3.4 adding and subtracting mixed · pdf fileadding and subtracting mixed numbers section 3.4...

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Adding and SubtractingMixed Numbers

3.4

3.4 OBJECTIVES

1. Add any two mixed numbers2. Add any group of mixed numbers3. Subtract any two mixed numbers4. Solve an application that involves mixed number

addition or subtraction

261

Once you know how to add fractions, adding mixed numbers should be no problem if youkeep in mind that addition involves combining groups of the same kind of objects. Becausemixed numbers consist of two parts—a whole number and a fraction—we could work withthe whole numbers and the fractions separately. Generally, it is easier to rewrite mixednumbers as improper fractions, then do the addition.

This suggests the following general rule.

518

53�

512

56

The sum of the The sum of thefractional parts

�

partswhole-number

�

�

or 3

Step 1 Change the mixed numbers to improper fractions.Step 2 Add the fractions.Step 3 Rewrite the result as a mixed number if required.

Step by Step: To Add Mixed Numbers

NOTE Step 2 requires that thefractional parts have the samedenominator.

262 CHAPTER 3 ADDING AND SUBTRACTING FRACTIONS

Our first example illustrates the use of this rule.

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Example 1

Adding Mixed Numbers

Add and write the result as a mixed numbers.

Rewrite as a mixed number.� 7

3

5

Add the numerators.�38

5

Rewrite as improper fractions.� 16

5�

22

53

1

5� 4

2

5

Example 2

Adding Mixed Numbers with Different Denominators

Add, and write the result as a mixed number.

� 5

13

24

�133

24

Then add as before.�76

24�

57

24

The LCD of the fractions is 24. Renamethem with that denominator.

�19

6�

19

83

1

6� 2

3

8

C H E C K Y O U R S E L F 1

Add � Write the result as a mixed number.3

410

.2

310

When the fractional portions of the mixed numbers have different denominators, wemust rename these fractions as equivalent fractions with the least common denominator toperform the addition in step 2. Consider Example 2.


Add � Write the result as a mixed number.3

56

.5

710

NOTE 5

12013

24B133

You follow the same procedure if more than two mixed numbers are involved in theproblem.

ADDING AND SUBTRACTING MIXED NUMBERS SECTION 3.4 263

Adding Mixed Numbers with Different Denominators

Add.

� 10

3

40

�403

40

�88

40�

150

40�

165

40

�11

5�

15

4�

33

8 2

1

5� 3

3

4� 4

1

8

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Example 3

NOTE The LCD of the threefractions is 40. Convert toequivalent fractions.


Add � � 3

34

.4

23

5

12

Step 1 Change the mixed numbers to improper fractions.Step 2 Subtract the fractions.Step 3 Rewrite the result as a mixed number if required.

Step by Step: To Subtract Mixed Numbers

We can use a similar technique for subtracting mixed numbers. The rule is similar to thatstated earlier for adding mixed numbers.

Example 4 illustrates the use of this rule.

Subtracting Mixed Numbers with Like Denominators

Subtract.

� 2

1

6

�13

6

�26

12

�67

12�

41

12 5

7

12� 3

5

12

Example 4


Again, we must rename the fractions if different denominators are involved. Thisapproach is shown in Example 5.

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Subtract � 5

38

.8

78

Example 5

Subtracting Mixed Numbers with Different Denominators

Subtract.

� 5

13

40

�213

40

Subtract as before.�348

40�

135

40

Write the fractions with denominator 40. �87

10�

27

8 8

7

10� 3

3

8


Subtract � 3

58

.7

1112

To subtract a mixed number from a whole number, we use the same techniques.

Example 6

Subtracting Mixed Numbers

Subtract.

� 3

1

4

�13

4

Write both the whole number and themixed number as improper fractions witha common denominator.

6 � 2

3

4�

24

4�

11

4

6 � 2

3

4


Subtract 7 � 3

25

.

NOTE

Multiply the numerator anddenominator by 4 to form acommon denominator.

6 �61

�244

ADDING AND SUBTRACTING MIXED NUMBERS SECTION 3.4 265

An Application of the Subtraction of Mixed Numbers

Linda was inches (in.) tall on her sixth birthday. By her seventh year she was in.

tall. How much did she grow during the year?

Because we want the difference in height, we must subtract.

Linda grew in. during the year.3

3

8

� 3

3

8 in.

�27

8 in.

�413

8�

386

8

51

5

8� 48

1

4�

413

8�

193

4

51

5

848

1

4

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Example 7


You use yards (yd) of fabric from a 50-yd bolt. How much fabric remains on the

bolt?

4

34

Often we will have to use more than one operation to find the solution to a problem.Consider Example 8.

Example 8

An Application Involving Mixed Numbers

A rectangular poster is to have a total length of in. We want a -in. border on the top

and a 2-in. border on the bottom. What is the length of the printed part of the poster?

