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2.3.1: Adding and Subtracting Fractions - Mathematics


Paul and Tony order a pizza which has been cut into eight equal slices. Tony eats three slices (shaded in light red (or a darker shade of gray in black-and-white printing) in Figure (PageIndex{1})), or 3/8 of the whole pizza.

It should be clear that together Paul and Tony eat five slices, or 5/8 of the whole pizza. This reflects the fact that

[ frac{2}{8} + frac{3}{8} = frac{5}{8}. onumber ]

This demonstrates how to add two fractions with a common (same) denominator. Keep the common denominator and add the numerators. That is,

[ egin{align*} frac{2}{8} + frac{3}{8} &= frac{2 + 3}{8} ~ && extcolor{red}{ ext{ Keep denominator; add numerators.}} &= frac{5}{8} ~ && extcolor{red}{ ext{ Simplify numerator.}} end{align*} ]

Adding Fractions with Common Denominators

Let a/c and b/c be two fractions with a common (same) denominator. Their sum is defined as

[ frac{a}{c} + frac{b}{c} = frac{a + b}{c} onumber ]

That is, to add two fractions having common denominators, keep the common denominator and add their numerators.

A similar rule holds for subtraction.

Subtracting Fractions with Common Denominators

Let a/c and b/c be two fractions with a common (same) denominator. Their difference is defined as

[ frac{a}{c} - frac{b}{c} = frac{a-b}{c}. onumber ]

That is, to subtract two fractions having common denominators, keep the common denominator and subtract their numerators.

Example (PageIndex{1})

Find the sum of 4/9 and 3/9.

Solution

Keep the common denominator and add the numerators.

[ egin{aligned} frac{4}{9} + frac{3}{9} &= frac{4+3}{9} ~ & extcolor{red}{ ext{ Keep denominator; add numerators.}} &= frac{7}{9} ~ & extcolor{red}{ ext{ Simplify numerator.}} end{aligned} onumber ]

Exercise (PageIndex{1})

Add:

[ frac{1}{8} + frac{2}{8} onumber ]

Answer

3/8

Example (PageIndex{2})

Subtract 5/16 from 13/16.

Solution

Keep the common denominator and subtract the numerators.

[ egin{aligned} frac{13}{16} - frac{5}{16} &= frac{13-5}{16} ~ & extcolor{red}{ ext{ Keep denominator; subtract numerators.}} &=frac{8}{16} ~ & extcolor{red}{ ext{ Simplify numerator.}} end{aligned} onumber ]

Of course, as we learned in Section 4.1, we should always reduce our final answer to lowest terms. One way to accomplish that in this case is to divide numerator and denominator by 8, the greatest common divisor of 8 and 16.

[ egin{aligned} = frac{8 div 8}{16 div 8} ~ & extcolor{red}{ ext{ Divide numerator and denominator by 8.}} = frac{1}{2} ~ & extcolor{red}{ ext{ Simplify numerator and denominator.}} end{aligned} onumber ]

Exercise (PageIndex{2})

Subtract:

[ frac{11}{12} - frac{7}{12} onumber ]

Answer

1/3

Example (PageIndex{3})

Simplify:

[ frac{3}{x} - left( - frac{7}{x} ight) . onumber ]

Solution

Both fractions share a common denominator.

[ egin{aligned} frac{3}{x} - left( - frac{7}{x} ight) &= frac{3}{x} + frac{7}{x} ~ & extcolor{red}{ ext{ Add the opposite.}} &= frac{3+7}{x} ~ & extcolor{red}{ ext{ Keep denominator, add numerators.}} &= frac{10}{x} ~ & extcolor{red}{ ext{ Simplify.}} end{aligned} onumber ]

Adding Fractions with Different Denominators

Consider the sum

[ frac{4}{9} + frac{1}{6}. onumber ]

We cannot add these fractions because they do not have a common denominator. So, what to do?

Goals

In order to add two fractions with different denominators, we need to:

  1. Find a common denominator for the given fractions.
  2. Make fractions with the common denominator that are equivalent to the original fractions.

If we accomplish the two items in the “Goal,” we will be able to find the sum of the given fractions.

So, how to start? We need to find a common denominator, but not just any common denominator. Let’s agree that we want to keep the numbers as small as possible and find a least common denominator.

Definition: Least Common Denominator

The least common denominator (LCD) for a set of fractions is the smallest number divisible by each of the denominators of the given fractions.

Consider again the sum we wish to find:

[ frac{4}{9} + frac{1}{6} . onumber ]

The denominators are 9 and 6. We wish to find a least common denominator, the smallest number that is divisible by both 9 and 6. A number of candidates come to mind: 36, 54, and 72 are all divisible by 9 and 6, to name a few. But the smallest number that is divisible by both 9 and 6 is 18. This is the least common denominator for 9 and 6.

We now proceed to the second item in “Goal.” We need to make fractions having 18 as a denominator that are equivalent to 4/9 and 1/6. In the case of 4/9, if we multiply both numerator and denominator by 2, we get

[ egin{aligned} frac{4}{9} &= frac{4 cdot 2}{9 cdot 2} ~ & extcolor{red}{ ext{ Multiply numerator and denominator by 2.}} &= frac{8}{18}. ~ & extcolor{red}{ ext{ Simplify numerator and denominator.}} end{aligned} onumber ]

In the case of 1/6, if we multiply both numerator and denominator by 3, we get

[ egin{aligned} frac{1}{6} &= frac{1 cdot 3}{6 cdot 3} ~ & extcolor{red}{ ext{ Multiply numerator and denominator by 3.}} &= frac{3}{18}. ~ & extcolor{red}{ ext{ Simplify numerator and denominator.}} end{aligned} onumber ]

Typically, we’ll arrange our work as follows.

