addition and subtraction of rational numbers

8/7/2019 Addition and Subtraction of Rational Numbers

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161

3.3 Addition and Subtraction of Rational NumbersIn this section we consider addition and subtraction of both fractions and decimals. We start with

addition and subtraction of fractions with the same denominator. Consider the sum1

8+3

8. If you

think of eighths as the quantity being added, it makes sense the sum is:

1

8+

3

8=

1+ 3

8

=4

8

=1 /4

2 /4

=1

2

Mathematically, we are actually using the distributive property. Since1

8= 1

1

8and

3

8= 3

1

8,

we have the sum:

1

8+

3

8= 1

1

8+ 3

1

8

= 4

1

8

=4

8

=1 /4

2 /4

=1

2

Regardless of the way you look at the problem, adding (or subtracting) two fractions with the

same denominator simply means to add or subtract their numerators, leaving the denominatoruntouched.


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Example 1 Add or subtract the fractions, as indicated. Be sure to simplify all answers.

a.

5

12+

1

12

b.3

8!

5

8

c. !4

15!

6

15

d. !9

16+

5

16!

3

16

Solution a. Add the two fractions, combine the numerators, then simplify:

5

12+

1

12=

5 +1

12combine fractions

=6

12add numerators

=1 /6

2 /6factor GCF

=1

2cancel common factors

b. Subtract the two fractions, combine the numerators, then simplify:

3

8!

5

8=

3! 5

8combine fractions

=3+ (!5)

8rewrite subtraction as addition

=!2

8add numerators

= !1 /2

4 /2factor GCF

= ! 14

cancel common factors

Note how we rewrite subtraction as addition in the second step.


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c. Subtract the two fractions, combine the numerators, then simplify:

!

4

15!

6

15 =

!4 ! 6

15 combine fractions

=!4 + (!6)

15rewrite subtraction as addition

=!10

15add numerators

= !2 /5

3 /5factor GCF

= !2

3cancel common factors

d. Add the three fractions, combine the numerators, then simplify:

!

9

16+

5

16!

3

16=!9 + 5 ! 3

16combine fractions

=!9 + 5 + (!3)

16add numerators

=!12 + 5

16add negatives

= !7

16add numbers

Notice that when working with negative fractions such as !9

16, we treat the negative as being

with the numerator. This is done to allow the denominator to always be positive, making it easierto compare denominators.When two denominators are not the same, we need to build each fraction to a common

denominator. For example, suppose we are adding the two fractions5

6+3

8. Can we find a

denominator that both fractions can be built up to? Since the least common multiple (LCM) of 6and 8 is 24, then both fractions can be converted to one with a denominator of 24. That willallow us to add the two fractions using the least common denominator (fraction terminologyfor LCM).


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Converting each fraction:

5

6 +3

8 =5

6

4

4 +3

8

3

3 converting to common denominators

=20

24+

9

24building fractions

=20 + 9

24combining fractions

=29

24or 1

5

24adding fractions

Fractions can actually be built to any common denominator (such as 48 in the previous example),however the LCM will provide the smallest denominator to use, which usually results in lesserrors and simplifying of answers. Note that we gave the mixed form of the answer also.Generally we do not give mixed form answers, unless they are asked for or mixed numbers wereused originally in the problem.Example 2 Add or subtract the fractions, as indicated. Be sure to simplify all answers.

a. !3

4+5

6

b. !7

8!

5

16

c. ! 512

!2

3+ 3

8

d. !7

10!

13

15!

9

20


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Solution a. The LCM of 4 and 6 is 12. Converting each fraction to the commondenominator of 12, then combining numerators and simplifying:

!

3

4+

5

6= !

3

4

3

3+

5

6

2

2converting to common denominators

= !9

12+

10


=!9 +10


=1

12adding fractions

b. The LCM of 8 and 16 is 16. Converting each fraction to the common

denominator of 16, then combining numerators and simplifying:

!

7

8!

5

16= !

7

8

2

2!

5


= !14

16!

5


=!14 ! 5


=!14 + (!5)

12

converting to addition

= !19

12or !1

7

12adding fractions


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c. The LCM of 12, 3, and 8 is 24. Converting each fraction to the commondenominator of 24, then combining numerators and simplifying:

!

5

12!

2

3+

3

8

= !5

12

2

2!

2

3

8

8+

3

8

3


= !10

24!

