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Operations withFractions?MODULE4LESSON 4.1ESSENTIAL QUESTIONApplying GCF andLCM to FractionOperationsHow can you use operationswith fractions to solvereal-world problems?6.NS.4LESSON 4.2Dividing Fractions6.NS.1LESSON 4.3Dividing MixedNumbers6.NS.1LESSON 4.4Solving MultistepProblems withFractions and MixedNumbers Houghton Mifflin Harcourt Publishing Company Image Credits: Patrick Batchelder/Alamy6.NS.1Real-World Videomy.hrw.commy.hrw.comTo find your average rate of speed, divide the distanceyou traveled by the time you traveled. If you ride in ataxi and drive 12 mile in 14 hour, your rate was 2 mi/hwhich may mean you were in heavy traffic.my.hrw.comMath On the SpotAnimated MathPersonal Math TrainerGo digital with yourwrite-in studentedition, accessible onany device.Scan with your smartphone to jump directlyto the online edition,video tutor, and more.Interactively explorekey concepts to seehow math works.Get immediatefeedback and help asyou work throughpractice sets.75
Are YOU Ready?PersonalMath TrainerComplete these exercises to review skills you will needfor this module.Write an Improper Fractionas a Mixed NumberEXAMPLE13 55 55 355 1 1 35 2 35 2 35my.hrw.comOnline Practiceand HelpWrite as a sum using names for one plus aproper fraction.Write each name for one as one.Add the ones.Write the mixed number.Write each improper fraction as a mixed number.1. 942. 83233.6114.2175.5156.8337.10298.12Multiplication FactsEXAMPLE7 6 7 6 42Use a related fact you know.6 6 36Think: 7 6 (6 6) 6 36 6 429. 6 510. 8 911. 10 1112. 7 813. 9 714. 8 615. 9 1116. 11 12Division FactsEXAMPLE63 7 Think:63 7 9So, 63 7 9.7 times what number equals 63?7 9 63Divide.76Unit 217. 35 718. 56 819. 28 720. 48 821. 36 422. 45 923. 72 824. 40 5 Houghton Mifflin Harcourt Publishing CompanyMultiply.
Reading Start-UpVisualize VocabularyUse the words to complete the triangle. Write the review wordthat fits the description in each section of the triangle.partof a wholetop numberof a fractionVocabularyReview Wordsarea (área) denominator(denominador) fraction (fracción)greatest common factor(GCF) (máximo comúndivisor (MCD))least common multiple(LCM) (mínimo comúnmúltiplo (m.c.m.))length (longitud) numerator (numerador)product (producto)width (ancho)Preview Wordsbottom number of a fractionmixed number (númeromixto)order of operations (ordende las operaciones)reciprocals (recíprocos)Understand Vocabulary Houghton Mifflin Harcourt Publishing CompanyIn each grouping, select the choice that is described by the givenvocabulary word.1. reciprocalsA 1:1531B 4 6C3and 5352. mixed number1 1A -3 51B 32C -53. order of operationsA 5-3 2 0B 5-3 2 4C 5-3 2 6Active ReadingLayered Book Before beginning the module,create a layered book to help you learn theconcepts in this module. Label each flap withlesson titles. As you study each lesson, writeimportant ideas, such as vocabulary andprocesses, under the appropriate flap. Referto your finished layered book as you work onexercises from this module.Module 477
GETTING READY FOROperations with FractionsUnderstanding the standards and the vocabulary terms in the standardswill help you know exactly what you are expected to learn in this module.6.NS.1Interpret and computequotients of fractions, andsolve word problems involvingdivision of fractions byfractions, e.g., by using visualfraction models and equationsto represent the problem.Key VocabularyWhat It Means to YouYou will learn how to divide two fractions. You will also understandthe relationship between multiplication and division.EXAMPLE 6.NS.1Zachary is making vegetable soup. The recipe makes 6 34 cups ofsoup. How many 1 12-cup servings will the recipe make?6 34 1 12quotient (cociente)The result when one number isdivided by another.27 324fraction (fracción)A number in the form ba , whereb 0. 9227 ·24 3 4 12The recipe will make 4 21 servings.Find the greatest commonfactor of two whole numbersless than or equal to 100 andthe least common multiple oftwo whole numbers less than orequal to 12. Use the distributiveproperty to express a sum oftwo whole numbers 1–100 witha common factor as a multipleof a sum of two whole numberswith no common factor.Visit my.hrw.comto see all CACommon CoreStandardsexplained.my.hrw.com78Unit 2What It Means to YouYou can use greatest common factors and least common multiplesto simplify answers when you calculate with fractions.EXAMPLE 6.NS.4Add. Write the answer in simplest form.1Use the LCM of 3 and 6 as 16 26 163a common denominator.2 1 6Add the numerators. 363 3 6 3Simplify by dividing by the GCF.The GCF of 3 and 6 is 3. 12Write the answer in simplest form. Houghton Mifflin Harcourt Publishing Company6.NS.4