1

3

812

1

4


First, we will draw a sketch of the poster:

Now, we will use that sketch to find the total width of the top and bottom borders.

Now subtract that sum (the top and bottom borders) from the total length of the poster.

The length of the printed part is in.8

7

8

� 8

7

8 in.

�71

8

121

4�

27

8�

49

4�

27

8�

98

8�

27

8

1

3

8� 2 �

11

8�

16

8�

27

8 in.

83

in.1

12

2 in.

in.41 ?

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You cut one shelf feet (ft) long and one ft long from a 12-ft piece of lumber.

Can you cut another shelf 4 ft long?

4

12

3

34

C H E C K Y O U R S E L F A N S W E R S

1. 2.

3. 4. 5.

6. 7. 8. No, only ft is “left over.”33

445

1

4 yd3

3

5

7

11

12� 3

5

8�

95

12�

29

8�

190

24�

87

24�

103

24� 4

7

243

1

213

11

12

5

7

10 � 3

5

6�

57

10�

23

6�

171

30�

115

30�

286

30� 9

16

30� 9

8

155

7

10

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Exercises

Do the indicated operations.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26. 1

1

15� 3

3

10� 2

4

52

3

8� 2

1

4� 1

5

6

1

5

6� 3

5

12� 2

1

43

3

4� 5

1

2� 2

3

8

4 � 1

2

35 � 2

1

4

9

3

7� 2

13

217

5

12� 3

11

18

5

4

5� 1

1

63

2

3� 2

1

4

5

3

7� 2

1

73

2

5� 1

4

5

3

5

6� 1

1

67

7

8� 3

3

8

4

5

6� 3

2

3� 7

5

93

3

5� 4

1

4� 5

3

10

3

1

5� 2

1

2� 5

1

42

1

4� 3

5

8� 1

1

6

2

1

4� 1

1

61

1

3� 2

1

5

5

8

9� 4

4

96

5

9� 4

7

9

11

6� 5

5

62

1

9� 5

5

9

5

2

9� 6

4

92

2

9� 3

5

9

3.4

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

267

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Solve the following applications.

27. Plumbing. A plumber needs pieces of pipe and inches (in.) long. What is

the total length of the pipe that is needed?

28. Postage. Marcus has to figure the postage for sending two packages. One weighs

pounds (lb), and the other weighs lb. What is the total weight?

29. Working hours. Franklin worked hours (h) on Monday, h on Wednesday and

h Friday. What was the total number of hours that he worked?

30. Distance. Robin ran mi on Sunday, mi on Tuesday, and mi on Friday. How

far did she run during the week?

31. Perimeter. Find the perimeter of the figure below.

32. Perimeter. Find the perimeter of the figure below.

33. Consumer purchases. Senta is working on a project that uses three pieces of fabric

with lengths and yd. She needs to allow for yd of waste. How much fabric

should she buy?

34. Construction. The framework of a wall is in. thick. We apply in. wallboard and

-in. paneling to the inside. Siding that is in. thick is applied to the outside. What

is the finished thickness of the wall?

35. Stocks. A stock was listed at points on Monday. By closing time Friday, it was

at . How much did it drop during the week?

36. Cooking. A roast weighed lb before cooking and lb after cooking. How many

pounds were lost during the cooking?

3

3

84

1

4

28

3

4

34

3

8

3

4

1

4

5

83

1

2

1

8

5

8

3

4, 1

1

4,

21

41

in.

in.

1

2

85

in.1

41

85

in.

in.

1

1

83 in.1

3

1

22

1

45

1

3

4

1

2

5

3

42

1

4

2

3

43

7

8

25

3

415

5

8

ANSWERS

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

268

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37. Quantity of material. A roll of paper contains yd. If yd is cut from theroll, how much paper remains?

38. Geometry. Find the missing dimension in the figure below.

39. Carpentry. A in. bolt is placed through a board that is in. thick. How far

does the bolt extend beyond the board?

40. Working hours. Ben can work 20 h per week on a part-time job. He works h

on Monday and h on Tuesday. How many more hours can he work during the week?

41. Geometry. Find the missing dimension in the figure below.

42. Carpeting. The Whites used square yards (yd2) of carpet for their living room,

yd2 for the dining room, and yd2 for a hallway. How much will remain if a

50 yd2 roll of carpeting is used?

43. Construction. A construction company has bids for paving roads of and

miles (mi) for the month of July. With their present equipment, they can pave 8 mi

in 1 month. How much more work can they take on in July?

3

1

3

1

1

2,

3

4,

6

1

415

1

2

20

3

4

?

in.85

in.415

3

3

4

5

1

2

3

1

24

1

4

3

in.415

?

in.83

16

7

830

1

4

ANSWERS

37.

38.

39.

40.

41.

42.

