d, ; œ œ Yes. 2. a and have no common factor other than. Yes, it is in lowest terms. b and have a common factor of. No, it is not in lowest terms. c and have a common factor of. No, it is not in lowest terms. 3. a d and have no common factor other than. Yes, it is in lowest terms. b c d e ƒ ƒ ƒ ƒ ƒ ƒ!! ƒ!!! ƒ! ƒ!! ƒ 4. a y y y œ y y y b y y y œ y y y y c œ y d y y œ! y y 5. a and y y y y œ y y y y y y y y œ y y y y The fractions are equivalent Ð œ ÑÞ! b and! y y œ! y y! y y œ y y The fractions are not equivalent Ð Á ÑÞ!! c and

17 2.4 Writing a Fraction in Lowest Terms 85! y y œ y y! y y œ y y The fractions are equivalent œ.!! d and!!! y y y œ! y y y!! œ y y y y y y œ œ The fractions are not equivalent Ð Á ÑÞ 2.4 Section Exercises !!!!! X X X X X X X X X X ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ!! ƒ!!! ƒ! ƒ!! ƒ! ƒ ƒ ƒ ƒ ƒ!!!! ƒ! ƒ!!!! ƒ! ƒ ƒ ƒ!! ƒ!! ƒ!! ƒ ƒ ƒ y y œ y y y y y y œ y y y y! œ y y œ œ! y y œ y y! y y y y œ! y y y y y y y œ y y y y y y œ y y y y y y y y œ y y y y y

18 86 Chapter 2 Multiplying and Dividing Fractions y y 37. œ y y y 38. œ y 39. and y y y y y y y y The fractions are equivalent Ð œ ÑÞ 40. and y y œ œ y y The fractions are equivalent Ð œ ÑÞ! 41. and!! y y y y! y y The fractions are not equivalent Ð Á ÑÞ 42. and! y y y! y! The fractions are not equivalent Ð Á! ÑÞ 43. and y y The fractions are not equivalent Ð Á ÑÞ 44. and y y y y The fractions are not equivalent Ð Á ÑÞ 45. and! y y y! y The fractions are equivalent Ð œ ÑÞ 46. and!! y y! y y! y y y œ œ! y y y! The fractions are equivalent Ð! œ! ÑÞ 47. and œ y y y œ y œ y y y y y y œ The fractions are not equivalent ÁÞ

19 2.5 Multiplying Fractions and œ y y œ œ y y y y y y y y œ The fractions are equivalent œþ 49. and! y! y y y The fractions are equivalent Ð œ ÑÞ! 50. and! y y y y œ œ y y y y! y y y! y y y The fractions are equivalent Ð œ ÑÞ 51. A fraction is in lowest terms when the numerator and the denominator have no common factors other than. Some examples are,, and Þ 52. Two fractions are equivalent when they represent the same portion of a whole. For example, the! fractions and are equivalent. 53. y y y y y y! œ œ œ œ The fractions are equivalent ˆ œ Þ! y y y y y œ œ y y y y y y y œ œ y y y y œ y y! œ y y y œ œ y 2.5 Multiplying Fractions 2.5 Margin Exercises 1. of as read from the figure is the darker shaded part of the second figure. One of eight equal parts is shaded, or Þ œ 2. a b y y c d 3. a y y y y y y b yy y! y c y! y y y d y y 4. y a œ œ y b y or y

20 88 Chapter 2 Multiplying and Dividing Fractions c d y! y! y y œ y y œ 5. a Area œ length width œ y y œ yd b Area œ length width œ y y œ mi c Area œ length width œ y y œ y y œ mi 2.5 Section Exercises y œ y y œ y y œ y y y y y!! œ œ œ y y y y y yy y y y y y y y y y œ y y y y y œ y y y œ œ œ œ! y y y! œ y! œ œ œ y y y y œ œ y y œ y y y y yy y y y y y yy y y y y y yyy yy y y y y y y y œ œ œ œ

