Problem Set: Fractions
Representing parts of a whole
Using Models to Represent Fractions
In the following exercises, name the fraction of each figure that is shaded.
Exercise 1
Answer:
ⓐ = 41
ⓑ = 43
ⓒ = 83
ⓓ = 95
Exercise 2
Using Models to Represent Fractions
In the following exercises, shade parts of circles or squares to model the following fractions.
- 21
Answer: 
- 31
- 43
Answer: 
- 52
- 65
Answer: 
- 87
- 85
Answer: 
- 107
Model Improper Fractions and Mixed Numbers
In the following exercises, use fraction circles to make wholes, if possible, with the following pieces.
- 3 thirds
Answer: 
- 8 eighths
- 7 sixths
Answer: 
- 4 thirds
- 7 fifths
Answer: 
- 7 fourths
Model Improper Fractions and Mixed Numbers
In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.
Exercise 1
Answer: ⓐ 45=141 ⓑ 47=143 ⓒ 811=183
Exercise 2
Exercise 3
Answer: ⓐ 411=243 ⓑ 819=283
Model Improper Fractions and Mixed Numbers
In the following exercises, draw fraction circles to model the given fraction.
- 33
- 44
Answer: 
- 47
- 35
Answer: 
- 611
- 813
Answer: 
- 310
- 49
Answer: 
Converting Between Improper Fractions and Mixed Numbers
Write an Improper Fraction as a Mixed Number
In the following exercises, rewrite the improper fraction as a mixed number.
- 23
- 35
Answer: 132
- 411
- 513
Answer: 253
- 625
- 928
Answer: 391
- 1342
- 1547
Answer: 3152
Write a Mixed Number as an Improper Fraction
In the following exercises, rewrite the mixed number as an improper fraction.
- 132
- 152
Answer: 57
- 241
- 265
Answer: 617
- 297
- 275
Answer: 719
- 374
- 395
Answer: 932
Modeling and Finding Equivalent Fractions
Modeling Equivalent Fractions
In the following exercises, use fraction tiles or draw a figure to find equivalent fractions.
- How many sixths equal one-third?
- How many twelfths equal one-third?
Answer: 4
- How many eighths equal three-fourths?
- How many twelfths equal three-fourths?
Answer: 9
- How many fourths equal three-halves?
- How many sixths equal three-halves?
Answer: 9
Finding Equivalent Fractions
In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.
- 41
- 31
Answer: Answers may vary. Correct answers include 62,93,124.
- 83
- 65
Answer: Answers may vary. Correct answers include 1210,1815,2420.
- 72
- 95
Answer: Answers may vary. Correct answers include 1810,2715,3620.
Locating and Ordering Fractions and Mixed Numbers on the Number Line
Locating Fractions on the Number Line
In the following exercises, plot the numbers on a number line.
- 32,45,512
- 31,47,513
Answer: 
- 41,59,311
- 107,25,813,3
Answer: 
- 231,−231
- 143,−153
Answer: 
- 43,−43,132,−132,25,−25
- 52,−52,143,−143,38,−38
Answer: 
Ordering Fractions on the Number Line
In the following exercises, order each of the following pairs of numbers, using < or >.
- -1\text{ __}-\Large\frac{1}{4}
- -1\text{ __}-\Large\frac{1}{3}
Answer: <
- -2\Large\frac{1}{2}\normalsize\text{ __}-3
- -1\Large\frac{3}{4}\normalsize\text{ __}-2
Answer: >
- -\Large\frac{5}{12}\text{ __}-\Large\frac{7}{12}
- -\Large\frac{9}{10}\text{ __}-\Large\frac{3}{10}
Answer: <
- -3\text{ __}-\Large\frac{13}{5}
- -4\text{ __}-\Large\frac{23}{6}
Answer: <
Everyday Math
Music Measures
A choreographed dance is broken into counts. A
11 count has one step in a count, a
21 count has two steps in a count and a
31 count has three steps in a count. How many steps would be in a
51 count? What type of count has four steps in it?
Music Measures
Fractions are used often in music. In
44 time, there are four quarter notes in one measure.
- How many measures would eight quarter notes make?
Answer: 2
- The song "Happy Birthday to You" has 24 quarter notes. How many measures are there in "Happy Birthday to You?"
Answer: 6
Baking
Nina is making five pans of fudge to serve after a music recital. For each pan, she needs
21 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.
[practice-area rows="4"][/practice-area]
Explain how you locate the improper fraction
421 on a number line on which only the whole numbers from
0 through
10 are marked.