[ egin{aligned} frac{4} + frac{1}{6} &= frac{4 cdot extcolor{red}{2}}{9 cdot extcolor{red}{2}} + frac{1 cdot extcolor{red}{3}}{6 cdot extcolor{red}{3}} ~ & extcolor{red}{ ext{ Equivalent fractions with LCD = 18.}} &= frac{8}{18} + frac{3}{18} ~ & extcolor{red}{ ext{ Simplify numerators and denominators.}} &= frac{8+3}{18} ~ & extcolor{red}{ ext{ Keep common denominator; add numerators.}} &= frac{11}{18} ~ & extcolor{red}{ ext{ Simplify numerator.}} end{aligned} onumber ]

Let’s summarize the procedure.

Adding or Subtracting Fractions with Different Denominators

  1. Find the LCD, the smallest number divisible by all the denominators of the given fractions.
  2. Create fractions using the LCD as the denominator that are equivalent to the original fractions.
  3. Add or subtract the resulting equivalent fractions. Simplify, including reducing the final answer to lowest terms.

Example (PageIndex{4})

Simplify: ( displaystyle frac{3}{5} - frac{2}{3}).

Solution

The smallest number divisible by both 5 and 3 is 15.

[ egin{aligned} frac{3}{5} - frac{2}{3} &= frac{3 cdot extcolor{red}{3}}{5 cdot extcolor{red}{3}} - frac{2 cdot extcolor{red}{5}}{3 cdot extcolor{red}{5}} ~ & extcolor{red}{ ext{ Equivalent fractions with LCD = 15.}} &= frac{9}{15} - frac{10}{15} ~ & extcolor{red}{ ext{ Simplify numerators and denominators.}} &= frac{9-10}{15} ~ & extcolor{red}{ ext{ Keep LCD; subtract numerators.}} &= frac{-1}{15} ~ & extcolor{red}{ ext{ Simplify numerator.}} end{aligned} onumber ]

Although this answer is perfectly acceptable, negative divided by positive gives us a negative answer, so we could also write

[ = - frac{1}{15}. onumber ]

Exercise (PageIndex{4})

Subtract:

[ frac{3}{4} - frac{7}{5} onumber ]

Answer

-13/20

Example (PageIndex{5})

Simplify: (-frac{1}{4} - frac{5}{6}).

Solution

The smallest number divisible by both 4 and 6 is 12.

[ egin{aligned} -frac{1}{4} - frac{5}{6} &= - frac{1 cdot extcolor{red}{3}}{4 cdot extcolor{red}{3}} - frac{5 cdot extcolor{red}{2}}{6 cdot extcolor{red}{2}} ~ & extcolor{red}{ ext{ Equivalent fractions with LCD =12.}} &= - frac{3}{12} - frac{10}{12} ~ & extcolor{red}{ ext{ Simplify numerators and denominators.}} &= frac{-3-10}{12} ~ & extcolor{red}{ ext{ Keep LCD; subtract numerators.}} &= frac{-13}{12} ~ & extcolor{red}{ ext{ Simplify numerator.}} end{aligned} onumber ]

Exercise (PageIndex{5})

Subtract: (-frac{3}{8} - frac{1}{12})

Answer

-11/24

Example (PageIndex{6})

Simplify: (frac{5}{x} + frac{3}{4}).

Solution

The smallest number divisible by both 4 and x is 4x.

[ egin{aligned} frac{5}{x} + frac{3}{4} = frac{5 cdot extcolor{red}{4}}{x cdot extcolor{red}{4}} + frac{3 cdot extcolor{red}{x}}{4 cdot extcolor{red}{x}} ~ & extcolor{red}{ ext{ Equivalent fractions with LCD = }4x.} = = frac{20}{4x} + frac{3x}{4x} ~ & extcolor{red}{ ext{ Simplify numerators and denominators.}} = frac{20 + 3x}{4x} ~ & extcolor{red}{ ext{ Keep LCD; add numerators.}} end{aligned} onumber ]

Exercise (PageIndex{6})

Add:

[ frac{5}{z} + frac{2}{3} onumber ]

Answer

[ frac{15+2z}{3z} onumber ]

Example (PageIndex{7})

Simplify: (frac{2}{3}- frac{x}{5}).

Solution

The smallest number divisible by both 3 and 5 is 15.

[ egin{aligned} frac{2}{3} - frac{x}{5} = frac{2 cdot extcolor{red}{5}}{3 cdot extcolor{red}{5}} - frac{x cdot extcolor{red}{3}}{5 cdot extcolor{red}{3}} ~ & extcolor{red}{ ext{ Equivalent fractions with LCD = 15.}} = frac{10}{15} - frac{3x}{15} ~ & extcolor{red}{ ext{ Simplify numerators and denominators.}} = frac{10 - 3x}{15} ~ & extcolor{red}{ ext{ Keep LCD; subtract numerators.}} end{aligned} onumber ]

Least Common Multiple

First we define the multiple of a number.

Definition: Multiples

The multiples of a number d are 1d, 2d, 3d, 4d, etc. That is, the multiples of d are the numbers nd, where n is a natural number.

For example, the multiples of 8 are 1 · 8, 2 · 8, 3 · 8, 4 · 8, etc., or equivalently, 8, 16, 24, 32, etc.

Definition: Least Common Multiple

The least common multiple (LCM) of a set of numbers is the smallest number that is a multiple of each number of the given set. The procedure for finding an LCM follows:

  1. List all of the multiples of each number in the given set of numbers.
  2. List the multiples that are in common.
  3. Pick the least of the multiples that are in common.

Example (PageIndex{7})

Find the least common multiple (LCM) of 12 and 16.

Solution

List the multiples of 12 and 16.