21 2.5 Multiplying Fractions y œ y 20. y!! œ y 21. œ y! y 22. y! œ! y 23. y œ y 24. y!! œ! y! y y y 25. œ y y y 26. y œ! y y y!! 27.!! œ! y! y 28. y!!!!! œ! y 29. y!!!!! œ! y 30. y! œ! y 31. œ œ œ 32.! œ y!! y y! y! y 33.! œ y y y œ!!!!! y y! 34.! œ! y y! œ y y 35. œ y y œ!! y y 36. œ y y œ!! 37. Area œ length width œ y œ œ mi y 38. Area œ length width œ ft 39. Area œ length width œ y œ meters y 40. Area œ length width y œ œ in. y 41. Area œ length width œ y œ œ in. y!

22 90 Chapter 2 Multiplying and Dividing Fractions 42. Area œ length œ width y y yd yy! 43. Multiply the numerators and multiply the denominators. An example is Þ 44. You must divide a numerator and a denominator by the same number. If you do all possible divisions, your answer will be in lowest terms. One example is y y œ Þ y y 45. Area œ length width y œ œ yd y 46. Area œ length width y œ œ yd y 47. Area œ length width y œ œ mi y 48. Area œ length width œ œ mi 49. Sunny Side Soccer Park Creekside Soc. Park Area œ length width Area œ length width œ œ œ mi œ mi They are both the same size. 50. Rocking Horse Ranch Silver Spur Ranch Area œ length width Area œ length width œ œ y y y y mi mi y y y y Neither ranch is larger in area. They are both the same size. 51.!,!!!!,!!!!!!!!!!!!!!! œ!,!!! The estimate of the total number of stores for the six largest chains is 6!,!!!. 52.!,,! œ,, which is the exact number of stores for the six largest chains. 53. An estimate of the number of stores is y!!!!!!!!! œ œ!!!þ y The exact value is! œ! œ,!. Þ Rounding gives us stores. 54. An estimate of the number of stores is y!!!!!!!! œ œ!!þ y The exact value is œ œ,! œ!.þ Rounding gives us! stores. 55. We need a multiple of with two nonzero digits that is close to!. A reasonable choice is!! and an estimate is!! y!!!! œ œ!! stores Þ y This value is closer to the exact value because using!! as a rounded guess is closer to! than using!!! as a rounded guess. 56. We need a multiple of with two nonzero digits that is close to. A reasonable choice is!! and an estimate is!! y!!!! œ œ!!! stores Þ y This value is closer to the exact value because using!! as a rounded guess is closer to than using!!! as a rounded guess. 2.6 Applications of Multiplication 2.6 Margin Exercises 1. a Find the amount they can save by multiplying and,.

23 2.6 Applications of Multiplication 91 œ!, y,, œ!, y They can save!, in a year. b To find her retirement income, multiply,!.,y!,! œ œ,! y and She will receive,! as retirement income. 2. Step 1 The problem asks for the number of prescriptions paid for by a third party. Step 2 A third party pays for prescriptions,. Step 3 An estimate is Step 4 The exact value is œ y y Step 5 A third party pays for of the total number of y!!!!!!!!þ y œ Þ prescriptions. Step 6 The answer is reasonably close to our estimate. 3. Step 1 The problem asks for the fraction of students who speak Spanish. Step 2 of the of the students who speak a foreign language, speak Spanish. Step 3 An estimate is y Þ y Step 4 The exact value is, which is the same as the estimate since we didnt round. Step 5 The fraction of students who speak Spanish is. Step 6 The answer,, matches our estimate. 4. a The fraction is Þ b Multiply by the number of people in the survey,!!þ y!!!!!!!! œ!! y!! children buy food from vending machines. c The fraction is! Þ d Multiply! by!!. y!! y!!! œ!! œ! œ!! children buy food from a convenience store or street vendor. 2.6 Section Exercises 1. Multiply the length depth and the width. y y y y The area of the file cabinet top is yd. 2. Multiply the length and width. y y y!! The area is yd. 3. Multiply the length and the width. œ The area of the cookie sheet is ft. 4. Multiply by,!!!,!!!þ,!!,!!!,!!! y,!!!,!!!,!!! œ œ,!!,!!! y,!!,!!! people who shop at flea markets on a daily basis purchase produce. 5. Multiply the length and the width. y y! The area of the top of the table is yd.!