LESSON4.1?Applying GCF andLCM to FractionOperationsESSENTIAL QUESTION6.NS.4Find the greatest common factorof two whole numbers less than orequal to 100 and the least commonmultiple of two whole numbers lessthan or equal to 12. Use the distributiveproperty to express a sum of two wholenumbers 1–100 with a common factoras a multiple of a sum of two wholenumbers with no common factor.How do you use the GCF and LCM when adding, subtracting,and multiplying fractions?Multiplying FractionsTo multiply two fractions you first multiply the numerators and then multiplythe denominators.numerator numeratornumerator denominator denominatordenominatorMath On the Spotmy.hrw.comThe resulting product may need to be written in simplest form. To writea fraction in simplest form, you can divide both the numerator and thedenominator by their greatest common factor.Example 1 shows two methods for making sure that the product of twofractions is in simplest form.EXAMPL 1EXAMPLE6.NS.4Multiply. Write the product in simplest form.A 13 35 Houghton Mifflin Harcourt Publishing Company1 31 35 33 5Write the problem as a single fraction.3 15Multiply numerators. Multiply denominators.3 3 15 3Simplify by dividing by the GCF.The GCF of 3 and 15 is 3. 15Write the answer in simplest form.6 in the numerator and 3 in thedenominator have a common factorother than one. Divide by the GCF 3.B 67 2366 2 23 77 3Write the problem as a single fraction.26 2 7 3Simplify before multiplying using the GCF.2 2 7 1Multiply numerators. Multiply denominators.1 47Math TalkMathematical PracticesCompare the methods inthe Example. How do youknow if you can use themethod in B ?Lesson 4.179
YOUR TURNPersonalMath TrainerMultiply. Write each product in simplest form.Online Practiceand Helpmy.hrw.com1. 16 352. 34 793. 37 234. 45 2787 5.10216. 67 16Multiplying Fractions andWhole NumbersMath On the Spotmy.hrw.comTo multiply a fraction by a whole number, you can rewrite the whole numberas a fraction and multiply the two fractions. Remember to use the GCF to writethe product in simplest form.EXAMPLE 26.NS.4A class has 18 students. The teacher asks how many students in the class havepets and finds 59 of the students have pets. How many students have pets?Estimate the product. Multiply the whole number by the nearestbenchmark fraction.5is close to 12 , so multiply 12 times 18.91 18 92STEP 2Multiply. Write the productin simplest form.5 189Math TalkMathematical PracticesHow can you checkto see if the answer iscorrect?518 18 59 19 18 59 1215 2 1 110 10 110 students have pets.80Unit 25times 18You can write9three ways.5 1895· 1895(18)9Rewrite 18 as a fraction.Simplify before multiplying using the GCF.Multiply numerators. Multiply denominators.Simplify by writing as a whole number. Houghton Mifflin Harcourt Publishing Company Image Credits: Life on white/AlamySTEP 1
Reflect7.Analyze Relationships Is the product of a fraction less than 1 and awhole number greater than or less than the whole number? Explain.YOUR TURNMultiply. Write each product in simplest form.9.3 20510.1 8311.1 14412.7 731013.3 102108.5 248PersonalMath TrainerOnline Practiceand Helpmy.hrw.comAdding and Subtracting FractionsYou have learned that to add or subtract two fractions, you can rewrite thefractions so they have the same denominator. You can use the least commonmultiple of the denominators of the fractions to rewrite the fractions.EXAMPL 3EXAMPLE6.NS.4Math On the Spotmy.hrw.com81Add15 6 . Write the sum in simplest form. Houghton Mifflin Harcourt Publishing CompanySTEP 1Rewrite the fractions as equivalent fractions. Use theLCM of the denominators.88 216 1515 2301 551 66 530STEP 2The LCM of 15 and 6 is 30.Mathematical PracticesAdd the numerators of the equivalent fractions. Then simplify.16521 30 3030 3 2130 37 10Math TalkCan you use anothercommon multiple ofthe denominators inExample 3 to findthe sum? Explain.Simplify by dividing by the GCF.The GCF of 21 and 30 is 3.Reflect14.Can you also use the LCM of the denominators of the fractions to81rewrite the difference15 - 6 ? What is the difference?Lesson 4.181