[practice-area rows="4"][/practice-area]
Multiplying and Dividing Fractions
Simplify Fractions
In the following exercises, simplify each fraction. Do not convert any improper fractions to mixed numbers.
- 217
Answer: 31
- 248
- 2015
Answer: 43
- 1812
- −8840
Answer: −115
- −9963
- −63108
Answer: −712
- −48104
- 252120
Answer: 2110
- 294182
- −192168
Answer: −87
- −224140
- 11y11x
Answer: yx
- 15b15a
- −12y3x
Answer: −4yx
- −32y4x
- 21y14x2
Answer: 3y2x2
- 32b224a
Multiply and Simplify Fractions
In the following exercises, multiply, and write the answer in simplified form.
- 52⋅31
Answer: 152
- 21⋅83
- 43⋅109
Answer: 4027
- 54⋅72
- −32(−83)
Answer: 41
- −43(−94)
- −95⋅103
Answer: −61
- −83⋅154
- 127(−218)
Answer: −92
- 125(−158)(−1514)(209)
Answer: −5021
- (−109)(3325)
- (−8463)(−9044)
Answer: 3011
- (−6033)(−8840)
- 4⋅115
- 1120
- 5⋅38
- 73⋅21n
Answer: 9n
- 65⋅30m
- −28p(−41)
Answer: 7p
- −51q(−31)
- −8(417)
Answer: −34
- 514(−15)
- −1(−83)
- 83
- (−1)(−76)
- (32)3
- 278
- (54)2
- (56)4
- 6251296
- (74)4
Find Reciprocals
In the following exercises, find the reciprocal.
- 43
Answer: 34
- 32
- −175
Answer: −517
- −196
- 811
Answer: 118
- −13
- −19
Answer: −191
- −1
Answer: 1
Divide and Simplify Fractions
In the following exercises, divide, and write the answer in simplified form.
- 21÷41
- 21÷81
Answer: 4
- 43÷32
- 54÷43
- 1516
- −54÷74
- −43÷53
- −45
- −97÷(−97)
- −65÷(−65)
Answer: 1
- 43÷11x
- 52÷9y
- 5y18
- 85÷10a
- 65÷15c
- 2c25
- 185÷(−2415)
- 187÷(−2714)
- −43
- 127p÷821p
- 125q÷815q
- 92
- 158u÷2512v
- 2512r÷3518s
- 15s14r
- −5÷21
- −3÷41
Answer: −12
- 43÷(−12)
- 52÷(−10)
- −251
- −18÷(−29)
- −15÷(−35)
Answer: 9
- 21÷(−43)÷87
- 211÷87⋅112
Everyday Math
Baking
- A recipe for chocolate chip cookies calls for 43 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 81,41,31,21, and 1 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
4 pans of fudge to serve after a music recital. For each pan, she needs
32 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.
Answer: 38=232
- Measuring cups usually come in sets of 81,41,31,21, and 1 cup. Draw a diagram to show two different ways that Nina could measure the condensed milk she needs.
Portions
Don purchased a bulk package of candy that weighs
5 pounds. He wants to sell the candy in little bags that hold
41 pound. How many little bags of candy can he fill from the bulk package?
Portions
Kristen has
43 yards of ribbon. She wants to cut it into equal parts to make hair ribbons for her daughter’s
6 dolls. How long will each doll’s hair ribbon be?
Answer: 81yard
Writing Exercises
Explain how you find the reciprocal of a fraction.
[practice-area rows="4"][/practice-area]
Explain how you find the reciprocal of a negative fraction.
[practice-area rows="4"][/practice-area]
Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into
6 or
8 slices. Would he prefer
3 out of
6 slices or
4 out of
8 slices? Rafael replied that since he wasn’t very hungry, he would prefer
3 out of
6 slices. Explain what is wrong with Rafael’s reasoning.
[practice-area rows="4"][/practice-area]
Give an example from everyday life that demonstrates how
21⋅32 is 31.
[practice-area rows="4"][/practice-area]
Multiplying and Dividing Mixed Numbers and Complex Fractions
Multiply Mixed Numbers
In the following exercises, multiply and write the answer in simplified form.
- 483⋅107
- 294⋅76
Answer: 2144
- 2215⋅353
- 3625⋅6103
Answer: 835
- 432(−181)
- 252(−292)
Answer: −316
- −494⋅51613
- −1207⋅21211
Answer: −1663
Divide Mixed Numbers
In the following exercises, divide, and write your answer in simplified form.