24 92 Chapter 2 Multiplying and Dividing Fractions 6. Multiply the items sold by the fraction which is junk food. y!! œ œ y There are junk food items. 7. Multiply by Þ y œ œ y He earned on his job. 8. Multiply the fraction that is personal earnings by the employers profits. y!! œ œ y Her earnings are Þ 9. Multiply the daily parking fee by the fraction. y y The daily parking fee in Boston is. 10. Multiply the daily parking fee by the fraction. y y The daily parking fee in San Francisco is. 11. of the! runners are women. y!!!! y! runners are women. 12. Multiply the fraction of rooms for nonsmokers by the number of rooms. y!! œ œ y There are rooms for nonsmokers. 13. The smallest sector of the circle graph is the Dont know group, so this response was given by the least number of people. To find how many people gave this response, multiply 0 by the total number of people,!!!.! y!!!!!!!! y!! people gave this response. 14. The largest sector of the circle graph is the Somewhat more group, so this response was given by the greatest number of people. To find how many people gave this response, multiply by the total number of people,!!!. y!!!!!!!!!! y!!! people gave this response. 15. There are two groups that watch less television. The Much less group corresponds to the fraction! Þ The Somewhat less group corresponds to the fraction Þ Much less: y!!!!!!!!!!! y! Somewhat less: y!!!!!!!! y Adding, we get!!! œ! people. 16. There are two groups that watch more television. The Much more group corresponds to the fraction! Þ The Somewhat more group was calculated in Exercise to be!!þ Much more: y!!!!!!!!!!! y! Adding, we get!!!! œ!! people. 17. Because everyone is included and fractions are given for all groups, the sum of the fractions must be 1, or allof the people. 18. Answers will vary. Some possibilities are 1. You made an addition error. 2. The fractions on the pie graph are incorrect. 3. The fraction errors were caused by rounding. 19. Add the income for all twelve months to find the income for the year.!!!! œ,!!! The Owens family had income of, for the!!! year.

25 2.6 Applications of Multiplication Multiply the fraction by the total income,!!!.,!!,y!!!,!!! œ œ,!! y Their taxes were,!!. 21. From Exercise 19, the total income is,!!!. The circle graph shows that of the income is for rent.,!!,y!!!,!!!,!! y The amount of their rent is,!!þ 22. Multiply the fraction by the total income.,y!!!,!!! œ œ, y They spent, on food. 23. Multiply the total income by the fraction saved.,y!!!,!!! œ y œ The Owens family saved for the year. 24. Multiply the fraction by the total income.!,y!!!,!!! œ œ! y They spent! on clothing. 25. The error was made when dividing by and writing instead of. The correct solution is! y y! œ œ Þ! y! y 26. Yes, the statements are true. Since whole numbers are or greater, when you multiply, the product will always be greater than either of the numbers multiplied. But, when you multiply two proper fractions, you are finding a fraction of a fraction, and the product will be smaller than either of the two proper fractions. 27. Multiply the cost in the United States by Þ!!!!!!! y! y The cost of Lasik eye surgery for one eye in Thailand is!. 28. Multiply the cost in the United States by! Þ!!,y!!!! y, The cost of a knee replacement in Mexico is. 29. We want of the actual length. œ y y œ The length of the scale model is feet. 30. We want of a quart for each gallonþ! y! œ y He will need to add 71 quarts to the! gallons of gasoline. 31. First multiply and,!!! to find the number of her votes from senior citizens.!!!,y!!!,!!! œ œ,!!! y To find the votes needed from voters other than the senior citizens, subtract:,!!!,!!! œ!!! votes. She needs!!! votes from voters other than the senior citizens. 32. Multiply the fraction by the cost,!!!.!!!,,y!!!!!!!!!, y To find the amount that is to be paid by the owner, subtract:,!!!,!!! œ,!!!,!!! will be needed to open the shop.

