Multiplying And Dividing Fractions MODULE 3

LESSON3.2?Multiplying MixedNumbersNumberand operations—6.3.E Multiply and dividepositive rational numbersfluently. Also 6.3.BESSENTIAL QUESTIONHow do you multiply mixed numbers?EXPLORE ACTIVITY6.3.BModeling Mixed-Number MultiplicationYour town is building a new playground. One third of the 1 34-acre playgroundwill be developed to include a swimming pool and splash pad. Find thenumber of acres that will be used for the swimming pool and splash pad.1 Follow these steps to model the situation and find3.A Shade the model to show the amount of land takenup by the entire playground.The whole playground3is 1acres.4334 . 11 4 .4 4 4 4 Houghton Mifflin Harcourt Publishing CompanyB Double shade 13 of 1 34 to show the portion of the land that is taken upby the swimming pool and splash pad. Divide the remaining fourthinto three parts so that all of the parts are equal.Cacre will be used for the swimming pool and splash pad.D Did the fraction representing the entire playground increase ordecrease when multiplied by 13? Explain.Reflect1. Make a Conjecture Will the product 1 12 12 be greater than orless than 12? Explain.Lesson 3.269

Multiplying Fractions andMixed NumbersTo rename a mixed number as a fraction, first multiply thedenominator of the fraction by the whole number. Then addthe product to the numerator.Math On the Spotmy.hrw.comThen add.2 13 73First multiply.EXAMPLE 1My NotesThe science club members are planting a garden. They used 2 14 bags ofgravel as a border for the flower section of the garden. They will need 13as much gravel for the vegetable section as for the flower section.How much gravel will they use in the vegetable section?STEP 1Estimate the product. Round the mixed number to the nearestwhole number. Find the nearest benchmark for the fraction.2 14 is close to 2, so multiply 13 times 2.1 2 233STEP 2Multiply. Write the product in simplest form.1 2 143Math Talk1 2 14 13 943Mathematical ProcessesHow can youdetermine if the answeris reasonable? 9 13 4Rewrite the mixed number as a fractiongreater than 1.3Simplify before multiplying using the GCF. 3 11 4Rewrite fraction. 34Multiply numerators.Multiply denominators.The science club members will need 34 bag of gravel for thevegetable section of the garden.YOUR TURNMultiply. Write each product in simplest form.PersonalMath TrainerOnline Assessmentand Interventionmy.hrw.com70Unit 22. 3 13 343. 1 45 124. 56 2 345. 35 2 1596. 4 13107. 5 16 18 Houghton Mifflin Harcourt Publishing Company1

Multiplying Mixed NumbersMultiplying a mixed number by another mixed number is the same asmultiplying a fraction by a mixed number. Rewrite each mixed number as afraction. Then multiply the numerators and multiply the denominators.Math On the Spotmy.hrw.comEXAMPL 2EXAMPLE6.3.EGrace is making 2 12 batches of muffins for her school’s annual bake sale.If one batch of muffins requires 1 14 cups of flour, how many cups of flourdoes Grace need to make 2 12 batches of muffins?STEP 1Estimate the product. Round each mixed number to the nearestwhole number.2 12 is close to 3 and 1 14 is close to 1, so multiply 3 by 1.3 1 3STEP 2Multiply. Write the product in simplest form.2 12 1 142 12 1 14 52 54Rewrite each mixed number as afraction greater than 1.5 5 2 4Multiply numerators.Multiply denominators.25 , or 3 188Write the fraction greater than 1 as amixed number.Grace will need 3 18 cups of flour.Reflect Houghton Mifflin Harcourt Publishing Company8. Analyze Relationships When you multiply two mixed numbers, will theproduct be less than or greater than the factors? Use an example to explain.YOUR TURNMultiply. Write each product in simplest form.9. 2 23 1 1710. 2 38 1 3511. 4 12 3 3712. 5 14 4 23PersonalMath TrainerOnline Assessmentand Interventionmy.hrw.comLesson 3.271

Guided Practice1. Mr. Martin’s yard is 1 13 acres. He wants to plant grass on 16 of his yard.(Explore Activity)a. Draw a model to show how many acres will be covered by grass.b. How many acres will be covered by grass?c. Write the multiplication shown by the model.d. Will the mixed number that represents the original size ofMr. Martin’s yard increase or decrease when multiplied by 16 ? Explain.2. 1 15 353. 1 34 474. 1 56 2575. 1 451096. 59 3107. 78 3 138. 2 15 2 359. 4 34 3 45?ESSENTIAL QUESTION CHECK-IN72Unit 210. How can you multiply two mixed numbers? Houghton Mifflin Harcourt Publishing CompanyMultiply. Write each product in simplest form. (Example 1 and Example 2)