16

24+

9


=!10 !16 + 9


=!10 + (!16) + 9

24

converting to addition

= !17

24adding fractions

d. The LCM of 10, 15, and 20 is 60. Converting each fraction to the common

denominator of 60, then combining numerators and simplifying:

!

7

10!

13

15!

9

20

= !7

10

6

6

!

13

15

4

4

!

9

20

3

3

converting to common denominators

= !42

60!

52

60!

27


=!42 ! 52 ! 27


=!42 + (!52) + (!27)

60converting to addition

= !121

60or ! 2

1

60adding fractions


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Recall that the least common multiple of numbers cannot always be found easily. In such cases,using primes to find the LCM is a faster method. Suppose we want to add the two fractions:

17

84+19

72

Start by finding the prime factorizations of 84 and 72:

84 = 4 21 = 2 2( ) 3 7( ) = 2 2 3 7

72 = 8 9 = 2 4( ) 3 3( ) = 2 2 2 3 3

Since the LCM must have three 2s, two 3s, and one 7, it is:

LCM = 2 2 2 3 3 7 = 504 Now build up the fractions using the prime factors:

17

84+

19

72=

17

84

6

6+

19

72

7


=102

504+

133


=102 +133


=235

504adding numerators

=5 47

2 2 2 3 3 7prime factorizations

=235

504multiplying factors

Note a few advantages in using primes for the common denominator. In building the fractions,

the forms of 1 used which were6

6and

7

7can be found by just looking at the prime

factorizations, rather than by using division. Also, in the step of simplifying the resultingfraction, the prime factorization for the denominator is already known (that is how we got thedenominator!), so only the numerator needs to be factored in order for the fraction to be reduced.For these reasons, many students find that using primes to obtain common denominators (ratherthan by guessing) is a better approach.


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Example 3 Add or subtract the fractions, as indicated. Use primes to find the least common

denominator. Be sure to simplify all answers.

a. !5

8+17

36

b. !31

40!

19

28

c. !5

12+13

20!

17

45

d.5

8x+

7

12y

Solution a. Start by finding the prime factorizations of 8 and 36:

8 = 2 4 = 2 2 2

36 = 4 9 = 2 2 3 3

The LCM must have three 2s and two 3s, which is:

LCM = 2 2 2 3 3 = 72 Now build the fractions to the LCM, combine numerators, and simplify:

!

5

8+

17

36= !

5

8

9

9+

17

36

2


= !45

72+

34


=!45 + 34


= !11

72adding numerators


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b. Start by finding the prime factorizations of 40 and 28:

40 = 4 10 = 2 2 2 5

28 = 4 7 = 2 2 7

The LCM must have three 2s, one 5, and one 7, which is:

LCM = 2 2 2 5 7 = 280 Now build the fractions to the LCM, combine numerators, and simplify:

!

31

40!

19

28= !

31

40

7

7!

19

28

10


= !217

280!

190


=!217 !190


=

!217 + (!190)

280changing to addition

= !407


c. Start by finding the prime factorizations of 12, 20, and 45:

12 = 4 3 = 2 2 3

20 = 4 5 = 2 2 5

45 = 9 5 = 3 3 5

The LCM must have two 2s, two 3s, and one 5, which is:

LCM = 2 2 3 3 5 = 180


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Now build the fractions to the LCM, combine numerators, and simplify:

!

5

12+

13

20!

17

45

= !5

12

15

15+

13

20

9

9!

17

45

4


= !75

180+

117

180!

68


=!75 +117 ! 68


=!75 +117 + (!68)

180changing to addition

= !26


= !/2 13

/2 2 3 3 5cancelling common factors

= !13

90multiplying factors

d. Start by finding the prime factorizations of 8x and 12y:

8x = 2 2 2 x

12y = 2 2 3 y

The LCM must have three 2s, one 3, one x, and one y, which is:

LCM = 2 2 2 3 x y = 24xyNow build the fractions to the LCM and combine numerators:

5

8x+

7

12y=

5

8x

3y

3y+

7

12y

2x

2xconverting to common denominators

=15y

24xy+

14x

24xybuilding fractions

=15y +14x

24xycombining fractions

Notice how we cannot do any further simplification of this resulting

fraction. In algebra you will learn some techniques which can be applied tosimplify fractions such as this one.