26 94 Chapter 2 Multiplying and Dividing Fractions 33. Multiply the remaining of the estate by the fraction going to the American Cancer Society. œ of the estate goes to the American Cancer Society. 34. Multiply the remaining of their total investments by the fraction invested in bonds. y y The couple invested bonds. 2.7 Dividing Fractions 2.7 Margin Exercises of their total investment in 1. a The reciprocal of is because. b The reciprocal of is because. c The reciprocal of is because. d The reciprocal of is because. 2. a ƒ œ b y ƒ œ y c d y œ ƒ œ 3. a! ƒ œ! œ ƒ œ y y œ œ y! œ!! y b ƒ œ ƒ œ y œ c ƒœ ƒ œ y y d ƒœ ƒ œ 4. a Divide the total amount of fuel by the size of the tanks. y! ƒ œ ƒ œ œ! y! fuel tanks can be filled. b Divide the total number of quarts in the cask by the size of the bottles.!! ƒ œ ƒ!! y!! œ y œ œ œ!! -quart bottles can be filled. 5. a Divide the fraction of the bonus money by the number of employees. ƒœ ƒ œ y y Each employee will receive money. b Since they donate have of the bonus of the winnings, they œ œ of the winnings left to divide. Divide the fraction of the winnings that remain by the number of employees. ƒœ ƒ œ y œ y! Each employee will receive 2.7 Section Exercises! 1. The reciprocal of is because 2. The reciprocal of is because of the prize money..!.!

27 2.7 Dividing Fractions The reciprocal of is because 4. The reciprocal of is because. 5. The reciprocal of is because! 6. The reciprocal of! is because!!!.! 7. The reciprocal of is because 8. The reciprocal of! is! because!.!! 9. ƒ œ y œ y 10. ƒ œ y œ y 11. ƒ œ œ œ 12. ƒ œ y œ œ y 13. ƒ œ œ! 14. ƒ œ œ ƒ œ y y y y ƒ œ y y y œ y y ƒ œ y y ƒ œ y y œ yy! y œ ƒ œ œ! y!!.!!.! ! y y œ ƒ œ! y! œ œ y y œ ƒ œ y y y œ y y œ ƒ œ œ y y y y œ œ 23. ƒ œ ƒ œ 24. ƒ œ ƒ œ œ 25. y œƒ œ ƒ œ y 26. y œƒ œ ƒ œ y 27. y œ ƒ œ ƒ œ œ y 28.! œ ƒ œ ƒ œ!!! œ! 29. of a quart divided into parts: ƒœ ƒ œ y y Each cat will get of a quart. 30. Divide the number of quarts of shampoo by the fraction of a quart each container holds. y ƒ œ y œ! Harold can fill! containers. 31. Divide the total number of cups by the size of the measuring cup. ƒ œ ƒ œ œ They need to fill the measuring cup times.