Multiples of 12 : 12, 24, 36, 48, 60, 72, 84, 96,...

Multiples of 16 : 16, 32, 48, 64, 80, 96, 112,...

Pick the common multiples.

Common Multiples : 48, 96,...

The LCM is the least of the common multiples.

LCM(12,16) = 48

Exercise (PageIndex{7})

Find the least common denominator of 6 and 9.

Answer

18

Important Observation

The least common denominator is the least common multiple of the denominators.

For example, suppose your problem is 5/12 + 5/16. The LCD is the smallest number divisible by both 12 and 16. That number is 48, which is also the LCM of 12 and 16. Therefore, the procedure for finding the LCM can also be used to find the LCD.

Least Common Multiple Using Prime Factorization

You can also find the LCM using prime factorization.

LCM By Prime Factorization

To find an LCM for a set of numbers, follow this procedure:

  1. Write down the prime factorization for each number in compact form using exponents.
  2. The LCM is found by writing down every factor that appears in step 1 to the highest power of that factor that appears.

Example (PageIndex{8})

Use prime factorization to find the least common multiple find the least common denominator of 18 and 24. (LCM) of 12 and 16.

Solution

Prime factor 12 and 16.

[ egin{aligned} 12 = 2 cdot 2 cdot 3 16 = 2 cdot 2 cdot 2 cdot 2 end{aligned} onumber ]

Write the prime factorizations in compact form using exponents.

[ egin{aligned} 12 = 2^2 cdot 3^1 16 = 2^4 end{aligned} onumber ]

To find the LCM, write down each factor that appears to the highest power of that factor that appears. The factors that appear are 2 and 3. The highest power of 2 that appears is 24. The highest power of 3 that appears is 31.

[ egin{aligned} ext{LCM} = 2^4 cdot 3^1 ~ & extcolor{red}{ ext{ Keep highest power of each factor.}} end{aligned} onumber ]

Now we expand this last expression to get our LCM.

[ egin{aligned} = 16 cdot 3 ~ & extcolor{red}{ ext{ Expand: } 2^4 = 16 ext{ and } 3^1 = 3.} = 48. ~ & extcolor{red}{ ext{ Multiply.}} end{aligned} onumber ]

Note that this answer is identical to the LCM found in Example 8 that was found by listing multiples and choosing the smallest multiple in common.

Exercise (PageIndex{8})

Use prime factorization to find the least common denominator of 18 and 24.

Answer

72

Example (PageIndex{10})

Simplify: (frac{5}{28} + frac{11}{42}).

Solution

Prime factor the denominators in compact form using exponents.

28 = 2 · 2 · 7=22 · 7

42 = 2 · 3 · 7=21 · 31 · 71

To find the LCD, write down each factor that appears to the highest power of that factor that appears. The factors that appear are 2, 3, and 7. The highest power of 2 that appears is 22. The highest power of 3 that appears is 31. The highest power of 7 that appears is 71.

[ egin{aligned} ext{LCM} = 2^2 cdot 3^1 cdot 7^1 ~ & extcolor{red}{ ext{ Keep highest power of each factor.}} = 4 cdot 3 cdot 7 ~ & extcolor{red}{ ext{ Expand: } 2^2 = 4, ~ 3^1 = 3, ~ 7^1 = 7.} = 84 ~ & extcolor{red}{ ext{ Multiply.}} end{aligned} onumber ]

Create equivalent fractions with the new LCD, then add.

[ egin{aligned} frac{5}{28} + frac{11}{42} = frac{5 cdot extcolor{red}{3}}{28 cdot extcolor{red}{3}} + frac{11 cdot extcolor{red}{2}}{42 cdot extcolor{red}{2}} ~ & extcolor{red}{ ext{ Equivalent fractions with LCD = 84.}} = frac{15}{84} + frac{22}{84} ~ & extcolor{red}{ ext{ Simplify numerators and denominators.}} = frac{37}{84} ~ & extcolor{red}{ ext{ Keep LCD; add numerators.}} end{aligned} onumber ]

Exercise (PageIndex{10})

Simplify: ( frac{5}{24} + frac{5}{36})

Answer

25/72

Example (PageIndex{11})

Simplify: (- frac{11}{24} - frac{1}{18}).

Solution

Prime factor the denominators in compact form using exponents.

24 = 2 · 2 · 2 · 3=23 · 31

18 = 2 · 3 · 3=21 · 32

To find the LCD, write down each factor that appears to the highest power of that factor that appears. The highest power of 2 that appears is 23. The highest power of 3 that appears is 32.

[ egin{aligned} ext{LCM} = 2^3 cdot 3^2 ~ & extcolor{red}{ ext{ Keep highest power of each factor.}} = 8 cdot 9 ~ & extcolor{red}{ ext{ Expand: } 2^3 = 8 ext{ and } 3^2 = 9.} = 72. ~ & extcolor{red}{ ext{ Multiply.}} end{aligned} onumber ]

Create equivalent fractions with the new LCD, then subtract.

[ egin{aligned} - frac{11}{24} - frac{1}{18} = - frac{11 cdot extcolor{red}{3}}{24 cdot extcolor{red}{3}} - frac{1 cdot extcolor{red}{4}}{18 cdot extcolor{red}{4}} ~ & extcolor{red}{ ext{ Equivalent fractions with LCD = 72.}} = - frac{33}{72} - frac{4}{72} ~ & extcolor{red}{ ext{ Simplify numerators and denominators.}} = frac{-33-4}{72} ~ & extcolor{red}{ ext{ Keep LCD; subtract numerators.}} = frac{-37}{72} ~ & extcolor{red}{ ext{ Simplify numerator.}} end{aligned} onumber ]

Of course, negative divided by positive yields a negative answer, so we can also write our answer in the form

[ - frac{11}{24} - frac{1}{18} = - frac{37}{72}. onumber ]

Exercise (PageIndex{11})

Simplify: ( - frac{5}{24} - frac{11}{36})

Answer

−37/72

Comparing Fractions

The simplest way to compare fractions is to create equivalent fractions.

