2.1 Basics of Fractions CHAPTER MULTIPLYING AND DIVIDING FRACTIONS 71 (b) Improper fractions: numerator greater than or equal to denominator 2.1 Section Exercises 2.1 Basics of Fractions Numerator Denominator Numerator Denominator One part is unshaded: (b) The figure has equal parts Numerator Denominator One part is shaded: Numerator Denominator Five parts are unshaded: The fraction represents of the into which a whole is divided The fraction represents of the into which a whole is divided equal parts The fraction represents of the into which a whole is divided equal parts The fraction represents of the into which a whole is divided The figure has equal parts Three parts are shaded: 2.1 Margin Exercises (a) The figure has equal parts Three parts are shaded: (a) An area equal to (b) An area equal to (a) of the of the parts is shaded parts is shaded Numerator Denominator 10 equal parts equal parts One part is unshaded: The figure has equal parts Five parts are shaded: (b) Numerator Denominator 11 Three parts are unshaded: The figure has equal parts One part is shaded: Two parts are unshaded: (c) Numerator Denominator 12 An area equal to of the parts is shaded: 13 One part is unshaded: Each of the two figures is divided into parts and are shaded: (d) Numerator Denominator Three are unshaded: (a) Proper fractions: numerator smaller than denominator 14 An area equal to of the parts is shaded: 15 One part is unshaded: 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: Copyright © 2014 Pearson Education, Inc 72 16 Chapter Multiplying and Dividing Fractions Four of the Three of the 17 coins are pennies: 24 coins are nickels: none Two of the coins are dimes: There are students, and are hearing impaired hearing impaired students (numerator) total students (denominator) 18 19 20 21 Improper fractions: numerator greater than or equal to denominator 25 There are shopping carts of which are in the parking lot ( are not in the parking lot, but are in the store) Fraction of carts in store: There are rooms are for nonsmokers, and are for smokers There are time Proper fractions: numerator smaller than denominator employees Answers will vary One possibility is Numerator Denominator The denominator shows the number of equal parts in the whole and the numerator shows how many of the parts are being considered 26 An example is as a proper fraction and improper fraction as an A proper fraction has a numerator smaller than the denominator are part- An improper fraction has a numerator that is greater than or equal to the denominator Proper fractions: numerator smaller than denominator Proper fraction Improper fraction 2.2 Mixed Numbers Improper fractions: numerator greater than or equal to denominator 2.2 Margin Exercises 22 (a) The figure shows whole object with equal parts, all shaded, and a second whole with parts shaded, so parts are shaded in all Proper fractions: numerator smaller than denominator (b) Since each of these diagrams is divided into Improper fractions: numerator greater than or equal to denominator pieces, the denominator will be The number of pieces shaded is • 23 Proper fractions: numerator smaller than denominator Multiply and (a) Add • Improper fractions: numerator greater than or equal to denominator (b) Copyright © 2014 Pearson Education, Inc Multiply and Add 2.2 Mixed Numbers • Multiply and (c) Add 73 proper fraction since the numerator is is a smaller than the denominator The statement is true • • Multiply and Multiply and (d) Add Add (a) The mixed number can be changed to the improper fraction , not The statement is false Divide by Whole number part Remainder The statement "Some mixed number cannot be changed to an improper fraction" is false since any mixed number can be changed to an improper fraction.