28 96 Chapter 2 Multiplying and Dividing Fractions 32. Divide the portion of the trip completed by the number of days to get the portion completed each day. ƒœ ƒ œ y œ y She completes 8 of the total trip each day. 33. Divide the amount of eye drops by the amount needed for each dispenser. ƒ œ ƒ œ œ dispensers can be filled. 34. Divide the number of pounds of peanuts by the fraction of pounds of peanuts each person will get. y!! ƒ œ y œ guests can be served with! pounds of peanuts. 35. Divide the total cords of wood by the amount per trip. y!!! ƒ œ! y œ œ! Pam has to make! trips. 36. Divide the 200-yard roll into -yard pieces.!! ƒ œ y!!! œ! y Manuel can cut! pieces from the roll. 37. Answers will vary. A sample answer follows: You can divide two fractions by using the reciprocal of the second fraction divisor and then multiplying. 38. Sometimes the answer is smaller and sometimes it is larger. ƒ œ y smaller than but œ y not smaller than Ñ ƒ œ y œ œ larger y 39. Each loafcake requires pound of jellybeans, so to make loafcakes, use multiplication. y y pounds will be needed. 40. We want of! patients use multiplication.! y!!! œ! y! patients were still taking their drugs. 41. Divide the cans of compound by the fraction of a can needed for each new home. y ƒ œ œ œ! y! homes can be plumbed. 42. Divide the gallons of gear lube by the fraction of a gallon needed for each tractor differential. y ƒ œ œ œ y tractors can be serviced. 43. In of the visits, doctors failed to discuss the issues use multiplication. y œ œ y The doctors failed to discuss the issues in visits. 44. of the hours have been completed use multiplication. y œ œ y She has worked hours. 45. Divide the yards of fabric by the fraction of a yard needed for each dish towel. y! ƒ œ œ œ y towels can be made.

29 2.8 Multiplying and Dividing Mixed Numbers Divide the! vials of medication by the fraction of a vial needed for each patient. y!!! ƒ œ!! y œ!!!! patients can be treated. 47. The indicator words for multiplication are underlined below. more than per double twice times product less than difference equals twice as much 48. The indicator words for division are underlined below. fewer sum of goes into divide per quotient equals double loss of divided by 49. To divide two fractions, multiply the first fraction by the reciprocal of the second fraction. 50. The reciprocal of is because Þ The reciprocal of is because Þ The reciprocal of is because œ œ Þ The reciprocal of is because Þ 51. a To find the perimeter of any regular 3-, 4-, 5-, or 6-sided figure, multiply the length of one side by 3, 4, 5, or 6, respectively. b The stamp has four sides, so multiply by. y œ in. y The perimeter of the stamp is inches. 52. Area œ length width œ œ The area is in.. Multiply the length by the width to find the area of any rectangle. 2.8 Multiplying and Dividing Mixed Numbers 2.8 Margin Exercises 1. a à is more than Þ Ã Half of is Þ rounds up to. b à is less than Þ Ã Half of is Þ rounds to. c à is more than Þ Ã Half of is Þ rounds up to. d à is more than Þ Ã Half of is Þ rounds up to. e à is the same as Þ Ã Half of is Þ rounds up to. f à is less than Þ Ã Half of is Þ rounds to.

30 98 Chapter 2 Multiplying and Dividing Fractions 2. a Estimate: rounds to. rounds to. œ œ œ œ b Estimate: rounds to. rounds to. œ y œ œ y c Estimate: rounds to. rounds to. œ! y y! œ œ œ y y d Estimate: rounds to. rounds to. œ! y œ œ y!! 3. a ƒ Estimate: rounds to. rounds to. ƒœ y y ƒ œ ƒ œ œ œ y y b! ƒ Estimate:! rounds to!. rounds to.! ƒ œ! ƒ c ƒ Estimate: œ ƒ œ rounds to. rounds to. ƒœ y ƒ œ ƒ œ y d ƒ Estimate: rounds to. rounds to. ƒ ƒ y œ ƒ y 4. a Multiply the amount of paint needed for each car by the number of cars. Estimate: rounds to. rounds to. œ œ œ œ quarts are needed for cars.

31 2.8 Multiplying and Dividing Mixed Numbers 99 b Multiply the amount that she earns per hour by the number of hours that she worked. Estimate: rounds to. rounds to. œ œ œ œ! Clare would earn!. 5. a Divide the total pounds of brass by the number of pounds needed for one engine. Estimate: rounds to. rounds to. ƒ ƒ œ ƒ œ y œ y œ propellers can be manufactured from pounds of brass. b Divide the total number of quarts by the number of quarts needed for each oil change. Estimate:! rounds to!!. rounds to.!! ƒ! ƒ! y œ! ƒ œ œ y oil changes can be made with! quarts of oil. 2.8 Section Exercises 1. Estimate: œ! œ 2. Estimate: œ œ 3.! Estimate: œ y y œ œ!! y y! 4. Estimate: œ! œ! 5. Estimate: œ y y œ œ y y 6. Estimate: œ yy! œ! y y 7. Estimate: œ 8. y!! y Estimate: œ y œ y 9. Estimate: œ! œ! y œ y 10. Estimate: œ continued