Example (PageIndex{12})

Arrange the fractions −1/2 and −4/5 on a number line, then compare them by using the appropriate inequality symbol.

Solution

The least common denominator for 2 and 5 is the number 10. First, make equivalent fractions with a LCD equal to 10.

[ egin{array}{c} - frac{1}{2} = - frac{1 cdot extcolor{red}{5}}{2 cdot extcolor{red}{5}} = - frac{5}{10} - frac{4}{5} = - frac{4 cdot extcolor{red}{2}}{5 cdot extcolor{red}{2}} = - frac{8}{10} end{array} onumber ]

To plot tenths, subdivide the interval between −1 and 0 into ten equal increments.

Because −4/5 lies to the left of −1/2, we have that −4/5 is less than −1/2, so we write

[ - frac{4}{5} < - frac{1}{2}. onumber ]

Exercise (PageIndex{12})

Compare −3/8 and −1/2.

Answer

[ - frac{1}{2} < - frac{3}{8} onumber ]

Exercises

In Exercises 1-10, list the multiples the given numbers, then list the common multiples. Select the LCM from the list of common multiples.

1. 9 and 15

2. 15 and 20

3. 20 and 8

4. 15 and 6

5. 16 and 20

6. 6 and 10

7. 20 and 12

8. 12 and 8

9. 10 and 6

10. 10 and 12


In Exercises 11-20, for the given numbers, calculate the LCM using prime factorization.

11. 54 and 12

12. 108 and 24

13. 18 and 24

14. 36 and 54

15. 72 and 108

16. 108 and 72

17. 36 and 24

18. 18 and 12

19. 12 and 18

20. 12 and 54


In Exercises 21-32, add or subtract the fractions, as indicated, and simplify your result.

21. (frac{7}{12} − frac{1}{12})

22. (frac{3}{7} − frac{5}{7})

23. (frac{1}{9} + frac{1}{9})

24. (frac{1}{7} + frac{3}{7})

25. (frac{1}{5} − frac{4}{5})

26. (frac{3}{5} − frac{2}{5})

27. (frac{3}{7} − frac{4}{7})

28. (frac{6}{7} − frac{2}{7})

29. (frac{4}{11} + frac{9}{11})

30. (frac{10}{11} + frac{4}{11})

31. (frac{3}{11} + frac{4}{11})

32. (frac{3}{7} + frac{2}{7})


In Exercises 33-56, add or subtract the fractions, as indicated, and simplify your result.

33. (frac{1}{6} − frac{1}{8})

34. (frac{7}{9} − frac{2}{3})

35. (frac{1}{5} + frac{2}{3})

36. (frac{7}{9} + frac{2}{3})

37. (frac{2}{3} + frac{5}{8})

38. (frac{3}{7} + frac{5}{9})

39. (frac{4}{7} − frac{5}{9})

40. (frac{3}{5} − frac{7}{8})

41. (frac{2}{3} − frac{3}{8})

42. (frac{2}{5} − frac{1}{8)

43. (frac{6}{7} − frac{1}{6})

44. (frac{1}{2} − frac{1}{4})

45. (frac{1}{6} + frac{2}{3})

46. (frac{4}{9} + frac{7}{8})

47. (frac{7}{9} + frac{1}{8})

48. (frac{1}{6} + frac{1}{7})

49. (frac{1}{3} + frac{1}{7})

50. (frac{5}{6} + frac{1}{4})

51. (frac{1}{2} − frac{2}{7})

52. (frac{1}{3} − frac{1}{8})

53. (frac{5}{6} − frac{4}{5})

54. (frac{1}{2} − frac{1}{9})

55. (frac{1}{3} + frac{1}{8})

56. (frac{1}{6} + frac{7}{9})


In Exercises 57-68, add or subtract the fractions, as indicated, by first using prime factorization to find the least common denominator.

57. (frac{7}{36} + frac{11}{54})

58. (frac{7}{54} + frac{7}{24})

59. (frac{7}{18} − frac{5}{12})

60. (frac{5}{54} − frac{7}{12})

61. (frac{7}{36} + frac{7}{54})

62. (frac{5}{72} + frac{5}{108})

63. (frac{7}{24} − frac{5}{36})

64. (frac{11}{54} + frac{7}{72})

65. (frac{11}{12} + frac{5}{18})

66. (frac{11}{24} + frac{11}{108})

67. (frac{11}{54} − frac{5}{24})

68. (frac{7}{54} − frac{5}{24})


In Exercises 69-80, add or subtract the fractions, as indicated, and simplify your result.

69. (frac{−3}{7} + left( frac{−3}{7} ight))

70. (frac{−5}{9} + left( frac{−1}{9} ight))

71. (frac{7}{9} − left( frac{−1}{9} ight) )

72. (frac{8}{9} − left( frac{−4}{9} ight))

73. (frac{7}{9} + left( frac{−2}{9} ight))

74. ( frac{2}{3} + left( frac{−1}{3} ight))

75. (frac{−3}{5} − frac{4}{5})

76. (frac{−7}{9} − frac{1}{9})

77. (frac{−7}{8} + frac{1}{8})

78. (frac{−2}{3} + (frac{1}{3})

79. (frac{−1}{3} − left( frac{−2}{3} ight))

80. (frac{−7}{8} − left( frac{−5}{8} ight))


In Exercises 81-104, add or subtract the fractions, as indicated, and simplify your result.