• Multiply and (b) Add Divide by Whole number part The mixed number can be changed to the improper fraction , not The statement is false • Remainder Multiply and (c) Divide Add The statement is true by Whole number part • Multiply and Remainder Add • (d) part DivideWhole by number Multiply and Add • Remainder Multiply and Add • 2.2 Section Exercises 10 mproper fraction since the numerator is is anthan i or equal to the denominator The greater statement is true Copyright © 2014 Pearson Education, Inc Multiply and Add 74 Chapter Multiplying and Dividing Fractions • • Multiply and 11 Multiply and 21 Add Add • 12 Multiply and • Multiply 22 Add Add • • Multiply 13 and and Multiply 23 and Add Add • Multiply 14 and 24 • Multiply and and and Add Add • 25 Multiply 15 • and Multiply Add Add • 26 • Add Multiply and 16 Multiply Add • • Multiply 27 Add Multiply and 17 and Add 28 • • Multiply and Add Multiply and 18 Add • Multiply and 29 • Add Multiply and 19 Add 30 • • 20 Multiply and Add Multiply and Add 31 The improper fraction can be changed to the false mixed number , not The statement is Copyright © 2014 Pearson Education, Inc 2.2 Mixed Numbers 32 33 34 The statement "An improper fraction cannot always be written as a whole number or a mixed number" is false since a mixed number always has a value equal to or greater than a whole number 40 Whole number part The statement "Some improper fractions can be written as a whole number with no fraction part" is true For example, The statement "The improper fraction written as the whole number " is true Remainder can be 41 35 Whole number part Whole number part Remainder Remainder 42 36 Whole number part Whole number part Remainder Remainder 43 Whole number part 37 Whole number part Remainder 38 Remainder Whole number part 44 Whole number part Remainder Remainder 39 45 Whole number part Remainder Whole number part Remainder Copyright © 2014 Pearson Education, Inc 75 76 Chapter Multiplying and Dividing Fractions 46 51 Whole number part Whole number part Remainder Remainder 47 Whole number part 52 Whole number part Remainder Remainder 48 Whole number part 53 Whole number part Remainder Remainder 49 Whole number part 54 Remainder Whole number part Remainder 50 Whole number part 55 Remainder Multiply the denominator by the whole number and add the numerator The result becomes the new numerator, which is placed over the original denominator Copyright © 2014 Pearson Education, Inc 2.2 Mixed Numbers 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 77 64 Whole number part Remainder • 57 65 The commands used will vary The following is from a TI-83 Plus: 66 The commands used will vary The following is from a TI-83 Plus: 67 The commands used will vary The following is from a TI-83 Plus: • 58 • 59 • 60 • 61 • 62 Whole number part 63 Remainder Note: You can use the following procedure on any calculator Divide by to get Subtract Multiply by to get The mixed number is Copyright © 2014 Pearson Education, Inc 78 68 Chapter Multiplying and Dividing Fractions (c) The improper fractions in Exercise 71 are all equal to or greater than The commands used will vary The following is from a TI-83 Plus: 73 The following fractions can be written as whole or mixed numbers Whole number part Remainder 69 The following fractions are proper fractions 70 (a) The proper fractions in Exercise 69 are the ones where the numerator is less than the denominator Whole number part Remainder ; (b) ; Whole number part ; Remainder ; (c) The proper fractions in Exercise 69 are all less than 71 The following fractions are improper fractions 72 (a) The improper fractions in Exercise 71 are the ones where the numerator is equal to or greater 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 ; (b) than the denominator ; ; (b) ; ; 2.3 Factors 2.3 Margin Exercises ; (a) Factorizations of • The factors of • : • are , , , , , and Copyright © 2014 Pearson Education, Inc 2.3 Factors (b) Factorizations of • • The factors of • are , , , , and (c) Factorizations of • • • , , and • Divide by Divide by Divide Quotient is by • , are , , , , , , , , • • (b) , , , , , , • , , , , , , , , , , , (a) Divide by Divide by Divide Quotient is •Divide • • , , , , , and each have no factor other than themselves or ; , , , , , , and each have a factor of ; , , and have a factor of So , , , , , , , , , and are composite by • , , , , and are prime because they are divisible only by themselves and Divide Either method is correct and yields the prime factorization as follows: : • The factors of and , , , • are , , , , , , (d) Factorizations of • • : • The factors of This division is done from the "top-down." : by by • (c) prime • Divide by Divide