32 100 Chapter 2 Multiplying and Dividing Fractions y y œ y y Estimate: œ œ y y œ y y Estimate: œ y y œ y y 13. ƒ Estimate: ƒœ y y ƒ œ ƒ œ œ y y 14. ƒ Estimate: ƒœ y ƒ œ ƒ œ y œ y y 15. ƒ Estimate: ƒœ ƒœ ƒ œ œ 16. ƒ Estimate: ƒ ƒ œ ƒ œ 17. ƒ Estimate: ƒœ ƒ œ ƒ œ 18. ƒ Estimate: ƒ y ƒ œ ƒ œ œ œ y 19. ƒ 20. Estimate: ƒœ y ƒ œ ƒ œ œ y ƒ Estimate: ƒœ y ƒ œ ƒ œ œ y! 21. ƒ Estimate: ƒ y ƒ œ ƒ œ y œ y y! 22. ƒ Estimate: ƒœ ƒ œ ƒ œ y y 23. ƒ Estimate: ƒœ ƒ œ ƒ œ œ 24. ƒ Estimate: ƒœ ƒ œ ƒ œ

33 2.8 Multiplying and Dividing Mixed Numbers Multiply each amount by Þ a Applesauce : cup Estimate: œ cups cups œ œ œ b Salt : tsp. Estimate: œ tsp. tsp. œ œ œ c Flour : cups Estimate: œ cups cups œ œ œ 26. Multiply each amount by Þ a Flour: cups Estimate: œ cups cups œ œ œ b Applesauce: cup Estimate: œ cups cups œ œ œ c Vegetable oil : cup Estimate:! œ! cups y œ œ œ cup y 27. Divide each amount by. a Vanilla extract : tsp. Estimate: ƒœ tsp. ƒœ ƒ œ œ tsp. b Applesauce : cup Estimate: ƒœ cup ƒœ ƒ œ œ cup c Flour : cups Estimate: ƒœcup ƒ œ ƒ œ œ cup 28. Divide each amount by. : cups a Flour Estimate: ƒœ cup ƒ œ ƒ œ œ cup b Salt : tsp. Estimate: ƒœ teaspoon ƒœ ƒ œ œ teaspoon c Applesauce: cup Estimate: ƒœ cup cup ƒ y œ ƒ œ œ y 29. Divide the number of gallons available by the number of gallons needed for each unit. Estimate: ƒ! units y ƒ œ ƒ œ œ y units can be painted with gallons of paint. 30. Divide the number of total minutes by the number of minutes per moment. Estimate:! ƒ œ! moments! y!!!! ƒ œ ƒ œ! y There are! moments in an 8-hour work day. 31. Each handle requires inches of steel tubing. Use multiplication. Estimate:! œ!! in. œ in. inches of steel tubing is needed to make jacks.