81. (frac{−2}{7}) + frac{4}{5})

82. (frac{−1}{4} + frac{2}{7})

83. (frac{−1}{4} − left( frac{−4}{9} ight))

84. (frac{−3}{4} −left( frac{−1}{8} ight))

85. (frac{−2}{7} + frac{3}{4})

86. (frac{−1}{3} + frac{5}{8})

87. (frac{−4}{9} − frac{1}{3})

88. (frac{−5}{6} − frac{1}{3})

89. (frac{−5}{7} − left( frac{−1}{5} ight))

90. (frac{−6}{7} − left( frac{−1}{8} ight))

91. (frac{1}{9} + left( frac{−1}{3} ight))

92. (frac{1}{8} + left( frac{−1}{2} ight))

93. (frac{2}{3} + left( frac{−1}{9} ight))

94. (frac{3}{4} + left( frac{−2}{3} ight))

95. (frac{−1}{2} + left( frac{−6}{7} ight))

96. (frac{−4}{5} + left( frac{−1}{2} ight))

97. (frac{−1}{2} + left( frac{−3}{4} ight))

98. (frac{−3}{5} + left( frac{−1}{2} ight))

99. (frac{−1}{4} − frac{1}{2})

100. (frac{−8}{9} − frac{2}{3})

101. (frac{5}{8} − left( frac{−3}{4} ight))

102. (frac{3}{4} − left( frac{−3}{8} ight))

103. (frac{1}{8} − left( frac{−1}{3} ight))

104. (frac{1}{2} − left( frac{−4}{9} ight))


In Exercises 105-120, add or subtract the fractions, as indicated, and write your answer is lowest terms.

105. (frac{1}{2} + frac{3q}{5})

106. (frac{4}{7} − frac{b}{3})

107. (frac{4}{9} − frac{3a}{4})

108. (frac{4}{9} − frac{b}{2})

109. (frac{2}{s} + frac{1}{3})

110. (frac{2}{s} + frac{3}{7})

111. (frac{1}{3} − frac{7}{b})

112. (frac{1}{2} − frac{9}{s})

113. (frac{4b}{7} + frac{2}{3})

114. (frac{2a}{5} + frac{5}{8})

115. (frac{2}{3} − frac{9}{t})

116. (frac{4}{7} − frac{1}{y})

117. (frac{9}{s} + frac{7}{8})

118. (frac{6}{t} − frac{1}{9})

119. (frac{7b}{8} − frac{5}{9})

120. (frac{3p}{4} − frac{1}{8})


In Exercises 121-132, determine which of the two given statements is true.

121. (frac{−2}{3} < frac{−8}{7}) or (frac{− 2}{3} > frac{−8}{7})

122. (frac{−1}{7} < frac{−8}{9}) or (frac{− 1}{7} > frac{−8}{9})

123. (frac{6}{7} < frac{7}{3}) or (frac{6}{7} > frac{7}{3})

124. (frac{1}{2} < frac{2}{7}) or (frac{1}{2} > frac{2}{7})

125. (frac{−9}{4} < frac{−2}{3}) or frac{− 9}{4} > frac{−2}{3})

126. (frac{−3}{7} < frac{−9}{2}) or (frac{− 3}{7} > frac{−9}{2})

127. (frac{5}{7} < frac{5}{9}) or frac{5}{7} > frac{5}{9})

128. (frac{1}{2} < frac{1}{3}) or (frac{1}{2} > frac{1}{3})

129. (frac{−7}{2} < frac{−1}{5}) or (frac{− 7}{2} > frac{−1}{5})

130. (frac{−3}{4} < frac{−5}{9}) or (frac{− 3}{4} > frac{−5}{9})

131. (frac{5}{9} < frac{6}{5}) or (frac{5}{9} > frac{6}{5})

132. (frac{3}{2} < frac{7}{9}) or (frac{3}{2} > frac{7}{9})


Answers

1. 45

3. 40

5. 80

7. 60

9. 30

11. 108

13. 72

15. 216

17. 72

19. 36

21. (frac{1}{2})

23. (frac{2}{9})

25. (frac{−3}{5})

27. (frac{−1}{7})

29. (frac{13}{11})

31. (frac{7}{11})

33. (frac{1}{24})

35. (frac{13}{15})

37. (frac{31}{24})

39. (frac{1}{63})

41. (frac{7}{24})

43. (frac{29}{42})

45. (frac{5}{6})

47. (frac{65}{72})

49. (frac{10}{21})

51. (frac{3}{14})

53. (frac{1}{30})

55. (frac{11}{24})

57. (frac{43}{108})

59. (frac{−1}{36})

61. (frac{35}{108})

63. (frac{11}{72})

65. (frac{43}{36})

67. (frac{−1}{216})

69. (frac{−6}{7})

71. (frac{8}{9})

73. (frac{5}{9})

75. (frac{− 7}{5})

77. (frac{− 3}{4})

79. (frac{1}{3})

81. (frac{18}{35})

83. (frac{7}{36})

85. (frac{13}{28})

87. (frac{− 7}{9})

89. (frac{−18}{35})

91. (frac{− 2}{9})

93. (frac{5}{9})

95. (frac{−19}{14})

97. (frac{− 5}{4})

99. (frac{− 3}{4})

101. (frac{11}{8})

103. (frac{11}{24})

105. (frac{5+6 q}{10})

107. (frac{16 − 27 a}{36})

109. (frac{6 + s}{3 s})

111. (frac{b − 21}{3b})

113. (frac{12 b + 14}{21})

115. (frac{2 t − 27}{3t})

117. (frac{72 + 7 s}{8 s})

119. (frac{63 b − 40}{72})

121. (frac{− 2}{3} > (frac{− 8}{7})

123. (frac{6}{7} < frac{7}{3})

125. (frac{− 9}{4} < frac{− 2}{3})

127. (frac{5}{7} > frac{5}{9})

129. (frac{− 7}{2 } < frac{− 1}{5})

131. (frac{5}{9} < frac{6}{5})


Adding Fractions with Common Denominators

Abigail, Hanna, and Naomi are studying for their midterm exam. The material they are required to study consists of 16 chapters of reading. The three of them realize that 16 chapters is a lot of reading for each of them to do, so they decide to study in a more efficient manner. They come up with a plan in which each of them reads a certain number of chapters and then summarizes it for the other two. They will share notes, and each will find online videos corresponding to their particular set of chapters.