by Divide Quotient is • Divide • • by • (b) prime • • (c) by • • prime • • (d) (d) prime • • • (a) This division is done from the "bottom-up." Quotient is Divide by Divide by Divide by Divide by Divide by Divide by Divide Quotient is Divide Copyright â 2014 Pearson Education, Inc by by 79 80 Chapter Multiplying and Dividing Fractions Divide (a) by (e) Divide by Divide by Divide by Divide by Divide by Divide by Divide by Quotient is • Divide • • • by • Divide Quotient is • • • by • • • (a) (b) Divide by Divide Quotient is • Divide • by • • • (b) • by • (c) (c) Divide by Divide by Divide Quotient is • Divide • • by • • 2.3 Section Exercises Factorizations of : • by• • • The factors of are , , , and The statement is false (missing ) Factorizations of : • by Divide by Divide by Divide Quotient is • Divide • • • by by • • • The factors of are , , , , , and statement is true (d) Divide • Factorizations of : • • The factors of are , , , and statement is true Factorizations of • The The : • • The factors of are , , , , , and statement is false (missing and ) Copyright © 2014 Pearson Education, Inc The 102 Chapter Multiplying and Dividing Fractions 55 48 (a) of the multiplication miles have been completed—use • (a) To find the perimeter of any flat equal-sided 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 He has gone in miles (b) The number of miles that remain is miles 49 by Divide the yards of fabric by the fraction of a yard needed for each dish towel • 56 The perimeter of the stamp is • width Area length inches • • The area is in Multiply the length by the width to find the area of any rectangle 2.8 Multiplying and Dividing Mixed Numbers towels can be made 50 Multiply the number of applicants by the fraction of jobs available per applicant • There are 51 2.8 Margin Exercises • is more than Half of is (a) job openings The indicator words for multiplication are underlined below more than double times less than equals rounds up to per twice product difference twice as much is less than Half of is (b) rounds down to 52 The indicator words for division are underlined below fewer goes into per equals loss of sum of divide quotient double divided by rounds up to 53 To divide two fractions, multiply the first fraction by the reciprocal of the second fraction • 54 The reciprocal of is because The reciprocal of is because The reciprocal of is because The reciprocal of is is more than Half of is (c) (d) is more than Half of is • rounds up to • is the same as Half of is (e) because rounds up to Copyright â 2014 Pearson Education, Inc 2.8 Multiplying and Dividing Mixed Numbers is less than Half of is (f) 103 (b) rounds to Estimate: rounds to Exact: rounds down to • • (a) rounds to Estimate: (c) rounds to • rounds to Estimate: • rounds to • Exact: Exact: • • (b) Estimate: rounds to rounds to (d) • Exact: rounds to Estimate: • • rounds to • Exact: • (c) • Estimate: rounds to rounds to • Exact: • • • Multiply the amount of paint needed for each car by the number of cars rounds to rounds to Estimate: • • • Exact: (d) Estimate: rounds to rounds to quarts are needed for cars The answer is reasonably close to the estimate • Exact: • • • (a) Divide the total pounds of brass by the number of pounds needed for one engine Estimate: • (a) Estimate: rounds to rounds to Exact: rounds to rounds to • Exact: • • propellers can be manufactured from pounds of brass The answer is reasonably close to the estimate Copyright © 2014 Pearson Education, Inc 104 Chapter Multiplying and Dividing Fractions • (b) Divide the total number of quarts by the number of quarts needed for each oil change Estimate: rounds to rounds to Estimate: • • • Exact: Exact: • Estimate: • • • oil changes can be made with quarts of oil The answer is reasonably close to the estimate Exact: 10 The statement "When multiplying two mixed numbers, the reciprocal of the second mixed number must be used." is false A reciprocal is used when dividing fractions, not multiplying fractions The statement "If you were dividing a mixed number by the whole number , the reciprocal of would be " is false The reciprocal of is The statement "To round mixed numbers before estimating the answer, decide whether the numerator of the fraction part