Problem Set 4: Fractions
Representing parts of a whole
In the following exercises, name the fraction of each figure that is shaded.



















Everyday Math
Music Measures A choreographed dance is broken into counts. A count has one step in a count, a count has two steps in a count and a count has three steps in a count. How many steps would be in a count? What type of count has four steps in it? Music Measures Fractions are used often in music. In time, there are four quarter notes in one measure. ⓐ How many measures would eight quarter notes make? ⓑ The song "Happy Birthday to You" has quarter notes. How many measures are there in "Happy Birthday to You?" ⓐ 8 ⓑ 4 Baking Nina is making five pans of fudge to serve after a music recital. For each pan, she needs cup of walnuts. ⓐ How many cups of walnuts does she need for five pans of fudge? ⓑ Do you think it is easier to measure this amount when you use an improper fraction or a mixed number? Why?Writing Exercises
Give an example from your life experience (outside of school) where it was important to understand fractions. Answers will vary. Explain how you locate the improper fraction on a number line on which only the whole numbers from through are marked.Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
- sists of a whole number and a fraction where . It is written as , where .
- proper and improper fractions
- The fraction is proper if and improper if .
Practice Makes Perfect
Simplify Fractions In the following exercises, simplify each fraction. Do not convert any improper fractions to mixed numbers. Multiply Fractions In the following exercises, use a diagram to model. In the following exercises, multiply, and write the answer in simplified form. 9n 7p −34 Find Reciprocals In the following exercises, find the reciprocal. 1 Fill in the chart.Opposite | Absolute Value | Reciprocal |
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Opposite | Absolute Value | Reciprocal |
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Everyday Math
Baking A recipe for chocolate chip cookies calls for cup brown sugar. Imelda wants to double the recipe. ⓐ How much brown sugar will Imelda need? Show your calculation. Write your result as an improper fraction and as a mixed number. ⓑ Measuring cups usually come in sets of cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the recipe. Baking Nina is making pans of fudge to serve after a music recital. For each pan, she needs cup of condensed milk. ⓐ How much condensed milk will Nina need? Show your calculation. Write your result as an improper fraction and as a mixed number. ⓑ Measuring cups usually come in sets of cup. Draw a diagram to show two different ways that Nina could measure the condensed milk she needs. ⓐ ⓑ Answers will vary. Portions Don purchased a bulk package of candy that weighs pounds. He wants to sell the candy in little bags that hold pound. How many little bags of candy can he fill from the bulk package? Portions Kristen has yards of ribbon. She wants to cut it into equal parts to make hair ribbons for her daughter’s dolls. How long will each doll’s hair ribbon be?Writing Exercises
Explain how you find the reciprocal of a fraction. Explain how you find the reciprocal of a negative fraction. Answers will vary. Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into or slices. Would he prefer out of slices or out of slices? Rafael replied that since he wasn’t very hungry, he would prefer out of slices. Explain what is wrong with Rafael’s reasoning. Give an example from everyday life that demonstrates how . Answers will vary.Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
Everyday Math
Baking A recipe for chocolate chip cookies calls for cups of flour. Graciela wants to double the recipe. ⓐ How much flour will Graciela need? Show your calculation. Write your result as an improper fraction and as a mixed number. ⓑ Measuring cups usually come in sets with cups for cup. Draw a diagram to show two different ways that Graciela could measure out the flour needed to double the recipe. Baking A booth at the county fair sells fudge by the pound. Their award winning "Chocolate Overdose" fudge contains cups of chocolate chips per pound. ⓐ How many cups of chocolate chips are in a half-pound of the fudge? ⓑ The owners of the booth make the fudge in -pound batches. How many chocolate chips do they need to make a -pound batch? Write your results as improper fractions and as a mixed numbers. ⓐ ⓑWriting Exercises
Explain how to find the reciprocal of a mixed number. Explain how to multiply mixed numbers. Answers will vary. Randy thinks that is . Explain what is wrong with Randy’s thinking. Explain why , and are equivalent. Answers will vary.Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
Practice Makes Perfect
Model Fraction Addition In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.