YOUR TURNPersonalMath TrainerOnline Practiceand Helpmy.hrw.comAdd or subtract. Write each sum or difference in simplest form.515. 16145316.-12 20517.- 3812318. 1 141019. 23 6 1520. 3 16 - 17Guided PracticeMultiply. Write each product in simplest form. (Example 1)1. 12 582. 35 593. 38 254. 2 38 1655. 1 45 122 56. 110Find each amount. (Example 2)7. 14 of 12 bottles of water bottles9. 35 of 40 restaurant bill 8. 23 of 24 bananas 10. 56 of 18 pencils bananaspencils511. 38 2451 12.20129- 113.20 493-14.10 14515. 3 38 1275-16. 510 18?ESSENTIAL QUESTION CHECK-IN17. How can using the GCF and LCM help you when you add, subtract, andmultiply fractions?82Unit 2 Houghton Mifflin Harcourt Publishing CompanyAdd or subtract. Write each sum or difference in simplest form.
NameClassDate4.1 Independent PracticePersonalMath Trainer6.NS.4Solve. Write each answer in simplest form.18. Erin buys a bag of peanuts that weighs3 of a pound. Later that week, the bag4is 23 full. How much does the bag ofpeanuts weigh now? Show your work.my.hrw.comOnline Practiceand Help21. Marcial found a recipe for fruit salad thathe wanted to try to make for his birthdayparty. He decided to triple the recipe.Fruit Salad3 12 cups thinly sliced rhubarb15 seedless grapes, halved19. Multistep Marianne buys 16 bags ofpotting soil that comes in 58 -pound bags.a. How many pounds of potting soil doesMarianne buy?1 orange, sectioned210 fresh strawberries, halved3 apple, cored and diced52 peach, sliced31 plum, pitted and sliced1 cup fresh blueberries4 Houghton Mifflin Harcourt Publishing Company Image Credits:b. If Marianne’s father calls and says heneeds 13 pounds of potting soil, howmany additional bags should she buy?a. What are the new amounts for theoranges, apples, blueberries, andpeaches?20. Music Two fifths of the instruments in themarching band are brass, one third arepercussion, and the rest are woodwinds.a. What fraction of the band iswoodwinds?b. One half of the woodwinds areclarinets. What fraction of the band isclarinets?b. Communicate Mathematical IdeasThe amount of rhubarb in the originalrecipe is 3 12 cups. Using what you knowof whole numbers and what you knowof fractions, explain how you couldtriple that mixed number.c. One eighth of the brass instrumentsare tubas. If there are 240 instrumentsin the band, how many are tubas?Lesson 4.183
22. One container holds 1 87 quarts of water and a second container holds5 34 quarts of water. How many more quarts of water does the secondcontainer hold than the first container?23. Each of 15 students will give a 1 12 -minute speech in English class.a. How long will it take to give the speeches?b. Suppose the teacher begins recording on a digital camera with anhour available. Show that there is enough time to record everyoneif she gives a 15-minute introduction at the beginning of class andevery student takes a minute to get ready.c. How much time is left on the digital camera?FOCUS ON HIGHER ORDER THINKING24. Represent Real-World Problems Kate wants to buy a new bicycle froma sporting-goods store. The bicycle she wants normally sells for 360. Thestore has a sale in which all bicycles cost 65 of the regular price. What is thesale price of the bicycle?26. Justify Reasoning To multiply a whole number by a fraction, you canfirst write the whole number as a fraction with a denominator of 1.Explain why following this step does not change the product.84Unit 2Work Area Houghton Mifflin Harcourt Publishing Company25. Error Analysis To find the product 73 94 , Cameron found the GCF of43 and 9. He then multiplied the fractions 17 and 49 to find the product63.What was Cameron’s error?