is less than or more than half of the denominator." is true Estimate: • 11 • Estimate: 12 • • Estimate: • • Exact: • • Estimate: • • • • • • • Exact: • 14 • • Estimate: • • • • Exact: • • Exact: • • Estimate: 15 • • Estimate: • • Exact: • • • • • • • Estimate: • Exact: Exact: • • 13 • Estimate: Exact: The statement "When rounding mixed numbers to estimate the answer to a problem, the estimated answer can vary quite a bit from the exact answer However, it can still show whether the exact answer is reasonable." is true • • • 2.8 Section Exercises • • • • • • Exact: Copyright © 2014 Pearson Education, Inc • • • • 2.8 Multiplying and Dividing Mixed Numbers • 16 • 26 Estimate: • • Estimate: • Exact: • • Estimate: • 17 • Exact: • Estimate: • 18 • The best estimate is choice (d) 27 • The best estimate is choice (a) Estimate: Estimate: 19 The best estimate is choice (b) Exact: Estimate: The best estimate is choice (c) • 28 20 Estimate: 21 Exact: • • Estimate: 29 Exact: Estimate: 22 Estimate: Exact: • Exact: • 30 Estimate: 23 Estimate: Exact: • • Exact: 31 24 Estimate: Estimate: Exact: Exact: 25 • • 32 Estimate: Estimate: Exact: • Exact: Copyright © 2014 Pearson Education, Inc • 105 106 Chapter Multiplying and Dividing Fractions 36 33 Divide each amount by Multiply each amount by (a) Applesauce: Estimate: • • cup cups (a) Flour: cups cup • Estimate: cups Exact: (b) Salt: Estimate: tsp Exact: tsp • • • (b) Salt: tsp tsp Exact: teaspoon Estimate: (c) Flour: cups Estimate: • cups • teaspoon Exact: • • cups Exact: (c) Applesauce: 34 Multiply each amount by • • • • 37 cup cups cup Exact: cups Exact: (b) Applesauce: Estimate: • cup cup Estimate: (a) Flour: cups Estimate: • cups Divide the number of gallons available by the number of gallons needed for each unit units Estimate: • cups Exact: (c) Vegetable oil: cup Estimate: • cups • • Exact: • Divide each amount by (a) Vanilla extract: • units can be painted with cup Exact: 35 cup • 38 tsp tsp gallons of paint Divide the number of total minutes by the number of minutes per moment moments Estimate: Exact: Estimate: tsp Exact: • • (b) Applesauce: cup cup There are Estimate: cup Exact: (c) Flour: Estimate: • cups 39 Each handle requires Use multiplication Estimate: cup Exact: cup Exact: moments in an 8-hour work day • jacks inches of steel tubing in • • • inches of steel tubing is needed to make Copyright © 2014 Pearson Education, Inc in 2.8 Multiplying and Dividing Mixed Numbers 46 40 Assume that the inch length listed in the overall dimensions is the length of the handle Use multiplication Estimate: Exact: , • • Exact: , • in • The amount of wood that is necessary to make handles is , inches 41 homes Estimate: in 107 Divide the total amount of roofing material by the amount of roofing material needed for each roof The answer should include: homes can be re-roofed with roofing material 47 Step Change mixed numbers to improper fractions squares of (a) The maximum height of the standard jack is inches Use multiplication Estimate: • Step Multiply the fractions • in • in Exact: Step Write the answer in lowest terms, changing to mixed or whole numbers where possible The hydraulic lift must raise the car 42 The additional step is to use the reciprocal of the second fraction (divisor) 43 Multiply the amount of money for each cell phone times the number of cell phones to get the total amount of money from the sale of gold (b) There are inches in a foot, so the -foottall mechanic is inches tall So no, the mechanic can not stand under the car without bending million Estimate: $ • • Exact: million million in Estimate: million million from the sale of the Exact: (b) No, because 49 in Multiply the swimming speed of the person times the number of times faster that a shark can swim than a person Estimate: • miles per hour • • • Exact: units can be carpeted Divide the total amount of firewood to be delivered by the amount of firewood that can be delivered per trip Estimate: in is greater than inches units Exact: 45 in • The low-profile lift must raise the car Divide the number of square yards of carpet by the amount of carpet needed for each apartment unit Estimate: (a) The maximum height of the low-profile jack is inches Use division • You