Operations With Fractions MODULE 4

22. One container holds 1 78 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. If the teacher begins recording on a digital camera with an houravailable, is there enough time to record everyone if she gives a15-minute introduction at the beginning of class and every studenttakes a minute to get ready? Explain.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 where all bicycles cost 56 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 by placing the whole numberin the numerator and 1 in the denominator. Does following this stepchange the product? Explain.84Unit 2Work Area Houghton Mifflin Harcourt Publishing Company25. Error Analysis To find the product 37 49 , Cameron simplified 37 to 17 and4then multiplied the fractions 17 and 49 to find the product63. What isCameron’s error?

LESSON4.2 Dividing Fractions?COMMONCORE6.NS.1Interpret and computequotients of fractions, ,e.g., by using visual fractionmodels .ESSENTIAL QUESTIONHow do you divide fractions?COMMONCOREEXPLORE ACTIVITY 16.NS.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 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 1COMMONCOREPrep for 6.NS.1Find 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 55 51Math TalkMathematical PracticesHow 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.com86Unit 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 2COMMONCORE6.NS.1Using 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 4.287

Using Reciprocals to Divide FractionsDividing by a fraction is equivalent tomultiplying by its reciprocal.Math On the Spotmy.hrw.com1 14 455EXAMPLE 241 4515COMMONCORE6.NS.1Divide 59 23 . Write the quotient in simplest form.STEP 1Rewrite as multiplication, using the reciprocal of the divisor.5 2 59 329 3AnimatedMathSTEP 2my.hrw.com2The reciprocal ofis 32 .3Multiply and simplify.515 32 918 565 23 569Multiply the numerators. Multiply the denominatorsWrite 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?8835. 45 10Unit 26. 12 25 Houghton Mifflin Harcourt Publishing CompanyGuided Practice

NameClassDate4.2 Independent PracticeCOMMONCORE6.NS.18. 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 of contains 32 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 4.289

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.90Unit 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 MixedNumbersLESSON4.3?COMMONCORE6.NS.1Interpret and computequotients of fractions,and solve word problemsinvolving division of fractionsby fractions .ESSENTIAL QUESTIONHow do you divide mixed numbers?COMMONCOREEXPLORE ACTIVITY6.NS.1Modeling 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 4.391

Using Reciprocals to DivideMixed NumbersMath On the Spotmy.hrw.comMy NotesDividing by a fraction is equivalent to multiplying by its reciprocal. You canuse this fact to divide mixed numbers. First rewrite the mixed numbers asfractions greater than 1. Then multiply the dividend by the reciprocal of thedivisor.EXAMPLE 1COMMONCORE6.NS.1One serving of Harold’s favorite cereal contains 1 25 ounces. How manyservings are in a 17 12 -ounce box?STEP 1Write a division statement to represent the situation.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 divisor.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.Reflect92Unit 24.Analyze Relationships Explain how can you check the answer.5.What If? Harold serves himself 1 12 -ounces 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 2EXAMPLECOMMONCORE6.NS.1The 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 32 Houghton Mifflin Harcourt Publishing CompanySTEP 3Rewrite the problem as multiplication using the reciprocal ofthe divisor.1701702 17 17 32310 2 1703 1720, or 6 23 3Mathematical PracticesExplain how to findthe length of a rectanglewhen you know the areaand the width.Multiply 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 4.393

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?94Unit 2 Houghton Mifflin Harcourt Publishing CompanyWrite each situation as a division problem. Then solve. (Example 2)

NameClassDate4.3 Independent PracticeCOMMONCOREPersonalMath Trainer6.NS.1OnlineAssessment andInterventionmy.hrw.com10. 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. Dao has 2 38 pounds of hamburger meat. He is making 14 -pound hamburgers.Does Dao have enough meat to make 10 hamburgers? Explain.13. Multistep Zoey made 5 12 cups of trail mix for a camping trip. She wantsto divide the trail mix into 34 -cup servings. Houghton Mifflin Harcourt Publishing Companya. Ten people are going on the camping trip. Can Zoey make enough3 -cup servings so that each person on the trip has one serving?4b. What size would the servings need to be for everyone to have aserving? Explain.c. If Zoey decides to use the 34 -cup servings, how much more trail mixwill she need? Explain.14. The area of a rectangular picture frame is 30 13 square inches. The length ofthe frame is 6 12 inches. Find the width of the frame.Lesson 4.395

1115. The area of a rectangular mirror is 1116 square feet. The width of the3mirror is 2 4 feet. If there is a 5 foot tall space on the wall to hang themirror, will it fit? Explain.16. Ramon has a rope that is 25 12 feet long. He wants to cut it into 6 piecesthat are equal in length. How long will each piece be?17. Eleanor and Max used two rectangular wooden boards to make a set forthe school play. One board was 6 feet long, and the other was 5 12 feetlong. The two boards had equal widths. The total area of the set was60 38 square feet. What was the width?FOCUS ON HIGHER ORDER THINKINGWork Area19. Explain the Error To divide 14 23 2 34 , Erik multiplied 14 23 43 . ExplainErik’s error.20. Analyze Relationships Explain how you can find the missing number 2 57 . Then find the missing number.in 3 45 96Unit 2 Houghton Mifflin Harcourt Publishing Company218. Draw Conclusions Micah divided 11 23 by 2 56 and got 417 for an answer.Does his answer seem reasonable? Explain your thinking. Then checkMicah’s answer.