- 531÷4
- 1321÷9
Answer: 23
- −12÷3113
- −7÷541
Answer: −34
- 683÷281
- 251÷1101
Answer: 2
- −953÷(−153)
- −1843÷(−343)
Answer: 5
Translate Phrases to Expressions with Fractions
In the following exercises, translate each English phrase into an algebraic expression.
- the quotient of 5u and 11
- the quotient of 7v and 13
Answer: 137v
- the quotient of p and q
- the quotient of a and b
Answer: ba
- the quotient of r and the sum of s and 10
- the quotient of A and the difference of 3 and B
Answer: 3−BA
Simplify Complex Fractions
In the following exercises, simplify the complex fraction.
- 9832
- 15854
Answer: 23
- 3512−218
- 4033−169
Answer: −2215
- 2−54
- 3−109
Answer: −103
- 852
- 1035
Answer: 61
- 2n3m
- 3s5r
Answer: 5s3r
- −98−6x
- −12y−83
Answer: 2y9
- 101254
- 61432
Answer: 28
- −25497
- −64383
Answer: −181
Simplify Expressions with a Fraction Bar
In the following exercises, identify the equivalent fractions.
- Which of the following fractions are equivalent to −115?
−11−5,11−5,115,−115
- Which of the following fractions are equivalent to 9−4?
−9−4,9−4,94,−94
Answer: 9−4,−94
- Which of the following fractions are equivalent to −311?
3−11,311,−3−11,−311
- Which of the following fractions are equivalent to −613?
613,−613,−6−13,6−13
Answer: −613,6−13
Simplify Fractions
In the following exercises, simplify.
- 84+11
- 79+3
Answer: 712
- 1022+3
- 619−4
Answer: 25
- 24−1548
- 4+446
Answer: 423
- 8+4−6+6
- 17−8−6+3
Answer: −31
- 19−1322−14
- 18+1215+9
Answer: 54
- −105⋅8
- −243⋅4
Answer: −21
- 6⋅64⋅3
- 9⋅26⋅6
Answer: 2
- 2542−1
- 6072+1
Answer: 65
- 14+38⋅3+2⋅9
- 22+39⋅6−4⋅7
Answer: 2526
- 2⋅1015⋅5−52
- 3⋅1812⋅9−32
Answer: 611
- 4⋅5−2⋅35⋅6−3⋅4
- 5⋅6−9⋅28⋅9−7⋅6
Answer: 25
- 3−552−32
- 4−662−42
Answer: −10
- −3−222+4(3)
- −2−327+3(5)
Answer: −2
- 9.3−3.57⋅4−2(8−5)
- 8.7−6.69⋅7−3(12−8)
Answer: 2051
- 6(7−1)−3(17−9)9(8−2)−3(15−7)
- 7(8−3)−3(16−9)8(9−2)−4(14−9)
Answer: 718
Everyday Math
Baking
A recipe for chocolate chip cookies calls for
241 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 81,41,31,21, and 1 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
232 cups of chocolate chips per pound.
- How many cups of chocolate chips are in a half-pound of the fudge
Answer: 34=131cups
- The owners of the booth make the fudge in 10 -pound batches. How many chocolate chips do they need to make a 10 -pound batch? Write your results as improper fractions and as a mixed numbers.
Answer: 380=2632cups
Writing Exercises
Explain how to find the reciprocal of a mixed number.
[practice-area rows="4"][/practice-area]
Explain how to multiply mixed numbers.
[practice-area rows="4"][/practice-area]
Randy thinks that
321⋅541 is
1581. Explain what is wrong with Randy’s thinking.
[practice-area rows="4"][/practice-area]
Explain why
−21,2−1, and
−21 are equivalent.
[practice-area rows="4"][/practice-area]
Adding and Subtracting Fractions With Common Denominators
Model Fraction Addition
In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.
- 52+51
- 103+104
Answer: 
- 107
- 61+63
- 83+83
Answer: 
- 43
Add Fractions with a Common Denominator
In the following exercises, find each sum.
- 94+91
- 92+95
Answer: 97
- 136+137
- 159+157
Answer: 1516
- 4x+43
- 3y+32
Answer: 3y+2
- p7+p9
- q8+q6
Answer: q14
- 98b+93b
- 75a+74a
Answer: 79a
- 8−12y+83y
- 5−11x+57x
Answer: 5−4x
- −81+(−83)
- −81+(−85)
Answer: −43
- −163+(−167)
- −165+(−169)
Answer: −87
- −178+1715
- −199+1917
Answer: 198
- 136+(−1310)+(−1312)
- 125+(−127)+(−1211)
Answer: −1213
Model Fraction Subtraction
In the following exercises, use a model to subtract the fractions. Show a diagram to illustrate your model.