would have $ gold 44 $ 48 inches trips 50 The shark can swim miles per hour Multiply the boxes of tile per floor times the number of floors (homes) to get the total number of boxes needed Estimate: Exact: • trips will be needed to deliver firewood cords of • • boxes • Exact: boxes of tile are needed Copyright © 2014 Pearson Education, Inc 108 Chapter Multiplying and Dividing Fractions 10 Chapter There Review Exercises are parts, and is shaded • There are parts, and are shaded The factors of 11 There are parts, and are shaded Proper fractions have numerator (top) smaller than denominator (bottom) are , , , and : • The factors of • Factorizations of : are , , Factorizations of • • They are: Proper fractions have numerator (top) smaller than denominator (bottom) , and • The factors of 13 • are , , , , , , • They are: Improper fractions have numerator (top) larger than or equal to the denominator (bottom) They are: • Factorizations of • 12 Factorizations of : : • • The factors of , and , and • • are , , , , , , , , , 14 Improper fractions have numerator (top) larger than or equal to the denominator (bottom) • They are: • • 15 • • • • • • • • • 16 Whole number part Remainder Whole number part 17 Remainder • • • • • 18 • • 19 • • 20 • • 21 All parts out of a possible Copyright © 2014 Pearson Education, Inc parts are gold , Chapter Review Exercises 22 18 of the possible parts are gold • • • • 35 23 24 of the possible of the possible parts are gold • • • 26 36 • 37 • • • 38 • 25 27 parts are gold 109 • • • • • • • • • • • • • • • • • • • • and 39 The fractions are equivalent 28 40 and 41 and • • • • The fractions are not equivalent 29 • • 42 • 43 • • • • • The fractions are equivalent • • • 44 • 30 45 31 • • • • • • • • • • • 47 The area is yd Multiply the length and width • 33 • 34 • • • • • • The area is ft To find the area, multiply the length and the width • • • To find the area, multiply the length and the width • • 46 32 • The area is Copyright © 2014 Pearson Education, Inc ft 110 48 Chapter Multiplying and Dividing Fractions Multiply the length and width • 55 Divide the total yardage by the amount needed for each pull cord pull cords Estimate: • Exact: The area is ft • • 49 Estimate: • pull cords can be made • 56 • Exact: • 50 Multiply the weight per gallon times the number of aquariums times the gallons per aquarium Estimate: • Estimate: • • • • • • • • Exact: The weight of the water is • • • , or pounds • 57 Exact: Ebony sold of • pounds of rice • • • pounds 51 Thus, pounds remain She gave of pounds to her parents Estimate: • • Exact: • • • pounds 52 Ebony gave pounds to her parents The amount she has left is pounds Estimate: 58 Exact: Sheila paid of $ for taxes, social security, and a retirement plan • • 53 Divide the total tons of almonds by the size of the bins • bins Estimate: Exact: She paid $ for taxes, social security, and a retirement plan She paid of the remainder, $ $ $ , for basic living expenses • bins will be needed to store the almonds 54 The • other equal partners own • She has $ • • • $ $ left of the business Divide that amount by 59 must be divided by • Each of the other partners owns • of the business Each school will receive Copyright © 2014 Pearson Education, Inc • • of the amount raised Chapter Review Exercises • 60 of the catch must be divided evenly among fishermen 71 [2.2] • Each fisherman receives • • 61 [2.5] • [2.5] • 63 [2.8] 64 65 [2.8] [2.7] • ton 72 • • • • • • • • • • • • 73 [2.4] 74 [2.4] 75 [2.4] 76 [2.4] 77 [2.4] 78 [2.4] 79 [2.8] Multiply ounces per gallon by the number of gallons • • • • • • • • • • • • • • • • • • • 66 [2.2] • • 62 111 [2.7] • 67 [2.5] 68 [2.8] • • • Estimate: ounces • • • • • • • Exact: • ounces of the product are needed 69 [2.2] Whole number part 80 [2.8] Multiply the number of tanks by the number of quarts needed for each tank Estimate: qt • • Remainder • Exact: quarts are needed 81 70 [2.2] [2.8] To find the area, multiply the length and the width • Whole number part 82 Remainder • The area of the stamp is in [2.8] To find the area, multiply the length and the width • • The area of the patio table top is Copyright © 2014 Pearson Education, Inc yd 112 Chapter Multiplying and Dividing Fractions Chapter There