NameClassDate3.2 Independent Practice6.3.B, 6.3.EEstimate. Then solve.11. Carly is making 3 12 batches of biscuits. Ifone batch calls for 2 13 cups of flour, howmuch flour will she need?PersonalMath Trainermy.hrw.comOnlineAssessment andIntervention16. Ron wants to make a rectangularbasketball court. What is the area ofRon’s court?3 25 yd12. Bashir collected 4 13 baskets of peachesat an orchard. If each basket holds21 peaches, how many peaches did hecollect in all?4 34 yd17. Each of 15 students will give a 1 12 -minutespeech in English class.a. How long will it take to give thespeeches? Houghton Mifflin Harcourt Publishing Company13. Jared used 1 25 bags of soil for his garden.He is digging another garden that willneed 15 as much soil as the original. Howmuch will he use total?14. Critical Thinking Is the product of twomixed numbers less than, between, orgreater than the two factors? Explain.15. There are approximately 402 14 metersaround a typical running track. Sandra haschallenged herself to run 10 laps a day for5 days. How many meters will Sandra run ifshe meets her challenge?b. If the teacher begins recording on adigital camera with an hour available, isthere enough time to record everyoneif she gives a 15-minute introductionat the beginning of class and everystudent takes a minute to get ready?Explain.c. How much time is left on the digitalcamera?Lesson 3.273

Communicate Mathematical Ideas How is multiplying a wholenumber by a mixed number the same as multiplying two mixednumbers?FOCUS ON HIGHER ORDER THINKING19.Critique Reasoning To find the product 3 38 4 19 , Tara rewrote 3 38 as1317221 .and 4 19 as. Then she multiplied the fractions to find the product8972What were her errors?20.Represent Real-World Problems Ian is making his special barbecuesauce for a party. His recipe makes 3 12 cups of barbecue sauce and uses2 41 tablespoons of soy sauce. He wants to increase his recipe to makefive times as much barbecue sauce. He checks his refrigerator and findsthat he has 8 tablespoons of soy sauce. Will he have enough soy sauce?Explain.21.Analyze Relationships Is it possible to find the product of two mixednumbers by multiplying the whole number parts together, thenmultiplying the two fractional parts together, and finally adding the twoproducts? Use an example to support your answer.74Unit 2Work Area Houghton Mifflin Harcourt Publishing Company18.

LESSON3.3 Dividing Fractions?Number andoperations—6.3.EMultiply and divide positiverational numbers fluently.Also 6.3.AESSENTIAL QUESTIONHow do you divide fractions?6.3.EEXPLORE ACTIVITY 1Modeling Fraction DivisionIn some division problems, you may know a number of groups and need tofind how many or how much are in each group. In other division problems,you may know how many there are in each group, and need to find thenumber of groups.A You have 34 cup of salsa for making burritos. Eachburrito requires 18 cup of salsa. How many burritoscan you make?To find the number of burritos that canbe made, you need to determine howmany 18 -cup servings are in 34 cups.Use the diagram. How many eighths34are there in 34 ? Houghton Mifflin Harcourt Publishing CompanyYou have enough salsa to makeburritos.18B Five people share 12 pound of cheeseequally. How much cheese does eachperson receive?To find how much cheese each personreceives, you need to determine howmuch is in each ofparts.How much is in each part?Each person will receivepound.Reflect1. Write the division shown by each model.Lesson 3.375

ReciprocalsAnother way to divide fractions is to use reciprocals. Two numbers whoseproduct is 1 are reciprocals.312 43 1412Math On the Spotmy.hrw.com3 and 4 are reciprocals.43To find the reciprocal of a fraction, switch the numerator and denominator.numerator ·denominator 1denominatornumeratorEXAMPLE 16.3.AFind the reciprocal of each number.A 2992Switch the numerator and denominator.The reciprocal of 29 is 92.B 1881Switch the numerator and denominator.The reciprocal of 18 is 81, or 8.C 5Math Talk5 51Mathematical ProcessesHow can you checkyour answer?51Rewrite as a fraction.15Switch the numerator and the denominator.The reciprocal of 5 is 15.Reflect3. Communicate Mathematical Ideas Does every number have areciprocal? Explain.4. The reciprocal of a whole number is a fraction withnumerator.in theYOUR TURNPersonalMath TrainerOnline Assessmentand Interventionmy.hrw.com76Unit 2Find the reciprocal of each number.75. 86. 917. 11 Houghton Mifflin Harcourt Publishing Company2. Is any number its own reciprocal? If so, what number(s)? Justify your answer.