Now, the chapters are not created equally. Some are quite easy, while others are much tougher. Their goal is to spread the workload evenly between the three of them. Remember that there are 16 chapters.

Abigail has the highest number of chapters to go through with 6. Hanna has 5, while Naomi has only 4. If you were to add those up, you would notice that that only comes to 15 chapters. The last chapter in the book is about troubleshooting electrical systems, and the apprentices decide that they will go through that one together.

We can represent each of their workloads as a fraction of a whole:

What if were to add those fractions? It would look something like this:

What you’ll note is that the numerators are all different, while the denominators are all the same (16). When adding or subtracting fractions, the denominators must be the same. We refer to this as having a common denominator.

So, in order to get the answer to the above question, you just add all the numerators. Adding fractions is very simple in this respect.

Notice that the denominator in the final answer is the same as that in the fractions being added. By the end, the apprentices will have gone through 15 of the 16 chapters separately, and then they will go through the last chapter together.

The concept of adding fractions with common denominators is easy enough, and we did enough adding whole numbers that going through examples at this point might not be worth it (but if you need a review, see Adding Whole Numbers). What we will do instead is write down some examples of adding fractions so you can see the idea.

Do you notice anything about the answer to the last one? It can be reduced.

Before we get going any further with work on fractions, this might be a good time to state that, when working with fractions, we generally want to put the answer in lowest terms.


How to Add and Subtract Fractions

Here you will find support pages about how to add and subtract fractions (with both like and unlike denominators).

Adding Subtracting Fractions Calculator

If you want to use our Free Fraction Calculator to do the work for you then use the link below.

Our Fraction calculator will allow you to add or subtract fractions and show you the steps to work it out.

Otherwise, for more detailed support and worksheets, keep reading!

How to Add and Subtract Fractions with Like Denominators Video

Find out how to add and subtract fractions with like denominators using the video below.

Adding and Subtracting Fractions with Like Denominators Worksheets

Here you will find a selection of Free Fraction worksheets designed to help your child understand how to add and subtract fractions with the same denominator. The sheets are graded so that the easier ones are at the top.

Using these sheets will help your child to:

All the free Fraction worksheets in this section support the Elementary Math Benchmarks for Third Grade.

Adding Fractions with Like Denominators

  • Adding Fractions With Like Denominators Using Circles 1
  • Answers
  • PDF version
  • Adding Fractions Like Denominators 1
  • Answers
  • PDF version
  • Adding Fractions Like Denominators 2
  • Answers
  • PDF version
  • Adding Fractions Like Denominators 3
  • Answers
  • PDF version

Subtracting Fractions with Like Denominators

  • Subtract Fractions with Like Denominators Using Circles 1
  • Answers
  • PDF version
  • Subtracting Fractions Like Denominators 1
  • Answers
  • PDF version
  • Subtracting Fractions Like Denominators 2
  • Answers
  • PDF version
  • Subtracting Fractions Like Denominators 3
  • Answers
  • PDF version

Adding & Subtracting Fractions with Like Denominators

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Adding Subtracting Fractions with unlike denominators

If you need to add and subtract fractions with unlike denominators, then we have a page dedicated to this skill.

More Adding & Subtracting Fractions with Like Denominators

How to find unit fractions of a number

Here you will find a selection of Fraction worksheets designed to help your child understand how to work out fractions of different numbers, where the numerator is equal to 1.

Using these sheets will help your child to:

  • develop an understanding of fractions as parts of a whole
  • know how to calculate unit fractions of a range of numbers.

Free Printable Fraction Flashcards

Here you will find a selection of Fraction Flash Cards designed to help your child learn their Fractions.

Using Flash Cards is a great way to learn your Fraction facts as parts of a whole. They can be taken on a journey, played with in a game, or used in a spare five minutes daily until your child knows their Fractions off by heart.

Using these flashcards will help your child to:

All the printable Math facts in this section support the Elementary Math Benchmarks.

Learning Fractions Math Help Page

Here you will find the Math Salamanders free online Math help pages about Fractions.

There is a wide range of help pages including help with:

  • fraction definitions
  • equivalent fractions
  • converting improper fractions
  • how to add and subtract fractions
  • how to convert fractions to decimals and percentages
  • how to simplify fractions.

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The Math Salamanders hope you enjoy using these free printable Math worksheets and all our other Math games and resources.

We welcome any comments about our site or worksheets on the Facebook comments box at the bottom of every page.


Subtracting Mixed Fractions

Just follow the same method, but subtract instead of add:

Example: What is 15 3 4 − 8 5 6 ?

Convert to Improper Fractions:

15 3 4 = 63 4

8 5 6 = 53 6

63 4 becomes 189 12

53 6 becomes 106 12

189 12106 12 = 83 12

Convert back to Mixed Fractions:

83 12 = 6 11 12


2.3.1: Adding and Subtracting Fractions - Mathematics

It's easy to add and subtract like fractions, or fractions with the same denominator. You just add or subtract the numerators and keep the same denominator. The tricky part comes when you add or subtract fractions that have different denominators. To do this, you need to know how to find the least common denominator. In an earlier lesson, you learned how to simplify, or reduce, a fraction by finding an equivalent, or equal, fraction where the numerator and denominator have no common factors. To do this, you divided the numerator and denominator by their greatest common factor.