LESSON4.2 Dividing Fractions?6.NS.1Interpret and computequotients of fractions, andsolve word problems involvingdivision of fractions by fractions,e.g., by using visual fractionmodels and equations torepresent the problem.ESSENTIAL QUESTIONHow do you divide fractions?EXPLORE ACTIVITY 16.NS.1Modeling Fraction DivisionFor some real-world problems, you may need to divide a fraction bya fraction. For others, you may need to divide a fraction by a wholenumber.A You have 34 cup of salsa for making burritos. Eachburrito requires 18 cup of salsa. How many burritoscan you make?34To find the number of burritos that canbe made, you need to determine howmany 18 -cup servings are in 34 cup.In the diagram, 34 of a whole is dividedinto quarters and into eighths. How many18eighths are there in 34 ?You have enough salsa to makeburritos. Houghton Mifflin Harcourt Publishing CompanyB Five people share 12 pound of cheeseequally. How much cheese does eachperson receive?To find how much cheese each personreceives, you can divide 12 poundinto 5 equal parts. Use the diagram todetermine what fraction of a whole poundeach person receives.Each person receivespound.Reflect1. Write the division shown by the models inAandB.Lesson 4.285
ReciprocalsAnother way to divide fractions is to use reciprocals. Two numbers whoseproduct is 1 are reciprocals.Math On the Spotmy.hrw.com312 43 14123 and 4 are reciprocals.43To find the reciprocal of a fraction, switch the numerator and denominator.numerator ·denominator 1denominatornumeratorEXAMPLE 1Prep for 6.NS.1Find the reciprocal of each number.A 2992Switch the numerator and denominator.The reciprocal of 92 is 92.Math TalkMathematical PracticesHow can you checkthat the reciprocal inA is correct?B 1881Switch the numerator and denominator.The reciprocal of 81 is 81, or 8.C 55 5151Rewrite as a fraction.15Switch the numerator and the denominator.The reciprocal of 5 is 51.Reflect2. Is any number its own reciprocal? If so, what number(s)? Justify your answer.4. The reciprocal of a whole number is a fraction withnumerator.in theYOUR TURNPersonalMath TrainerOnline Practiceand Helpmy.hrw.com86Unit 2Find the reciprocal of each number.75. 86. 917. 11 Houghton Mifflin Harcourt Publishing Company3. Communicate Mathematical Ideas Does every number have areciprocal? Explain.