EXPLORE ACTIVITY 26.3.AUsing Reciprocals to FindEquivalent ValuesA Complete the table below.DivisionMultiplication6 2 37 76 7 7 25 3 58 8 35 8 8 31 5 16 6 561 6 51 1 34 3 41 3 4 1B How does each multiplication problem compare to its correspondingdivision problem?C How does the answer to each multiplication problem compare to theanswer to its corresponding division problem? Houghton Mifflin Harcourt Publishing CompanyReflect8. Make a Conjecture Use the pattern in the table to make a conjectureabout how you can use multiplication to divide one fraction by another.9. Write a division problem and a corresponding multiplication problem likethose in the table. Assuming your conjecture in 8 is correct, what is theanswer to your division problem?Lesson 3.377

Using Reciprocals to Divide FractionsMath On the Spotmy.hrw.comDividing by a fraction is equivalent tomultiplying by its reciprocal. So, to divideby a fraction, multiply by its reciprocal.1 14 45541 4515EXAMPLE 26.3.ADivide 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 918565 23 569Multiply the numerators. Multiply the denominators.Write the answer in simplest form.15 3 518 3 6YOUR TURNPersonalMath TrainerOnline Assessmentand InterventionDivide.911. 35 10910. 25 10my.hrw.comFind the reciprocal of each fraction. (Example 1)1. 252. 19103.3Divide. (Explore 1, Explore 2, and Example 2)4. 43 53 ?ESSENTIAL QUESTION CHECK-IN7. How do you divide fractions?7835. 45 10Unit 26. 12 25 Houghton Mifflin Harcourt Publishing CompanyGuided Practice

NameClassDate3.3 Independent Practice6.3.A, 6.3.E8. 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/Fotolia9. A team of runners is needed to run a 14 -mile1relay race. If each runner must runmile,16how many runners will be needed?PersonalMath Trainermy.hrw.comOnlineAssessment andIntervention13. 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?14. A pitcher contains 23 quart of lemonade.If an equal amount of lemonade ispoured into each of 6 glasses, how muchlemonade will each glass contain?10. Trevor paints 16 of the fence surrounding hisfarm each day. How many days will it takehim to paint 34 of the fence?15. How many tenths are there in 45 ?11. Six people share 35 pound of peanutsequally. What fraction of a pound ofpeanuts does each person receive?16. You make a large bowl of salad to sharewith your friends. Your brother eats 13 of itbefore they come over.112. Biology If one honeybee makes12teaspoon of honey during its lifetime, howmany honeybees are needed to make 12teaspoon of honey?a. You want to divide the leftover saladevenly among six friends. Whatexpression describes the situation?Explain.b. What fractional portion of the originalbowl of salad does each friend receive?Lesson 3.379

FOCUS ON HIGHER ORDER THINKINGWork Area17. 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.918. Represent Real-World Problems Liam hasgallon of paint for painting101the birdhouses he sells at the craft fair. Each birdhouse requiresgallon20of paint. How many birdhouses can Liam paint? Show your work.19. 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.121. 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.80Unit 2 Houghton Mifflin Harcourt Publishing Company20. Communicate Mathematical Ideas The reciprocal of a fraction less than1 is always a fraction greater than 1. Why is this?

Dividing MixedNumbersLESSON3.4?Number andoperations—6.3.EMultiply and divide positiverational numbers fluently.Also 6.3.AESSENTIAL QUESTIONHow do you divide mixed numbers?6.3.EEXPLORE ACTIVITYModeling Mixed Number DivisionAntoine is making sushi rolls. He has 2 12 cups of rice and will use 14 cupof rice for each sushi roll. How many sushi rolls can he make?A To find the number of sushi rolls that can be made, youneed to determine how many fourths are in 2 12. Use fractionpieces to represent 2 12 on the model below.114141211414141414141414B How many fourths are in 2 12? Houghton Mifflin Harcourt Publishing Company Image Credits: Stockbyte /GettyImagesAntoine has enough rice to makesushi rolls.Reflect1.Communicate Mathematical Ideas Which mathematical operationcould you use to find the number of sushi rolls that Antoine can make?Explain.2.Multiple Representations Write the division shown by the model.3.What If? Suppose Antoine instead uses 18 cup of rice for each sushi roll.How would his model change? How many rolls can he make? Explain.Lesson 3.481