In this lesson, you'll learn that you can also multiply the numerator and denominator by the same factor to make equivalent fractions.

In this example, since 12 divided by 12 equals one, and any number multiplied by 1 equals itself, we know 36/48 and 3/4 are equivalent fractions, or fractions that have the same value. In general, to make an equivalent fraction you can multiply or divide the numerator and denominator of the fraction by any non-zero number.

Since only like fractions can be added or subtracted, we first have to convert unlike fractions to equivalent like fractions. We want to find the smallest, or least, common denominator, because working with smaller numbers makes our calculations easier. The least common denominator, or LCD, of two fractions is the smallest number that can be divided by both denominators. There are two methods for finding the least common denominator of two fractions:

Method 1:
Write the multiples of both denominators until you find a common multiple.

The first method is to simply start writing all the multiples of both denominators, beginning with the numbers themselves. Here's an example of this method. Multiples of 4 are 4, 8, 12, 16, and so forth (because 1 × 4=4, 2 × 4=8, 3 × 4=12, 4 × 4=16, etc.). The multiples of 6 are 6, 12,…--that's the number we're looking for, 12, because it's the first one that appears in both lists of multiples. It's the least common multiple, which we'll use as our least common denominator.

Method 2:
Use prime factorization.

For the second method, we use prime factorization-that is, we write each denominator as a product of its prime factors. The prime factors of 4 are 2 times 2. The prime factors of 6 are 2 times 3. For our least common denominator, we must use every factor that appears in either number. We therefore need the factors 2 and 3, but we must use 2 twice, since it's used twice in the factorization for 4. We get the same answer for our least common denominator, 12.

prime factorization of 4 = 2 × 2
prime factorization of 6 = 2 × 3
LCD = 2 × 2 × 3 = 12

Now that we have our least common denominator, we can make equivalent like fractions by multiplying the numerator and denominator of each fraction by the factor(s) needed. We multiply 3/4 by 3/3, since 3 times 4 is 12, and we multiply 1/6 by 2/2, since 2 times 6 is 12. This gives the equivalent like fractions 9/12 and 2/12. Now we can add the numerators, 9 + 2, to find the answer, 11/12.


1.6 Add and Subtract Fractions

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Fractions.

Add or Subtract Fractions with a Common Denominator

When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.

Fraction Addition and Subtraction

To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.

Manipulative Mathematics

Example 1.77

Solution

Example 1.78

Find the difference: − 23 24 − 13 24 . − 23 24 − 13 24 .

Solution

Find the difference: − 19 28 − 7 28 . − 19 28 − 7 28 .

Find the difference: − 27 32 − 1 32 . − 27 32 − 1 32 .

Example 1.79

Solution

Find the difference: − 9 x − 7 x . − 9 x − 7 x .

Find the difference: − 17 a − 5 a . − 17 a − 5 a .

Now we will do an example that has both addition and subtraction.

Example 1.80

Simplify: 3 8 + ( − 5 8 ) − 1 8 . 3 8 + ( − 5 8 ) − 1 8 .

Solution

Simplify: − 2 9 + ( − 4 9 ) − 7 9 . − 2 9 + ( − 4 9 ) − 7 9 .

Simplify: 5 9 + ( − 4 9 ) − 7 9 . 5 9 + ( − 4 9 ) − 7 9 .

Add or Subtract Fractions with Different Denominators

As we have seen, to add or subtract fractions, their denominators must be the same. The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.

Least Common Denominator

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

Manipulative Mathematics

After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!

Example 1.81

How to Add or Subtract Fractions

Solution

How To

Add or Subtract Fractions.

  1. Step 1. Do they have a common denominator?
    • Yes—go to step 2.
    • No—rewrite each fraction with the LCD (least common denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.
  2. Step 2. Add or subtract the fractions.
  3. Step 3. Simplify, if possible.

When finding the equivalent fractions needed to create the common denominators, there is a quick way to find the number we need to multiply both the numerator and denominator. This method works if we found the LCD by factoring into primes.

Look at the factors of the LCD and then at each column above those factors. The “missing” factors of each denominator are the numbers we need.

In Example 1.81, the LCD, 36, has two factors of 2 and two factors of 3 . 3 .

The numerator 12 has two factors of 2 but only one of 3—so it is “missing” one 3—we multiply the numerator and denominator by 3.

The numerator 18 is missing one factor of 2—so we multiply the numerator and denominator by 2.


How to Subtract Fractions

Once you’ve mastered adding fractions, subtracting fractions will be a breeze! The process is exactly the same, though you’ll naturally be subtracting instead of adding.

#1: Find a Common Denominator

Let’s look at the following example:

We need to find the least common multiple for the denominators, which will look like this:

3 : 3, 6, 9, 12, 15, 18, 21, 24, 27, 30

10 : 10, 20, 30

The first number they have in common is 30, so we’ll be putting both numerators over a denominator of 30.

#2: Multiply to Get Both Numerators Over the Same Denominator

First, we need to figure out how much we’ll need to multiply both the numerator and denominator of each fraction by to get a denominator of 30. For $2/3$, what number times 3 equals 30? In equation form:

Our answer is 10, so we’ll multiply both the numerator and denominator by 10 to get $20/30$.

Next, we’ll repeat the process for the second fraction. What number do we need to multiply by 10 to get 30? Well, $30÷10=3$, so we’ll multiply the top and bottom by 3 to get $9/30$.

This makes our problem $20/30-9/30$, which means we’re ready to continue!

#3: Subtract the Numerators

Just as we did with addition, we’ll subtract one numerator from the other but leave the denominators alone.

Since we found the least common multiple, we already know that the problem can’t be reduced any further.