43.

269

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44. Travel. On an 8 h trip, Jack drives h and Pat drives h. How many hours areleft to drive?

45. Distance. A runner has told herself that she will run 20 mi each week. She runs

mi on Sunday, mi on Tuesday, mi on Wednesday, and mi on Friday.

How far must she run on Saturday to meet her goal?

46. Environment. If paper takes up of the space in a landfill and plastic takes up of

the space, how much of the landfill is used for other materials?

47. Environment. If paper takes up of the space in a landfill and organic waste takes

up of the space, how much of the landfill is used for other materials?

48. Interest. The interest rate on an auto loan in May was %. By September the rate

was up to %. How much did the interest rate increase over the period?

Answers

1. 3. 5. 7. 9. 11.

13. 15. 17. 19.

21. 23. 25. 27. 29. 31.

33. yd 35. points 37. yd 39. 41. 4 in.

43. 45. 47. 3

83

3

8 mi2

5

12 mi

3

4 in.13

3

85

5

82

3

4

4

1

4 in.12

1

2 h41

3

8 in.2

19

246

7

82

3

4

3

29

361

5

123

2

5� 1

4

5� 2

7

5� 1

4

5� 1

3

54

1

2

13

3

207

1

243

8

1511

1

37

2

35

7

9

14

1

4

123

8

1

8

1

2

1

10

1

2

2

1

84

3

44

1

45

1

2

2

1

22

3

4

ANSWERS

44.

45.

46.

47.

48.

270

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Using Your Calculator to Addand Subtract Mixed Numbers

We have already seen how to add, multiply, and divide fractions using our calculators. Nowwe will use our calculators to add and subtract mixed numbers.

Scientific Calculator

To enter a mixed number on a scientific calculator, press the fraction key between both the

whole number and the numerator and denominator. For example, to enter , press

3 7 12a b/ca b/c

3

7

12

271

Example 1

Adding Mixed Numbers

Add.

The keystroke sequence is

3 7 12 2 11 16

The result is

Graphing Calculator

As with multiplying and dividing fractions, when using a graphing calculator, you must

choose the fraction option from the math menu before pressing .

For the problem in Example 1, the keystroke sequence is

3 7 12 2 11 16

The display will read 301

48.

Enter�Frac��

3

7

12� 2

11

16,

Enter

6

13

48.

�a b/ca b/c�a b/ca b/c

3

7

12� 2

11

16


Find the sum.

4

3

7�4

5

6


1. 9

11

42

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Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

272

Calculator ExercisesAdd or subtract the following.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

Answers

1. 3. 5. 7. 9. 11.

13. 15. 224

5

6

3

13

482

1

637

31

5416

11

128

1

95

11

12

7

1

5� 3

2

3� 1

1

510

2

3� 4

1

5� 7

2

15

131

43

45� 99

27

606

2

3� 1

5

6

18

5

24� 11

3

405

11

16� 2

5

12

7

8

11� 4

13

224

7

9� 2

11

18

82

41

45� 97

25

2714

13

18� 22

23

27

6

3

8� 14

5

1211

2

3� 5

1

4

2

3

7� 4

9

145

4

9� 2

2

3

6

1

6� 8

2

33

2

3� 2

1

4

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Estimation Applications3.5

3.5 OBJECTIVES

1. Use estimation to solve application problems

273

Units AnalysisEvery denominate number with a fractional unit, such as

25

has a dual denominate number, which is the reciprocal number and thereciprocal units. In this case we have

The 25 indicates that we can drive 25 miles on 1 gallon. The

indicates that we use of a gallon for each mile.

Examples

Denominate Number Dual

55

minpage

32

pagemin

23

hmi

155

mih

125

125

galmi

migal

125

galmi

migal

Of all the skills you develop in the study of arithmetic, perhaps the most useful is that ofestimation. Every day, you have occasion to estimate. Here are a few estimation exercisesthat you may have gone through this morning.

How much flour should I put in the pancakes?

How much cash am I likely to need today?

How long will it take me to walk to the bus stop?

How long is the ride to school?

How many miles can I get on just over tank of gas?

Although you may not think of these as arithmetic problems, they are. As your estima-tion skills improve, so will your ability to come up with good answers to everydayproblems.

1

4


Estimating Measurements

Based on the gauge, is the gas tank closer to or full?

At slightly more than , the gauge indicates close to of a tank remains.2

3

1

2

E F

12

1

4

3

4,

2

3,

1

3,

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Example 1


Is the reading of the thermometer closer to 72º, 74º, or 76º?

1101009080706050403020100

–10–20

Cooking provides many opportunities for estimation. Our second example illustrates one.