LESSON4.4?Solving Multistep Problemswith Fractions and MixedNumbersESSENTIAL QUESTIONCOMMONCORE6.NS.1.Solve word problemsinvolving division of fractionsby fractions.How can you solve word problems involving more than onefraction operation?Solving Problems withRational NumbersSometimes more than one operation will be needed to solve a multistepproblem. You can use parentheses to group different operations. Recallthat according to the order of operations, you perform operations inparentheses first.EXAMPL 1EXAMPLEProblemSolvingCOMMONCOREMath On the Spotmy.hrw.com6.NS.1Jon is cooking enough lentils for lentil barley soup and lentil salad. Thelentil barley soup recipe calls for 34 cup of dried lentils. The lentil saladrecipe calls for 1 12 cups of dried lentils. Jon has a 18 -cup scoop. How manyscoops of dried lentils will Jon need to have enough for the soup andthe salad? Houghton Mifflin Harcourt Publishing Company Image Credits:Photodisc/Getty ImagesAnalyze InformationIdentify the important information. Jon needs 34 cup of dried lentils for soup and 1 12 cups for salad. Jon has a 18 -cup scoop. You need to find the total number of many scoops of lentils he needs.Formulate a Plan()You can use the expression 34 1 12 18 to find the number of scoops ofdried lentils Jon will need for the soup and the salad.Justify and EvaluateSolveFollow the order of operations. Perform the operations in parentheses first.First add to find the total amount of dried lentils Jon will need.3 1 12 34 324 34 64 94 2 14John needs12cups of4lentils.Lesson 4.497

Jon needs 2 14 cups of dried lentils for both the soup and the salad.To find how many 18 -cup scoops he needs, divide the total amount of driedlentils into groups of 18 .2 14 18 94 18 94 81 8 94 12Simplify beforemultiplying usingthe GCF.118 181Jon will need 18 scoops of dried lentils to have enough for both the lentilbarley soup and the lentil salad.Justify and EvaluateYou added 34 and 1 12 first to find the total number of cups of lentils. Thenyou divided the sum by 18 to find the number of 18 -cup scoops.YOUR TURNPersonalMath TrainerOnline Assessmentand Intervention1. Before conducting some experiments, a scientist mixes1 gram of Substance A with 3 gram of Substance B. If42the scientist uses 18 gram of the mixture for eachexperiment, how many experiments can be conducted?my.hrw.com1. An art student uses a roll of wallpaper to decorate two gift boxes. Thestudent will use 1 13 yards of paper for one box and 56 yard of paper for theother box. The paper must be cut into pieces that are 16 yard long. Howmany pieces will the student cut to use for the gift boxes? (Example 1)?ESSENTIAL QUESTION CHECK-IN2. How can you solve a multistep problem that involves fractions?98Unit 2 Houghton Mifflin Harcourt Publishing CompanyGuided Practice

NameClassDate4.4 Independent PracticeCOMMONCORE6.NS.13. Naomi has earned 54 mowing lawnsthe past two days. She worked 2 12 hoursyesterday and 4 14 hours today. If Naomi ispaid the same amount for every hour sheworks, how much does she earn per hourto mow lawns? (Example 2) Houghton Mifflin Harcourt Publishing Company Image Credits: NicholasEveleigh/The Image Bank/Getty Images4. An art teacher has 1 12 pounds of red clayand 34 pound of yellow clay. The teachermixes the red clay and yellow claytogether. Each student in the class needs1 pound of the clay mixture to finish the8assigned art project for the class. Howmany students can get enough clay tofinish the project?5. A hairstylist schedules 14 hour to trim acustomer’s hair and 61 hour to style thecustomer’s hair. The hairstylist plans towork 3 13 hours each day for 5 days eachweek. How many appointments can thehairstylist schedule each week if eachcustomer must be trimmed and styled?PersonalMath Trainermy.hrw.comOnlineAssessment andIntervention16. A picture framer has a thin board 10123feet long. The frame

fractions so they have the same denominator. You can use the least common multiple of the denominators of the fractions to rewrite the fractions. Add _8 15 1 _ 6. Write the sum in simplest form. Rewrite the fractions as equivalent fractions. Use the LCM as the denominator of both fractions