- 85−82
- 65−62
Answer: 
- 21
Subtract Fractions with a Common Denominator
In the following exercises, find the difference.
- 54−51
- 54−53
Answer: 51
- 1511−157
- 139−134
Answer: 135
- 1211−125
- 127−125
Answer: 61
- 214−2119
- −98−916
Answer: −38
- 17y−179
- 19x−198
Answer: 19x−8
- 85y−87
- 1311z−138
Answer: 1311z−8
- −d8−d3
- −c7−c7
Answer: −c14
- −u23−u15
- −v29−v26
Answer: −v55
- 76c−75c
- 1112d−119d
Answer: 113d
- 13−4r−135r
- 3−7s−37s
Answer: −314s
- −53−(−54)
- −73−(−75)
Answer: 72
- −97−(−95)
- −118−(−115)
Answer: −113
Identify and Use Fraction Operations
In the following exercises, perform the indicated operation and write your answers in simplified form.
- −185⋅109
- −143⋅127
Answer: −81
- 5n−54
- 116−11s
Answer: 116−s
- −247+242
- −185+181
Answer: −92
- 158÷512
- 127÷289
Answer: 2749
Everyday Math
Trail Mix
Jacob is mixing together nuts and raisins to make trail mix. He has
106 of a pound of nuts and
103 of a pound of raisins. How much trail mix can he make?
Baking
Janet needs
85 of a cup of flour for a recipe she is making. She only has
83 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?
Answer: 41cup
Writing Exercises
Greg dropped his case of drill bits and three of the bits fell out. The three bits were
167,
163, and
41. 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:
161 ,
81 , ___, ___,
165 ,
83 , ___,
21 ,
169 ,
85 .
[practice-area rows="4"][/practice-area]
Answer: Bits that fell out: 167 , 163 , 41 .
After a party, Lupe has
125 of a cheese pizza,
124 of a pepperoni pizza, and
124 of a veggie pizza left. Will all the slices fit into
1 pizza box? Explain your reasoning.
[practice-area rows="4"][/practice-area]
Adding and Subtracting Fractions with Different Denominators
Find the Least Common Denominator (LCD)
In the following exercises, find the least common denominator (LCD) for each set of fractions.
- 32 and 43
- 43 and 52
Answer: 20
- 127 and 85
- 169 and 127
Answer: 48
- 3013 and 4225
- 3023 and 485
Answer: 240
- 3521 and 5639
- 3518 and 4933
Answer: 245
- 32,61, and 43
- 32,41, and 53
Answer: 60
Convert Fractions to Equivalent Fractions with the LCD
In the following exercises, convert to equivalent fractions using the LCD.
- 31 and 41,LCD=12
- 41 and 51,LCD=20
Answer: 205,204
- 125 and 87,LCD=24
- 127 and 85,LCD=24
Answer: 2414,2415
- 1613 and -1211,LCD=48
- 1611 and -125,LCD=48
Answer: 4833,−4820
- 31,65, and 43,LCD=12
- 31,43, and 53,LCD=60
Answer: 6020,6045,6036
Add and Subtract Fractions with Different Denominators
In the following exercises, add or subtract. Write the result in simplified form.
- 31+51
- 41+51
Answer: 209
- 21+71
- 31+81
Answer: 2411
- 31−(−91)
- 41−(−81)
Answer: 83
- 51−(−101)
- 21−(−61)
Answer: 32
- 32+43
- 43+52
Answer: 2023
- 127+85
- 125+83
Answer: 2419
- 127−169
- 167−125
Answer: 481
- 1211−83
- 85−127
Answer: 241
- 32−83
- 65−43
Answer: 121
- −3011+4027
- −209+3017
Answer: 607
- −3013+4225
- −3023+485
Answer: −8053
- −5639−3522
- −4933−3518
Answer: −245291
- −32−(−43)
- −43−(−54)
Answer: 201
- −169−(−54)
- −207−(−85)
Answer: 4011
- 1+87
- 1+65
Answer: 611
- 1−95
- 1−103
Answer: 107
- 3x+41
- 2y+32
Answer: 63y+4
- 4y−53
- 5x−41
Answer: 204x−5
Identify and Use Fraction Operations
In the following exercises, perform the indicated operations. Write your answers in simplified form.