34 102 Chapter 2 Multiplying and Dividing Fractions 32. Assume that the inch length listed in the overall dimensions is the length of the handle. Use multiplication. Estimate: œ, in. y œ œ, in. y The amount of wood that is necessary to make handles is, inches. 33. The answer should include: Step 1 Change mixed numbers to improper fractions. Step 2 Multiply the fractions. Step 3 Write the answer in lowest terms, changing to mixed or whole numbers where possible. 34. The additional step is to use the reciprocal of the second fraction divisor. 35. Divide the total pounds of rubber by the pounds of rubber required for a tire. Estimate:,! ƒ! tires,! ƒ! œ,! ƒ!,y! œ œ! y! tires can be manufactured. 36. Divide the number of square yards of carpet by the amount of carpet needed for each apartment unit. Estimate:! ƒ! units! y!! ƒ œ ƒ œ! units can be carpeted. œ! y 37. Divide the length of the tube by the length of each spacer. Estimate:! ƒ œ! spacers y ƒ œ ƒ œ y y y œ spacers can be cut from the tube. 38. Divide the amount of sand tons by the amount his truck can carry each trip. Estimate: ƒ œ trips y ƒ œ ƒ œ y œ trips must be made. 39. a The maximum height of the standard jack is inches. Use multiplication. Estimate: œ in. y œ in. y The hydraulic lift must raise the car inches. b There are inches in a foot, so the -foot- tall mechanic is œ inches tall. So no, the mechanic can not stand under the car without bending. 40. a The maximum height of the low-profile jack is inches. Use division. Estimate: ƒ œ in. ƒ œ œ œ in. The low-profile lift must raise the car inches. b No, because in. is greater than in. 41. Multiply the gallons per acre times the number of acres to get the total number of gallons needed. Estimate: œ gallons œ œ œ gallons of the chemical are needed. 42. Multiply the boxes of tile per floor times the number of floors homes to get the total number of boxes needed. Estimate: œ boxes œ! œ!! œ boxes of tile are needed. Chapter 2 Review Exercises There are parts, and is shaded. There are parts, and are shaded. There are parts, and are shaded.

35 Chapter 2 Review Exercises Proper fractions have numerator top smaller than denominator bottom. They are: ß ß Þ Improper fractions have numerator top larger than or equal to the denominator bottom. They are: ß Þ 5. Proper fractions have numerator top smaller than denominator bottom. They are: ß Improper fractions have numerator top larger than or equal to the denominator bottom. They are: ß ß 6. œ œ œ 7. œ œ œ 8. à Whole number part à Remainder 9. œ œ 10. Factorizations of : ÃWhole number part! à Remainder œ œ The factors of are,,, and. 11. Factorizations of : œ œ œ œ The factors of are,,,,,,, and. 12. Factorizations of : œ œ The factors of are,,, and. 13. Factorizations of!:! œ! œ!! œ! œ! œ!! œ! The factors of! are,,,,,,!,,,!,, and!. 14. œ œ 15.!! œ œ 16.!!! á! œ œ 17. œ œ 18. œ œ 19. œ œ 20. œ œ! à 21. All parts out of a possible parts are gold. œ of the possible œ parts are gold. ƒ ƒ

36 104 Chapter 2 Multiplying and Dividing Fractions of the possible! œ parts are gold. ƒ ƒ 24. 1! of the possible! œ parts are gold.!! ƒ ƒ 25. y! y 26. y y y y y y œ y y y y y y 27. and ƒ ƒ The fractions are equivalent Ð œ ÑÞ! 28. and!!! ƒ!!! ƒ! The fractions are not equivalent Ð Á ÑÞ! 29. and!!! ƒ! œ œ!! ƒ! The fractions are equivalent Ð œ ÑÞ!! y œ y y œ y! œ œ y y! y œ y y y y y œ y y y y y!!!!!! œ œ y ƒ œ œ ƒ œ y œ œ y!! œ! ƒ! y œ yy! yy! y y y œ ƒ œ y y y 40. ƒ œ ƒ œ y 41. y ƒ œ y 42. ƒœ ƒ œ 43. ƒœ ƒ œ 44. y œ ƒ œ ƒ œ y 45. To find the area, multiply the length and the width.! œ!! The area is ft. 46. To find the area, multiply the length and the width. y y The area is in..