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When dealing with mixed numbers, two different approaches can be used. If we are adding twomixed numbers, both of which are positive, the easiest approach is to add the whole number and

fraction portions separately. For example, to add 4

2

3+ 3

1

2 , we first add the two fractions:

2

3+

1

2=

2

3

2

2+

1

2

3


=4

6+

3

6building fractions

=4 + 3


=7

6

adding fractions

= 11

6converting to mixed number

Now adding the mixed numbers:

42

3+ 3

1

2= 7 +1

1

6= 8

1

6


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However, when negative numbers become involved, this method becomes rather tricky. Thus, to

compute the subtraction 31

4! 6

3

5, it is best to convert the mixed numbers to fractions and

compute directly:

31

4! 6

3

5=

13

4!

33

5converting to fractions

=

13

4

5

5!

33

5

4


=65

20!

132


=

65 !132

20

combining fractions

= !

67

20subtracting fractions

= !37


Unless we are adding positive mixed numbers, it is this second approach we will use to combinemixed numbers.Example 4 Combine the mixed numbers, as indicated. Be sure to simplify any answers and

convert answers to mixed numbers.

a. 85

6+ 5

3

4

b. 31

8! 7

9

16

c. !41

3+ 2

3

5

d. !53

4! 3

2

3


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Solution a. Since we are adding positive mixed numbers, we can use the first approach.Start by adding the two fractions:

5

6 +3

4 =5

6

2

2 +3

4

3

3 converting to common denominators

=10

12+

9


=10 + 9


=19

12adding fractions

= 17


Now adding the mixed numbers:

85

6+ 5

3

4= 13+1

7

12= 14

7

12

b. Converting the mixed numbers to fractions, then combining:

31

8! 7

9

16=

25

8!

121


=

25

8

2

2!

121


=50

16

!

121

16

building fractions

=

50 !121


= !

71

16subtracting fractions

= !47



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c. Converting the mixed numbers to fractions, then combining:

!41

3

+ 23

5

= !13

3

+13

5

converting to fractions

= !13

3

5

5+

13

5

3


= !65

15+

39


=!65 + 39


= !26

15adding fractions

= !111


d. Converting the mixed numbers to fractions, then combining:

!53

4! 3

2

3= !

23

4!

11


= !23

4

3

3!

11

3

4


= !69

12!

44


=!69 ! 44

12 combining fractions

=!69 + (!44)

12converting to addition

= !113

12adding fractions

= !95



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Whereas adding and subtracting fractions and mixed numbers involves a number of steps infinding the common denominator, the same operations for decimals are fairly easy to apply.Since the decimal system involves tenths, hundredths, thousandths, etc, the place-values used

already represent common denominators. Thus, to compute 15.89 + 7.643, we only need to besure the decimal points are lined up so that the place-values are also lined up. Usually we insertplace-value holders (0), line up the decimal points, then just add as with whole numbers. Thesum is therefore:

11 1

15.890

+7.643

23.533

Subtraction is performed similarly, except that borrowing (rather than carrying) is involved.

Example 5 Perform the following additions and subtractions.

a. 45.982 + 6.57 b. 9.9 + 23.864 c. 5.07 ! 3.295 d. 6.4 ! 9.86

Solution a. Lining up the decimal and inserting place-value holders:

111

45.982

+6.570

52.552

b. Lining up the decimal and inserting place-value holders:

11

9.900

+23.864

33.764


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c. Lining up the decimal and inserting place-value holders:

496

/5. /0 /70

!3.295

1.775

d. This is actually trickier than it looks. Since 9.86 is larger than 6.4, this

subtraction will result in a negative number. To find out how much it will benegative, we actually need to reverse the subtraction:

9.86

!6.40

3.46

Since the value is actually negative, 6.4 ! 9.86 = !3.46 .

Terminology

least common denominator

Exercise Set 3.3

Add or subtract the fractions, as indicated. Be sure to simplify all answers.

1.7

12+

1

12 2.

4

15+

8

15

3.5

16!

11

16 4.

7

24!

13

24

5. !17

25!

8

25 6. !

19

30!

11

30

7. !23

30+

7

30 8. !

13

24+

5

24

9. !5

12+

7

12!

11

12 10. !

13

24!

7

24+11

24

11.7

30!

11

30!

17

30 12.

13

48!

17

48!

5

48


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13.3x

14!