- 43+61
- 43÷61
- 32+61
Answer: 65
- 32÷61
Answer: 4
- -52−81
- -52⋅81
- -54−81
Answer: −4037
- -54⋅81
Answer: −101
- n5÷158
- n5−158
- a3÷127
Answer: 149a
- a3−127
Answer: 249a−14
- 109⋅(−d11)
- 109+(−d11)
- 154⋅(−q5)
Answer: −3q4
- 154+(−q5)
Answer: 4512−25q
- −83÷(−103)
- −125÷(−95)
- −83+125
- −81+127
Answer: 2411
- 65−91
- 95−61
Answer: 187
- 83⋅(−2110)
- 127⋅(−358)
Answer: −152
- −157−4y
- −83−11x
Answer: 88−33−8x
- 12a11⋅169a
- 1310y⋅15y8
Answer: 3916
Use the Order of Operations to Simplify Complex Fractions
In the following exercises, simplify.
- 2+32(51)2
- 5+22(31)2
Answer: 811
- (32)223+42
- (43)233−32
Answer: 32
- (73)2(53)2
- (85)2(43)2
Answer: 2536
- 31+512
- 41+315
Answer: 760
- 43−3232+21
- 65−3243+21
Answer: 215
- 21+8387−32
- 41+5243−53
Answer: 133
Identify and Use Fraction Operations
In the following exercises, simplify.
- 21+32⋅125
- 31+52⋅43
Answer: 3019
- 1−53÷101
- 1−65÷121
Answer: −9
- 32+61+43
- 32+41+53
Answer: 6091
- 83−61+43
- 52+85−43
Answer: 4011
- 12(209−154)
- 8(1615−65)
Answer: 65
- 241985+61
- 301461+103
Answer: 1
- (95+61)÷(32−21)
- (43+61)÷(85−31)
Answer: 722
Evaluate Variable Expressions with Fractions
In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.
Exercise 1
x+21 when
- x=−81
- x=−21
Exercise 2
x+32 when
- x=−61
- x=−35
Answer:
- 21
- −1
Exercise 3
x+(−65) when
- x=31
- x=−61
Exercise 4
x+(−1211) when
- x=1211
- x=43
Answer:
- 0
- −61
Exercise 5
x−52 when
- x=53
- x=−53
Exercise 6
x−31 when
- x=32
- x=−32
Answer:
- 31
- −1
Exercise 7
107−w when
- w=21
- w=−21
Exercise 8
125−w when
- w=41
- w=−41
Answer:
- 61
- 32
Evaluate Variable Expressions with Fractions
In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.
- 4p2q when p=−21 and q=95
- 5m2n when m=−52 and n=31
Answer: 154
- 2x2y3 when x=−32 and y=−21
- 8u2v3 when u=−43 and v=−21
Answer: −169
- wu+v when u=−4,v=−8,w=2
- pm+n when m=−6,n=−2,p=4
Answer: −2
- a−ba+b when a=−3,b=8
- r+sr−s when r=10,s=−5
Answer: 3
Everyday Math
Decorating
Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs
163 yard of print fabric and
83 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
141 cups of sugar for the chocolate chip cookies, and
181 cups for the oatmeal cookies How much sugar does she need altogether?
Answer: She needs283cups
Writing Exercises
Explain why it is necessary to have a common denominator to add or subtract fractions.
[practice-area rows="4"][/practice-area]
Explain how to find the LCD of two fractions.
[practice-area rows="4"][/practice-area]
Adding and Subtracting Mixed Numbers
Model Addition of Mixed Numbers
In the following exercises, use a model to find the sum. Draw a picture to illustrate your model.
- 151+351
- 231+131
Answer: 
- 332
- 183+187
- 165+165
Answer: 
- 332
Add Mixed Numbers with a Common Denominator
In the following exercises, add.
- 531+631
- 294+591
Answer: 795
- 485+983
- 7109+3101
Answer: 11
- 354+654
- 932+132
Answer: 1131
- 6109+8103
- 894+298
Answer: 1131
Model Subtraction of Mixed Numbers
In the following exercises, use a model to find the difference. Draw a picture to illustrate your model.
- 161−65
- 181−85
Answer: 
- 21
Subtract Mixed Numbers with a Common Denominator
In the following exercises, find the difference.
- 287−183
- 2127−1125
Answer: 161
- 82017−4209
- 191513−13157
Answer: 652
- 873−474
- 592−394
Answer: 197
- 285−187
- 2125−1127
Answer: 65
Add and Subtract Mixed Numbers with Different Denominators
In the following exercises, write the sum or difference as a mixed number in simplified form.