37 Chapter 2 Review Exercises Multiply the length and width.! y! y The area is ft. 48. Multiply the length and width. y y The area is ft. 49. Estimate: œ œ œ œ 50. Estimate: œ y y œ y y 51. ƒ Estimate: ƒ ƒ œ œ œ 52. ƒ Estimate: ƒœ ƒ œ ƒ œ y œ y 53. Divide the total tons of almonds by the size of the bins. y!! ƒ œ y œ bins will be needed to store the almonds. 54. of the estate is to be divided into parts. ƒœ ƒ œ Each child will receive of the estate. 55. Divide the total yardage by the amount needed for each pull cord. Estimate: ƒ! pull cords y y ƒ œ ƒ œ œ y y pull cords can be made. 56. Multiply the weight per gallon times the number of aquariums times the gallons per aquarium. Estimate:! œ!!! œ! œ!! The weight of the water is!!, or pounds. 57. Ebony sold of!! pounds of rice. y!!!! œ œ pounds y Thus,!! œ pounds remain. She gave of to her parents. y! œ œ! pounds y Ebony gave! pounds to her parents. The amount she has left is! œ pounds. 58. Sheila paid of for taxes, social security, and a retirement plan. y œ œ y She paid for taxes, social security, and a retirement plan.! She paid of the remainder, œ!, for basic living expenses.! y! œ!! y She has! œ left. 59. must be divided by. ƒœ ƒ œ Each park will receive of the budget.

38 106 Chapter 2 Multiplying and Dividing Fractions 60. of the catch must be divided evenly among fishermen. ƒœ ƒ œ œ Each fisherman receives ton. 61. [2.5] 62. [2.5] y œ y 63. [2.8] œ œ 64. [2.8] œ œ 65. [2.7] 66. [2.7] 67. [2.5] 68. [2.8] y œ ƒ œ ƒ œ y! œ ƒ œ œ y!! y y ƒ œ ƒ œ œ y 69. [2.2] à Whole number part à Remainder œ 70. [2.2] ÃWhole number part à Remainder œ 71. [2.2] œ œ œ 72. [2.2] œ!! œ!! œ 73. [2.4] y y œ y y 74. [2.4]! y y œ! y y 75. [2.4] ƒ!! ƒ 76. [2.4] ƒ ƒ 77. [2.4] ƒ!! ƒ 78. [2.4] ƒ ƒ 79. [2.8] Multiply the area by the amount needed for each square yard. Estimate: œ ounces y y œ œ y y ounces of glue are needed. 80. [2.8] Multiply the number of in-ground tanks by the number of quarts needed for each in-ground tank. Estimate: œ qt œ œ œ quarts are needed. 81. [2.8] To find the area, multiply the length and the width. œ The area is in..

39 Chapter 2 Test [2.5] To find the area, multiply the length and the width. œ The area is yd. Chapter 2 Test There are parts, 1. and are shaded. There are parts, 2. and are shaded. 3. Proper fractions have the numerator top smaller than the denominator bottom. ß ß ß 4. œ œ œ 5.! ÃWhole number part à Remainder œ! 6. Factorizations of : œ œ œ The factors of are,,,,, and Þ 7. œ œ 8. œ œ 9.!!! á!! œ œ à ƒ 10. ƒ!! ƒ 11. ƒ 12. Write the prime factorization of both numerator and denominator. Divide the numerator and denominator by any common factors. Multiply the remaining factors in the numerator and denominator. y y y y y y 13. Multiply fractions by multiplying the numerators and multiplying the denominators. Divide the two fractions by using the reciprocal of the second fraction divisor and then multiplying y y y y y y 16. Multiply the length and the width. y œ y y y œ œ The area is yd. 17. First, find the number of seedlings that dont survive. 18.! y!! y Next, subtract to find the number that do survive.! œ seedlings do survive. y ƒ œ y!

40 108 Chapter 2 Multiplying and Dividing Fractions 19. œ ƒ œ ƒ œ œ œ 20. Divide the total length by the length of the pieces. 24. y ƒ œ ƒ œ y œ pieces can be cut. 21. Estimate: œ œ œ œ 22. Estimate: œ œ œ œ 23. ƒ Estimate:! ƒ œ y ƒ œ ƒ œ œ œ y Estimate: ƒ œ ƒ œ ƒ œ œ!! 25. If grams can be synthesized per day, multiply to find the amount synthesized in days. Estimate: œ grams! grams can be synthesized.!