5y

14 14.

17a

25!

9b

25

Add or subtract the fractions, as indicated. Be sure to simplify all answers.

15.2

3+5

6 16.

3

4+5

8

17.1

4!

7

16 18.

2

5!

11

20

19.5

8!

13

15 20.

5

8!

9

10

21. !5

6+2

3 22. !

7

9+5

6

23. !3

4+1

6 24. !

2

3+1

4

25. !5

8!

3

4 26. !

5

6!

4

9

27. !3

8!

7

12 28. !

5

7!

9

14

29. !5

8!

5

12+17

24 30. !

7

10!

11

15+

7

25

31. !7

20+13

30!

11

15 32. !

5

6+7

8!

11

12

Add or subtract the fractions, as indicated. Use primes to find the least common denominator.Be sure to simplify all answers.

33.7

8+19

36 34.

13

32+17

36

35. !26

35+11

15 36. !

23

35!

13

15

37. !27

40!

16

30 38. !

29

40!

18

25

39. !23

48+17

30 40.

13

48!

23

30

41. !7

12+11

20!

19

45 42. !

11

12+17

20!

24

35


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43. !13

18!

11

12+1

8 44. !

17

18!

19

24+

8

27

45.3

8x!

5

12x 46.

7

10x!

11

15x

47.5

12a!

8

15b 48.

3

8a!

7

12b

Combine the mixed numbers, as indicated. Be sure to simplify any answers and convert answersto mixed numbers.

49. 52

3+ 7

1

2 50. 9

3

4+ 8

7

8

51. 6

1

3 + 8

3

4 + 5

5

6 52. 9

1

2 + 7

9

10 + 6

3

5

53. 51

4! 9

7

8 54. 3

2

3! 8

5

6

55. 57

12! 9

13

15 56. 4

1

12! 9

7

15

57. !61

2+1

2

3 58. !8

1

3+ 3

3

4

59. !95

6+ 4

5

8 60. !7

5

8+ 9

7

12

61. !43

4!

5

1

6 62.!

5

3

5!

7

5

8

63. !76

7! 8

9

14 64. !9

3

8! 5

9

24

Perform the following additions and subtractions.

65. 18.95 + 9.473 66. 23.876 + 8.49 67. 6.99 + 25.808 68. 7.98 + 24.376 69. 14.07 ! 9.683 70. 103.62 ! 56.954 71. 25 !14.46 72. 32 !16.85

73. 102!

28.407 74. 115!

65.749 75. 8.3!12.473 76. 6.7 !14.826 77. 5.2 !13.104 78. 4.7 ! 26.43 79. !8.5 ! 25.77 80. !14.56 ! 29.859


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Answer each of the following application questions. Be sure to read the question, interpret theproblem mathematically, solve the problem, then answer the question. You should answer thequestion in the form of a sentence.

81. Maurice has $458.62 in his checking account, and writes checks for $15.87, $132.45,

and $88.60. What is his new balance in the account? 82. Sylvia has $682.36 in her checking account, and writes checks for $45.86, $102.39,

$23.69, and $16.70. What is her new balance in the account? 83. After writing a check for $78.97, Carolyn has $196.87 in her checking account. How

much was in her account before writing the check? 84. After writing a check for $199.68, Mary has $679.54 in her checking account. How

much was in her account before writing the check? 85. After depositing a check for $795.84 in his checking account, Alfred has $1669.86 in

his savings account. How much was in his account before depositing the check?

86. After depositing two checks for $186.52 and $337.50 in her account, Norma has$1156.40 in her savings account. How much was in her account before depositing thechecks?

87. John buys a stock at a price of1463

8. During the next day it rises 2

1

4, then it drops

67

8the following day. What is the price of the stock after these two days?

88. Dennis buys a stock at a price of461

2. During the next day it drops 1

5

16, then it rises

31

4

the following day. What is the price of the stock after these two days?

89. Three pieces of lumber are stacked on top of each other. The first piece is 31

2inches

thick, the next piece is 13

4inches thick, and the third piece is

7

8inches thick. How

thick is the stack of three pieces of lumber?

90. Three pieces of lumber are stacked on top of each other. The first piece is 53

4inches

thick, the next piece is 11

2inches thick, and the third piece is 2

1

8inches thick. How

thick is the stack of three pieces of lumber?

addition and subtraction of rational numbers

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