- 341+631
- 261+543
Answer: 71211
- 185+421
- 732+821
Answer: 1661
- 9107−231
- 654−141
Answer: 52011
- 232−321
- 287−431
Answer: −12411
Identify and Use Fraction Operations
In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.
- 285⋅143
- 132⋅461
Answer: 61817
- 72+74
- 92+95
Answer: 97
- 1125÷121
- 2103÷101
Answer: 23
- 13125−9127
- 1585−687
Answer: 843
- 95−94
- 1511−157
Answer: 154
- 4−43
- 6−52
Answer: 553
- 209÷43
- 247÷314
Answer: 161
- 9116+71110
- 8135+4139
Answer: 13131
- 352+543
- 265+451
Answer: 7301
- 158⋅1910
- 125⋅98
Answer: 2710
- 687−231
- 695−452
Answer: 2457
- 592−454
- 483−332
Answer: 2417
Everyday Math
Sewing
Renata is sewing matching shirts for her husband and son. According to the patterns she will use, she needs
283 yards of fabric for her husband’s shirt and
181 yards of fabric for her son’s shirt. How much fabric does she need to make both shirts?
Sewing
Pauline has
341 yards of fabric to make a jacket. The jacket uses
232 yards. How much fabric will she have left after making the jacket?
Answer: 127yards
Printing
Nishant is printing invitations on his computer. The paper is
821 inches wide, and he sets the print area to have a
121 -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
8 inches wide. The picture frame is
285 inches wide on each side. How wide will the framed picture be?
Answer: 1341inches
Writing Exercises
Draw a diagram and use it to explain how to add
185+287.
[practice-area rows="4"][/practice-area]
Edgar will have to pay
$3.75 in tolls to drive to the city.
- Explain how he can make change from a $10 bill before he leaves so that he has the exact amount he needs.
- How is Edgar’s situation similar to how you subtract 10−343?
[practice-area rows="4"][/practice-area]
Add
4125+387 twice, first by leaving them as mixed numbers and then by rewriting as improper fractions. Which method do you prefer, and why?
[practice-area rows="4"][/practice-area]
Subtract
387−4125 twice, first by leaving them as mixed numbers and then by rewriting as improper fractions. Which method do you prefer, and why?
[practice-area rows="4"][/practice-area]
Solving Equations That Contain 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.
Exercise 1
x−52=101:
- x=1
- x=21
- x=−21
Exercise 2
y−31=125:
- y=1
- y=43
- y=−43
Answer:
- no
- yes
- no
Exercise 3
h+43=52:
- h=1
- h=207
- h=−207
Exercise 4
k+52=65:
- k=1
- k=3013
- k=−3013
Answer:
- no
- yes
- no
Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality
In the following exercises, solve.
- y+31=34
- m+83=87
Answer: m=21
- f+109=52
- h+65=61
Answer: h=−32
- a−85=−87
- c−41=−45
Answer: c = −1
- x−(−203)=−2011
- z−(−125)=−127
Answer: z = −1
- n−61=43
- p−103=85
Answer: p=4037
- s+(−21)=−98
- k+(−31)=−54
Answer: k=−157
- 5j=17
- 7k=18
Answer: k=718
- −4w=26
- −9v=33
Answer: v=−311
Solve Equations with Fractions Using the Multiplication Property of Equality
In the following exercises, solve.
- 4f=−20
- 3b=−9
Answer: b = −27
- 7y=−21
- 8x=−32
Answer: x = −256
- −5p=−40
- −4q=−40
Answer: q = 160
- −12r=−6
- −15s=−3
Answer: s = 45
- −x=23
- −y=42
Answer: y = −42
- −h=−125
- −k=−2017
Answer: k=2017
- 54n=20
- 103p=30
Answer: p = 100
- 83q=−48
- 25m=−40
Answer: m = −16
- −92a=16
- −73b=9
Answer: b = −21
- −116u=−24
- −125v=−15
Answer: v = 36
Mixed Practice
In the following exercises, solve.
- 3x=0
- 8y=0
Answer: y = 0
- 4f=54
- 7g=97
Answer: g=91
- p+32=121
- q+65=121
Answer: q=−43
- 87m=101
- 41n=107
Answer: n=514
- −52=x+43
- −32=y+83
Answer: y=−2425
- 2011=-f
- 158=-d
Answer: d=−158
Translate Sentences to Equations and Solve
In the following exercises, translate to an algebraic equation and solve.