Example 2

Estimating Measurement

A recipe calls for cup of sugar. The only measuring cups you have are 1 cup, cup, and

cup. What should you do?1

4

1

2

1

3

ESTIMATION APPLICATIONS SECTION 3.5 275

To solve this problem you must believe that recipes are only approximations anyway. Once

you accept that idea, you can estimate cup. It is between cup and cup, so fill the

cup, dump it into the cup, and add a little more sugar.1

2

1

4

1

2

1

4

1

3

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A recipe calls for cup of flour, but you have only a 1 cup, a cup, and a cup

measure. How should you proceed?

13

23

34

Almost every shopping trip presents many estimation opportunities. Whether you are esti-mating the impact of car payments on your monthly budget, or estimating the number ofbananas you can buy for $3, you are practicing your arithmetic skills. Example 3 providesan opportunity to practice these skills.

Estimating Total Cost

Dog food is on special at three cans for a dollar. Your puppy eats about can per day. How

much should you budget for dog food over the next semester (almost 5 months)?

First, estimate the number of days in the semester. At 30 days per month, we’ll use

30 � 5 months � 150 days

Next, estimate the amount of food to be consumed. We could try using can per day, whichyields

� 150 days � 75 cans

but remember that this is a puppy, so we will assume his food intake will increase over thenext 5 months. To be safe, we will add 25 cans and estimate the total to be 100 cans.

Finally, we can estimate the total cost. If we buy all of the food today, we can buy it at

3 cans per dollar. At dollar per can, we get

� 100 cans � $33

The cost will be about $33.

dollarcan

1

3

1

3

canday

1

2

1

2

daysmonth

1

2

Example 3


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1. It is closer to 74°.

2. Fill the cup measure, dump it into the 1 cup measure, and add a little flour.

3. The cost is a little under $25.

2

3


Cat food is also on special at seven cans for $2 ( ). If the cat eats about

can per day, what is the approximate cost of the cat food for a 5-month semester?12

cansdollar

72

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Exercises

Find the dual of each denominate number.

1. 36 2. 42

3. 5 4. 125

5. 18 6. 84

7. 68 8. 55

Solve the following applications.

9. Mark wants to make $300 by selling cupcakes at a bake sale, and his mother hasoffered to pay for the ingredients. He plans to sell them for $5 per dozen. A boxof cake mix makes about 50 cupcakes. Approximately how many boxes shouldMark buy?

10. Amy is traveling to a city 418 miles away at a speed of roughly 55 miles per hour.About how long should her trip take?

11. The map tells Manuel that it is 423 miles from Eastwick to West Goshen. He knowsthat, with mixed city and freeway driving, he can average about 50 miles per hour. Heis currently in Eastwick and needs to be in West Goshen by noon. Make a roughestimate of the time that Manuel should leave Eastwick.

12. Sam works in a shipping department for a manufacturer. He is filling a customer’sorder for several small items. The items ordered weigh 21 ounces, 23 ounces,18 ounces, and 7 ounces. The company policy is to use a stronger box to shipproducts totaling more than 3 pounds. Knowing 16 ounces is 1 pound, estimatewhether Sam should use the stronger box.

13. Lauren is a contractor for a roofing job. She estimates that she will need about 4800roofing nails. According to a handbook, the roofing nails that are needed count out atabout 189 nails per pound. Estimate how many pounds Lauren will need for the job.

ergsmin

lumenscm2

coulombss

migal

jouless

pagesmin

kilowatth

fts

3.5

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

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14. Julio works as a quality control expert in a beverage factory. The assembly line thathe monitors produces about 20,000 bottles in a 24-hour period. Julio samples about

120 bottles an hour and rejects the line if he finds more than of the sample to be

defective. About how many defective bottles should Julio allow before rejecting the entire line?

15. Based on the amount of liquid in the pitcher, is the pitcher closer to or full?

16. Lunch meat sells for about $3 per pound. You use about pound per day for

sandwiches. How much should you budget for lunch meat over the next month (about20 work days)?

Estimate the following sums or differences.

17. 18.

19. 20.

Estimate the following products or quotients.

21. 22.

23. 24.

25.

Answers

1. 3. 5. 7. 9. 14 boxes

11. 3 A.M. 13. 24 lb 15. 17. 20 19. 6 21. 12

23. 3 25. 8

3

4

1

68

cm2

lumen

1

18 gal

mi

1

5 min

page

1

36 s

ft

156

7 4

1

8� 1

3

4

1711

12 6

1

109

2

9 2

6

7

31

7� 6

8

92

5

6� 4

1

3

181

5� 11

7

8� 14

10

11� 10

6

715

4

5� 3

9

10� 6

1

7

45

6� 3

1

5� 8

1

7� 11

8

92

1

8� 7

9

11� 4

6

7� 5

1

12

1

2

3

4

1

3,

2

3,

1

4

2

100

ANSWERS

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

278

3.4 adding and subtracting mixed · pdf fileadding and subtracting mixed numbers section 3.4...

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