Using Reciprocals to DivideMixed NumbersMath On the Spotmy.hrw.comMy NotesDividing by a fraction is equivalent to multiplying by its reciprocal. You can usethis fact to divide mixed numbers. First rewrite the mixed numbers as fractionsgreater than 1. Then multiply the first fraction by the reciprocal of the secondfraction.EXAMPLE 16.3.AOne serving of Harold’s favorite cereal contains 1 25 ounces. How manyservings are in a 17 12 -ounce box?STEP 1Write the situation as a division problem.17 12 1 25STEP 2You need to find how many2117groups of 12.5 are inRewrite the mixed numbers as fractions greater than 1.35 7517 12 1 25 2Rewrite the problem as multiplication using the reciprocal ofthe second fraction.3535 75 5722STEP 475The reciprocal of5 is 7 .Multiply.355 5 35 5772215 5 2 125, or 12 12 2Simplify first using the GCF.Multiply numerators. Multiply denominators.Write the result as a mixed number.There are 12 12 servings of cereal in the box.Reflect82Unit 24.Analyze Relationships Explain how you can check the answer.5.What If? Harold serves himself 1 12 -ounce servings of cereal eachmorning. How many servings does he get from a box of his favoritecereal? Show your work. Houghton Mifflin Harcourt Publishing CompanySTEP 3

YOUR TURN6.Sheila has 10 12 pounds of potato salad. She wants to divide the potatosalad into containers, each of which holds 1 14 pounds. How many containersdoes she need? Explain.PersonalMath TrainerOnline Assessmentand Interventionmy.hrw.comSolving Problems Involving AreaRecall that to find the area of a rectangle, you multiply length width. If youknow the area and only one dimension, you can divide the area by the knowndimension to find the other dimension.Math On the Spotmy.hrw.comEXAMPL 2EXAMPLE6.3.EThe area of a rectangular sandbox is 56 23 square feet. The length of thesandbox is 8 12 feet. What is the width?STEP 1Write the situation as a division problem.56 23 8 12STEP 2Math TalkRewrite the mixed numbers as fractions greater than 1.170 1756 23 8 12 32STEP 3Rewrite the problem as multiplication using the reciprocal ofthe second fraction.Mathematical ProcessesExplain how to findthe length of a rectanglewhen you know the areaand the width. Houghton Mifflin Harcourt Publishing Company1701702 17 17323STEP 4Multiply.10 170 21702 1733 1720, or 6 23 3Multiply numerators. Multiply denominators.1Simplify and write as a mixed number.The width of the sandbox is 6 23 feet.Reflect7.Check for Reasonableness How can you determine if your answeris reasonable?Lesson 3.483

YOUR TURNPersonalMath TrainerOnline Assessmentand Intervention8.The area of a rectangular patio is 12 38 square meters.The width of the patio is 2 34 meters. What is the length?9.1The area of a rectangular rug is 1412 square yards.The length of the rug is 4 13 yards. What is the width?my.hrw.comGuided PracticeDivide. Write each answer in simplest form. (Explore Activity and Example 1)1. 4 14 342. 1 12 2 143 4442423. 4 1 18 4. 3 15 1 17 5. 8 13 2 12 6. 15 13 3 56 7. A sandbox has an area of 26 square feet, andthe length is 5 21 feet. What is the width of thesandbox?8. Mr. Webster is buying carpet for an exerciseroom in his basement. The room will have anarea of 230 square feet. The width of the roomis 12 12 feet. What is the length?ESSENTIAL QUESTION CHECK-IN9. How does dividing mixed numbers compare with dividing fractions?84Unit 2 Houghton Mifflin Harcourt Publishing CompanyWrite each situation as a division problem. Then solve. (Example 2)

NameClassDate3.4 Independent PracticePersonalMath Trainer6.3.Emy.hrw.comOnlineAssessment andIntervention10. Jeremy has 4 12 cups of iced tea. He wants to divide the tea into 34 -cupservings. Use the model to find the number of servings he can 11. A ribbon is 3 23 yards long. Mae needs to cut the ribbon into pieces that are2 yard long. Use the model to find the number of pieces she can cut.3113131131313113131311313131312.

Multiplying Fractions and Whole Numbers To multiply a fraction by a whole number, you rewrite the whole number as a fraction and multiply the two fractions. A class has 18 students. The teacher asks how many students in the class have pets and finds _ 5 9 of the students have pets.