Test are parts, are shaded 10 There are parts, and are shaded 11 and Proper fractions have the numerator (top) smaller than the denominator (bottom) 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 13 Whole number part • • • • • • Multiply fractions by multiplying the numerators and multiplying the denominators Divide two fractions by using the reciprocal of the divisor (the second fraction) and then changing division to multiplication • • 14 Remainder 15 Factorizations of • : • • The factors of • • 16 Multiply the length and the width • are , , , , , and • • • The area of the grill is 17 • • • yd First, find the number of seedlings that don't survive • • Next, subtract to find the number that survive seedlings survive 18 • • • • • • • • • • 19 20 Divide the total length by the length of the pieces • • • • • • pieces can be cut Copyright © 2014 Pearson Education, Inc Cumulative Review Exercises (Chapters 1–2) • 21 Estimate: • • , , millions: ten-thousands: • Exact: • 22 Estimate: • • • Exact: , 23 Estimate: , , Exact: , , , 24 Estimate: Exact: , , , • • 25 / / • • • • • • • multiply to If grams can be synthesized per day, find the amount synthesized in days Estimate: • grams 10 Exact: • • • , • grams can be synthesized 11 Cumulative Review Exercises (Chapters 1–2) 12 hundreds: tens: Copyright © 2014 Pearson Education, Inc Attach 113 114 13 Chapter Multiplying and Dividing Fractions , R 16 , , To the nearest ten: Next digit is or less Tens place does not change All digits to the right of the underlined place change to zero , To the nearest hundred: Next digit is or more Hundreds place changes ( Check: To the nearest thousand: , 14 ) All digits to the right of the underlined place change to zero , , Next digit is R , or less Thousands place does not change , All digits to the right of the underlined place change to zero , Exponent Multiply Subtract 17 Check: 18 Square root Multiply Subtract Add • • , , 15 , 19 Multiply to find the amount used for the half-day and full-day tours; then add to find the total To the nearest ten: Next digit is or less Tens place does not change All digits to the right of the underlined place change to zero gallons of fuel are needed 20 Subtract to find the difference in cases To the nearest hundred: Next digit is or more Hundreds place changes ( , , , ) All digits to the right of the underlined place change to zero 21 To the nearest thousand: Next digit is There were mumps more cases of pertussis than Find the number of hairs lost in years and subtract to find the hairs remaining or more Thousands place changes ( , , ) All digits to the right of the underlined place change to zero , , , hairs remain Copyright © 2014 Pearson Education, Inc , , , Cumulative Review Exercises (Chapters 1–2) 22 Divide the total number of hours by the number of workers Whole number part 31 Remainder Each health care worker will work 23 Multiply the number of flushes and the amount of water used per flush to find the number of gallons of water used • • 32 • gallons of water are used in 24 hours flushes Divide the total length by the length of the pieces • • • • • • 33 25 26 27 pieces can be cut proper because the numerator is than the denominator improper because the numerator is than or the same as the denominator is proper because the numerator than the• denominator is smaller is larger is smaller • • • • • • • • • • 34 28 • 29 35 • 36 • 37 • • • • • • • 30 Whole number part Remainder 38 39 40 Copyright © 2014 Pearson Education, Inc 115 116 Chapter Multiplying and Dividing Fractions • • • 41 42 • • • • • • • • • 43 44 • 45 46 Copyright â 2014 Pearson Education, Inc ... Factorizations of : • • The factors of , and 13 , and • are , , , , , Factorizations of • • , • • are , , 17 • are , , , , Factorizations of : • The factors of Factorizations of The factors of , and. .. Multiplying and Dividing Fractions and 48 and • • • • • • • • • • • The fractions are not equivalent The fractions are equivalent 44 49 and • and • • • • • • • • • and • • • • • • • • • • • 51 and •... Multiply and Add 74 Chapter Multiplying and Dividing Fractions • • Multiply and 11 Multiply and 21 Add Add • 12 Multiply and • Multiply 22 Add Add • • Multiply 13 and and Multiply 23 and