- n divided by eight is −16.
- n divided by six is −24.
Answer: 6n=−24;n=−144
- m divided by −9 is −7.
- m divided by −7 is −8.
Answer: −7m=−8;m=56
- The quotient of f and −3 is −18.
- The quotient of f and −4 is −20.
Answer: −4f=−20;f=80
- The quotient of g and twelve is 8.
- The quotient of g and nine is 14.
Answer: 9g=14;g=126
- Three-fourths of q is 12.
- Two-fifths of q is 20.
Answer: 52q=20;q=50
- Seven-tenths of p is −63.
- Four-ninths of p is −28.
Answer: 94p=−28;p=−63
- m divided by 4 equals negative 6.
- The quotient of h and 2 is 43.
Answer: 2h=43
- Three-fourths of z is the same as 15.
- The quotient of a and 32 is 43.
Answer: 32a=43
- The sum of five-sixths and x is 21.
- The sum of three-fourths and x is 81.
Answer: 43+x=81;x=−85
- The difference of y and one-fourth is −81.
- The difference of y and one-third is −61.
Answer: y−31=−61;y=61
Everyday Math
Shopping
Teresa bought a pair of shoes on sale for
$48. The sale price was
32 of the regular price. Find the regular price of the shoes by solving the equation
32p=48
Playhouse
The table in a child’s playhouse is
53 of an adult-size table. The playhouse table is
18 inches high. Find the height of an adult-size table by solving the equation
53h=18.
Answer: 30 inches
Writing Exercises
There are three methods to solve the equation
−y=15. Which method do you prefer? Why?
[practice-area rows="4"][/practice-area]
Richard thinks the solution to the equation
43x=24 is
16. Explain why Richard is wrong.
[practice-area rows="4"][/practice-area]
Chapter Review Exercises
Visualize Fractions
In the following exercises, name the fraction of each figure that is shaded.
Using Models to Represent Fractions
Exercise 1
Exercise 2
Answer: 95
Using Models to Represent Mixed Numbers
In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.
Exercise 1
Exercise 2
Answer: 23
Write an Improper Fraction as a Mixed Number
In the following exercises, convert the improper fraction to a mixed number.
- 1558
- 1163
Answer: 5118
Write a Mixed Number as an Improper Fraction
In the following exercises, convert the mixed number to an improper fraction.
- 1241
- 954
Answer: 549
- Find three fractions equivalent to 52. Show your work, using figures or algebra.
- Find three fractions equivalent to −34. Show your work, using figures or algebra.
Locating Fractions on the Number Line
In the following exercises, locate the numbers on a number line.
- 85,34,343,4
- 41,−41,131,−131,27,−27
Answer: 
Ordering Fractions on the Number Line
In the following exercises, order each pair of numbers, using
< or
>.
- -1\text{ ___}-\Large\frac{2}{5}
- -2\Large\frac{1}{2}\normalsize\text{ ___}- 3
Multiply and Divide Fractions
Simplify Fractions
In the following exercises, simplify.
- −8463
- −12090
Answer: −43
- −14b14a
- −8y8x
Answer: −yx
Multiply Fractions
In the following exercises, multiply.
- 52⋅138
- −31⋅712
Answer: −74
- 92⋅(−3245)
- 6m⋅114
Answer: 1124m
- −41(−32)
- 351⋅187
Answer: 6
Find Reciprocals
In the following exercises, find the reciprocal.
- 92
- 415
Answer: 154
- 3
- −41
Answer: −4
Exercise 5
Fill in the chart.
|
Opposite |
Absolute Value |
Reciprocal |
−135 |
|
|
|
103 |
|
|
|
49 |
|
|
|
−12 |
|
|
|
Divide Fractions
In the following exercises, divide.
- 32÷61
Answer: 4
- (−53x)÷(−32y)
- 54÷3
Answer: 154
- 8÷232
- 832÷1121
Answer: 8
Multiply and Divide Mixed Numbers and Complex Fractions
In the following exercises, perform the indicated operation.
- 351⋅187
- −5127⋅4114
Answer: −11268
- 8÷232
- 832÷1121
Answer: 8
Translate Phrases to Expressions with Fractions
In the following exercises, translate the English phrase into an algebraic expression.
- the quotient of 8 and y
- the quotient of V and the difference of h and 6
Answer: h−6V
Simplify Complex Fractions
In the following exercises, simplify the complex fraction
- 5485
- −498
Answer: −92
- 834n
- −121−165
Answer: 22
Simplify Fractions
In the following exercises, simplify.