Everyday Math
Trail Mix Jacob is mixing together nuts and raisins to make trail mix. He has of a pound of nuts and of a pound of raisins. How much trail mix can he make? Baking Janet needs of a cup of flour for a recipe she is making. She only has of a cup of flour and will ask to borrow the rest from her next-door neighbor. How much flour does she have to borrow?Writing Exercises
Greg dropped his case of drill bits and three of the bits fell out. The case has slots for the drill bits, and the slots are arranged in order from smallest to largest. Greg needs to put the bits that fell out back in the case in the empty slots. Where do the three bits go? Explain how you know. Bits in case: , , ___, ___, , , ___, , , . Bits that fell out: , , . After a party, Lupe has of a cheese pizza, of a pepperoni pizza, and of a veggie pizza left. Will all the slices fit into pizza box? Explain your reasoning. Answers will vary.Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
add subtract mixed numbers, simplify
Find the Least Common Denominator (LCD)
In the following exercises, find the least common denominator (LCD) for each set of fractions. 20 48 240 245 60 Convert Fractions to Equivalent Fractions with the LCD In the following exercises, convert to equivalent fractions using the LCD. Add and Subtract Fractions with Different Denominators In the following exercises, add or subtract. Write the result in simplified form. Identify and Use Fraction Operations In the following exercises, perform the indicated operations. Write your answers in simplified form. ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ Use the Order of Operations to Simplify Complex Fractions In the following exercises, simplify. 32 Mixed Practice In the following exercises, simplify. −9 1 In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. when ⓐ ⓑ when ⓐ ⓑ ⓐ ⓑ when ⓐ ⓑ when ⓐ ⓑ ⓐ ⓑ when ⓐ ⓑ when ⓐ ⓑ ⓐ ⓑ when ⓐ ⓑ when ⓐ ⓑ ⓐ ⓑ −2 3Everyday Math
Decorating Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs yard of print fabric and yard of solid fabric. What is the total amount of fabric Laronda needs for each pillow cover? Baking Vanessa is baking chocolate chip cookies and oatmeal cookies. She needs cups of sugar for the chocolate chip cookies, and cups for the oatmeal cookies How much sugar does she need altogether?Writing Exercises
Explain why it is necessary to have a common denominator to add or subtract fractions. Explain how to find the LCD of two fractions. Answers will vary.Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
Practice Makes Perfect
Model Addition of Mixed Numbers In the following exercises, use a model to find the sum. Draw a picture to illustrate your model.


Everyday Math
Sewing Renata is sewing matching shirts for her husband and son. According to the patterns she will use, she needs yards of fabric for her husband’s shirt and yards of fabric for her son’s shirt. How much fabric does she need to make both shirts? Sewing Pauline has yards of fabric to make a jacket. The jacket uses yards. How much fabric will she have left after making the jacket? Printing Nishant is printing invitations on his computer. The paper is inches wide, and he sets the print area to have a -inch border on each side. How wide is the print area on the sheet of paper? Framing a picture Tessa bought a picture frame for her son’s graduation picture. The picture is inches wide. The picture frame is inches wide on each side. How wide will the framed picture be?Writing Exercises
Draw a diagram and use it to explain how to add . Edgar will have to pay \text{$3.75} in tolls to drive to the city. ⓐ Explain how he can make change from a \text{$10} bill before he leaves so that he has the exact amount he needs. ⓑ How is Edgar’s situation similar to how you subtract Answers will vary. Add twice, first by leaving them as mixed numbers and then by rewriting as improper fractions. Which method do you prefer, and why? Subtract twice, first by leaving them as mixed numbers and then by rewriting as improper fractions. Which method do you prefer, and why? Answers will vary.Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
solving equations with fractions
Determine Whether a Fraction is a Solution of an Equation In the following exercises, determine whether each number is a solution of the given equation. ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ ⓐ no ⓑ yes ⓒ no ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ ⓐ no ⓑ yes ⓒ no Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality In the following exercises, solve. c = −1 z = −1 Solve Equations with Fractions Using the Multiplication Property of Equality In the following exercises, solve. b = −27 x = −256 q = 160 s = 45 y = −42 p = 100 m = −16 b = −21 v = 36 Mixed Practice In the following exercises, solve. y = 0 Translate Sentences to Equations and Solve In the following exercises, translate to an algebraic equation and solve. divided by eight is . divided by six is . divided by is . divided by is . The quotient of and is . The quotient of and is . The quotient of and twelve is . The quotient of and nine is . Three-fourths of is . Two-fifths of is . Seven-tenths of is . Four-ninths of is . divided by equals negative . The quotient of and is . Three-fourths of is the same as . The quotient of and is . The sum of five-sixths and is . The sum of three-fourths and is . The difference of and one-fourth is . The difference of and one-third is .Everyday Math
Shopping Teresa bought a pair of shoes on sale for \text{$48}. The sale price was of the regular price. Find the regular price of the shoes by solving the equation Playhouse The table in a child’s playhouse is of an adult-size table. The playhouse table is inches high. Find the height of an adult-size table by solving the equation . 30 inchesWriting Exercises
There are three methods to solve the equation . Which method do you prefer? Why? Richard thinks the solution to the equation is . Explain why Richard is wrong. Answers will vary.Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
Chapter Review Exercises
Visualize Fractions
In the following exercises, name the fraction of each figure that is shaded.




Multiply and Divide Fractions
In the following exercises, simplify. In the following exercises, multiply. 6 In the following exercises, find the reciprocal. −4 Fill in the chart.Opposite | Absolute Value | Reciprocal | |
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