EXPLORE ACTIVITY 26.NS.1Using Reciprocals to DivideA The diagram below models the division problem 5 58.55858585858585858Each unit is divided into eighths. How many sections representing 58 arethere in 5?sections5 5 8B Use the diagrams to complete the division problems.54585858585 5 5 5 4 88 8C Complete the table. What do you notice about theQuotientquotient and product in each row? What do you notice5 about the divisor in column 1 and the multiplier in5 8column 2?5 5 4 85 5 Houghton Mifflin Harcourt Publishing Company88Product8 5 55 8 455 8 85Reflect8. Make a Conjecture Use the pattern in the table to make a conjectureabout how you can use multiplication to divide a number by a fraction.9. Sketch a model on a separate piece of paper that you could use to solvethe division problem 67 27. How can you use the pattern you see in thetable in C to solve this division problem?Lesson 4.287
Using Reciprocals to Divide FractionsDividing by a fraction is equivalent tomultiplying by its reciprocal.1 14 45541 4515Math On the Spotmy.hrw.comEXAMPLE 26.NS.1Divide 59 23 . Write the quotient in simplest form.STEP 1AnimatedMathRewrite as multiplication, using the reciprocal of the divisor.5 2 59 329 32The reciprocal ofis 32 .3my.hrw.comSTEP 2Multiply and simplify.515 32 918 565 23 569Multiply the numerators. Multiply the denominators.Write the answer in simplest form.15 3 518 3 6YOUR TURNPersonalMath TrainerOnline Practiceand HelpDivide.911. 35 10910. 25 10my.hrw.comGuided Practice1. 252. 9103.3Divide. (Explore 1, Explore 2, and Example 2)4. 43 53 ?35. 45 10ESSENTIAL QUESTION CHECK-IN7. Explain how to divide a number by a fraction.88Unit 26. 12 25 Houghton Mifflin Harcourt Publishing CompanyFind the reciprocal of each fraction. (Example 1)
NameClassDate4.2 Independent Practice6.NS.18. 8 23 9. 7 34 4 25 10.1511. 58 78 7 38 12.167 13. 45 1014. Alison has 12 cup of yogurt for making fruitparfaits. Each parfait requires 18 cup ofyogurt. How many parfaits can she make? Houghton Mifflin Harcourt Publishing Company Image Credits: Irochka/Fotolia15. How many tenths are there in 45 ?PersonalMath Trainermy.hrw.comOnline Practiceand Help18. A pitcher contains 23 quart of water. If anequal amount of water is poured into eachof 6 glasses, how much water will eachglass contain?19. Jackson wants to divide a 34 -pound box oftrail mix into small bags. Each of the bags1will holdpound of trail mix. How many12bags of trail mix can Jackson fill?20. You make a salad to share with friends.Your sister eats 13 of it before they arrive.a. You want to divide the leftover saladevenly among six friends. Whatexpression describes the situation?Explain.16. Biology Suppose one honeybee makes1teaspoon of honey during its lifetime.12How many honeybees are needed tomake 12 teaspoon of honey?b. What fraction of the original salad doeseach friend receive?17. A team of runners is needed to run a 14 -mile1relay race. If each runner must runmile,16how many runners will be needed?21. Six people share 53 pound of peanutsequally. What fraction of a pound ofpeanuts does each person receive?Lesson 4.289
922. Liam hasgallon of paint for painting the birdhouses he sells at the craft101fair. Each birdhouse requiresgallon of paint. How many birdhouses can20Liam paint? Show your work.FOCUS ON HIGHER ORDER THINKINGWork Area23. Interpret the Answer The length of a ribbon is 34 meter. Sun Yi needspieces measuring 13 meter for an art project. What is the greatest numberof pieces measuring 13 meter that can be cut from the ribbon? How muchribbon will be left after Sun Yi cuts the ribbon? Explain your reasoning.24. Represent Real-World Problems Write and solve a real-world problemthat can be solved using the expression 34 16.25. Justify Reasoning When Kaitlin divided a fraction by 12, the result wasa mixed number. Was the original fraction less than or greater than 12 ?Explain your reasoning.127. Make a Prediction Susan divides the fraction 58 by. Her friend Robyn1651divides 8 by 32 . Predict which person will get the greater quotient. Explainand check your prediction.90Unit 2 Houghton Mifflin Harcourt Publishing Company26. Communicate Mathematical Ideas The reciprocal of a fraction less than1 is always a fraction greater than 1. Why is this?
Personal Math Trainer Interactively explore key concepts to see how math works. Animated Math Go digital with your write-in student edition, accessible on any device. Scan with your smart phone to jump directly to the online edition, video tutor, and more. Math On the Spot MODULE my.hrw.com