- 55+16
- 3⋅128⋅4−52
Answer: 367
- 9⋅3−6⋅48⋅7+5(8−10)
Add and Subtract Fractions with Common Denominators
Add Fractions with a Common Denominator
In the following exercises, add.
- 83+82
Answer: 85
- 54+51
- 52+51
Answer: 53
- 3215+329
- 10x+107
Answer: 10x+7
Subtract Fractions with a Common Denominator
In the following exercises, subtract.
- 118−116
- 1211−125
Answer: 21
- 54−5y
- −3031−307
Answer: −1519
- 23−(23)
- 1511−155−(−152)
Answer: 158
Add and Subtract Fractions with Different Denominators
Find the Least Common Denominator (LCD)
In the following exercises, find the least common denominator.
- 31 and 121
- 31 and 54
Answer: 15
- 158 and 2011
- 43,61, and 105
Answer: 60
Convert Fractions to Equivalent Fractions with the LCD
In the following exercises, change to equivalent fractions using the given LCD.
- 31 and 51,LCD=15
- 83 and 65,LCD=24
Answer: 249 and 2420
- −169 and 125,LCD=48
- 31,43 and 54,LCD=60
Answer: 6020,6015 and 6048
Identify and Use Fraction Operations
In the following exercises, perform the indicated operations and simplify.
- 51+32
- 1211−32
Answer: 41
- −109−43
- −3611−2011
Answer: −9077
- −2522+409
- 10y−31
Answer: 303y−10
- 52+(−95)
- 114÷7d2
Answer: 1114d
- 52+(−83n)(−9n2)
- (85)2(32)2
Answer: 225256
- (1211+83)÷(65−101)
Mixed Practice
In the following exercises, evaluate.
Exercise 1
y−54 when
- y=−54
- y=41
Answer:
- −58
- −2011
Exercise 2
6mn2[/latex]when[latex]m=43 and n=−31
Add and Subtract Mixed Numbers
In the following exercises, perform the indicated operation.
- 431+931
Answer: 1332
- 652+753
- 5118+2114
Answer: 8111
- 385+387
- 92013−42011
Answer: 5101
- 2103−1109
- 21211−1127
Answer: 310
- 8116−2119
Solve Equations with Fractions
Determine Whether a Fraction is a Solution of an Equation
In the following exercises, determine whether the each number is a solution of the given equation.
Exercise 1
x−21=61:
- x=1
- x=32
- x=−31
Answer:
- no
- yes
- no
Exercise 2
y+53=95:
- y=21
- y=4552
- y=−452
Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality
In the following exercises, solve the equation.
- n+119=114
Answer: n=−115
- x−61=67
- h−(−87)=−52
Answer: h=−4051
- 5x=−10
- −z=23
Answer: z = −23
Translate Sentences to Equations and Solve
In the following exercises, translate and solve.
- The sum of two-thirds and n is −53.
- The difference of q and one-tenth is 21.
Answer: q−101=21;q=53
- The quotient of p and −4 is −8.
- Three-eighths of y is 24.
Answer: 83y=24;y=64
Chapter Practice Test
Convert the improper fraction to a mixed number.
- 519
Convert the mixed number to an improper fraction.
- 372
Answer: 723
Locate the numbers on a number line.
- 21,132,−243, and 49
In the following exercises, simplify.
- 205
Answer: 41
- 27s18r
- 31⋅43
Answer: 41
- 53⋅15
- −36u(−94)
Answer: 16u
- −5127⋅4114
- −65÷125
Answer: −2
- 117÷(−117)
- 109a÷815a
Answer: 2512
- −652÷4
- (−1565)÷(−361)
Answer: 5
- 116−6
- 5q2p
Answer: 2q5p
- −232−154
- 9−492−42
Answer: 13
- d2+d9
- −133+(−134)
Answer: −137
- −2522+409
- 52+(−57)
Answer: −1
- −103+(−85)
- −43÷3x
Answer: −4x9
- (43)223−22
- 569145+81
Answer: 3
Evaluate
x+31 when
- x=32
- x=−65
In the following exercises, solve the equation.
- y+53=57
Answer: y=54
- a−103=−109
- f+(−32)=125
Answer: f=1213
- −2m=−16
- −32c=18
Answer: c = −27
Translate and solve: The quotient of
p and
−4 is
−8. Solve for [latex]p[/late
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