However, let’s say that we just multiplied 3 by 10 to get the denominator of 30, so we need to check if we can reduce. Let’s use that little trick we learned to find the greatest possible common factor. Whatever factors 11 and 30 share, they can’t be greater than $30-11$, or 19.

30 : 2, 3, 5, 6, 10, 15

Since they don’t share any common factors, the answer cannot be reduced any further.


MathHelp.com

The basic idea with converting to common denominators is to multiply fractions by useful forms of 1 . What does this mean? Take a look:

Simplify

Before I can add these fractions, I have to find their common denominator. The lowest (smallest) common denominator is just the Least Common Multiple (LCM) of the two denominators, 4 and 5 . The prime factorizations and LCM of the denominators 4 and 5 are:

In other words, I have to convert the fourths and fifths into twentieths. I'll do this by multiplying by a useful form of 1 . In the case of the first fraction, 1 /4 , the 4 needs to become a 20 , so I need to multiply the 4 by 5 . To keep the fraction equal to its original value, I'll have to multiply the top by 5 , too. In other words, I'll multiply the fraction by 5 /5 , which is just a useful form of the number 1 :

Because I multiplied by (a useful form of) 1 , I haven't changed the actual value of the fraction. All I've changed is how the value is stated.

In the case of the second fraction, 2 /5 , the 5 needs to become a 20 , so I have to multiply the 5 by 4 . To keep the fraction equal to the same value, I also have to multiply the top by 4 , too. In other words, I'll multiply by 4 /4 , which is just a useful form of 1 :

The fourths and fifths are now both twentieths I'm finally in an all-apples situation. Only now can I actually add the fractions. To add these "apples", I add the numerators:

The numerator, 13 , is prime, and it isn't a factor of 20 , so there's no cancellation that I can do.

My simplified final answer is .

By the way, your calculator may be able to do all of this for you check your manual. But make sure you at least understand the basic idea, because you'll need this process later in algebra, when you get to fractions with polynomials, called "rational expressions".

Simplify

First, I'll find the LCM of the two denominators:

Since 5 is a factor of 15 , then the LCM is 15 in particular, one of the fractions is already in LCM form. I'll convert the other fraction to this common denominator, add, and, if possible, simplify:

There are no common factors, so nothing simplifies.

Simplify

First I'll find the LCM of the two denominators:

Notice that 8 and 6 both have 2 as a factor. The point of lining the factors up nice and neatly in columns, as I've done above, is to help avoid over-duplication of factors when finding the LCM. Be careful: there are only three 2 's in the LCM, not four.

To convert the first fraction to a denominator of 24 , I'll multiply, top and bottom, by 3 . To convert the second fraction's denominator, I'll multiply, top and bottom, by 4 .

The instructions don't say to express the answer in mixed-number form, so I'll leave it as an improper fraction. There are no common factors between the numerator and denominator, so I can't simplify any further.

Simplify

First, I'll find the LCM of the three denominators:

Now I'll convert the three fractions to the common denominator, add, and then see if I can simplify.

Because 4 was a common factor of 1072 and 364 , I was able to cancel this out and simplify to get my final answer:

Simplify

First, I'll find the LCM of the two denominators:

To convert to the LCM, I'll multiply the first fraction, top and bottom, by 7 , and the second fraction, top and bottom, by 5 .

The numerator, 106 , factors as 2×53 , and 53 is prime, so there's nothing I can cancel the fraction can't be further simplified.

You can use the Mathway widget below to practice adding and subtracting fractions. Try the entered exercise, or type in your own exercise. Then click the button to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)

(Clicking on "Tap to view steps" on the widget's answer screen will take you to the Mathway site for a paid upgrade.)


COMPLEX FRACTIONS

OBJECTIVES

Fractions are defined as the indicated quotient of two expressions. In this section we will present a method for simplifying fractions in which the numerator or denominator or both are themselves composed of fractions. Such fractions are called complex fractions.

Thus if the numerator and denominator of a complex fraction are composed of single fractions, it can be simplified by dividing the numerator by the denominator.

A generally more efficient method of simplifying a complex fraction involves using the fundamental principle of fractions. We multiply both numerator and denominator by the common denominator of all individual fractions in the complex fraction.

Recall that the fundamental principle of fractions states

We will use the fundamental principle to again simplify

The LCD of 3 and 4 is 12. Thus

The individual fractions are

This answer could be written as the mixed number

Make sure that each term in both numerator and denominator is multiplied by the LCD.

We need the LCD of individual fractions, y is not a fraction.


Fractions - Grade 5 Maths Questions

Solutions and explanations to grade 5 fractions questions are presented.


  1. 3 1/2 + 5 1/3 =
    Solution
    Add whole numbers together and fractions together
    3 1/2 + 5 1/3 = (3 + 5) + (1/2 + 1/3)
    Write fractions with the same denominator
    = 8 + (3/6 + 2/6) = 8 5/6

  2. It takes Julia 1/2 hour to wash, comb her hair and put on her clothes, and 1/4 hour to have her breakfast. How much time does it take Julia to be ready for school?
    Solution
    The total time for Julia to be ready for school is
    1/2 + 1/4
    Write fractions with the same denominator
    = 2/4 + 1/4 = 3/4 of an hour.

  3. Which two fractions are equivalent?
    1. 5/2 and 2/5
    2. 4/3 and 8/6
    3. 1/4 and 2/4
    4. 2/3 and 1/3

    .
    Solution
    There are two whole shaded items above and one shaded at 3/4. Hence the mixed number
    2 3/4 represents the shaded parts.

    .
    Solution
    1 7/10 in decimal form is
    1 7/10 = 1 + 7/10 = 1 + 0.7 = 1.7 and corresponds to point W.


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