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Easy Way to Add and Subtract Fractions

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CHAPTER 2 Operations with Rational Numbers and Introduction to Real Numbers

2.2 Add and Subtract Fractions

By the end of this section, you will be able to:

  • Add or subtract fractions with a common denominator
  • Add or subtract fractions with different denominators
  • Use the order of operations to simplify complex fractions
  • Evaluate variable expressions with 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.

If a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c are numbers where c\ne 0, then

\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\frac{a}{c}-\phantom{\rule{0.2em}{0ex}}\frac{b}{c}=\frac{a-b}{c}

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

Find the sum: \frac{x}{3}+\frac{2}{3}.

Solution

\frac{x}{3}+\frac{2}{3}
Add the numerators and place the sum over the common denominator. \frac{x+2}{3}

Find the sum: \frac{x}{4}+\frac{3}{4}.

Show answer

\frac{x+3}{4}

Find the sum: \frac{y}{8}+\frac{5}{8}.

Show answer

\frac{y+5}{8}

Find the difference: -\phantom{\rule{0.2em}{0ex}}\frac{19}{28}-\phantom{\rule{0.2em}{0ex}}\frac{7}{28}.

Show answer

-\phantom{\rule{0.2em}{0ex}}\frac{26}{28}

Find the difference: -\phantom{\rule{0.2em}{0ex}}\frac{27}{32}-\phantom{\rule{0.2em}{0ex}}\frac{1}{32}.

Show answer

-\phantom{\rule{0.2em}{0ex}}\frac{7}{8}

Find the difference: -\phantom{\rule{0.2em}{0ex}}\frac{9}{x}-\phantom{\rule{0.2em}{0ex}}\frac{7}{x}.

Show answer

-\phantom{\rule{0.2em}{0ex}}\frac{16}{x}

Find the difference: -\phantom{\rule{0.2em}{0ex}}\frac{17}{a}-\phantom{\rule{0.2em}{0ex}}\frac{5}{a}.

Show answer

-\phantom{\rule{0.2em}{0ex}}\frac{22}{a}

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

Simplify: \frac{3}{8}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\right)-\phantom{\rule{0.2em}{0ex}}\frac{1}{8}.

Simplify: \frac{2}{5}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{4}{9}\right)-\phantom{\rule{0.2em}{0ex}}\frac{7}{9}.

Show answer

-1

Simplify: \frac{5}{9}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{4}{9}\right)-\phantom{\rule{0.2em}{0ex}}\frac{7}{9}.

Show answer

-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}

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.

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

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!

Add: \frac{7}{12}+\frac{5}{18}.

Add: \frac{7}{12}+\frac{11}{15}.

Show answer

\frac{79}{60}

Add: \frac{13}{15}+\frac{17}{20}.

Show answer

\frac{103}{60}

  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. Add or subtract the fractions.
  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.

The number 12 is factored into 2 times 2 times 3 with an extra space after the 3, and the number 18 is factored into 2 times 3 times 3 with an extra space between the 2 and the first 3. There are arrows pointing to these extra spaces that are marked

In (Example 5), the LCD, 36, has two factors of 2 and two factors of 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

We will apply this method as we subtract the fractions in (Example 6).

Subtract: \frac{7}{15}-\phantom{\rule{0.2em}{0ex}}\frac{19}{24}.

Subtract: \frac{13}{24}-\phantom{\rule{0.2em}{0ex}}\frac{17}{32}.

Show answer

\frac{1}{96}

Subtract: \frac{21}{32}-\phantom{\rule{0.2em}{0ex}}\frac{9}{28}.

Show answer

\frac{75}{224}

In the next example, one of the fractions has a variable in its numerator. Notice that we do the same steps as when both numerators are numbers.

Add: \frac{3}{5}+\frac{x}{8}.

Add: \frac{y}{6}+\frac{7}{9}.

Show answer

\frac{9y+42}{54}

Add: \frac{x}{6}+\frac{7}{15}.

Show answer

\frac{15x+42}{135}

We now have all four operations for fractions. The table below summarizes fraction operations.

Simplify: a) \frac{5x}{6}-\phantom{\rule{0.2em}{0ex}}\frac{3}{10} b) \frac{5x}{6}\cdot\frac{3}{10}.


Solution

First ask, "What is the operation?" Once we identify the operation that will determine whether we need a common denominator. Remember, we need a common denominator to add or subtract, but not to multiply or divide.

a) What is the operation? The operation is subtraction.
Do the fractions have a common denominator? No. \frac{5x}{6}-\frac{3}{10}
Rewrite each fraction as an equivalent fraction with the LCD. \frac{5x\cdot5}{6\cdot5}-\frac{3\cdot3}{10\cdot3}

\frac{25x}{30}-\frac{9}{30}

Subtract the numerators and place the difference over the common denominators. \frac{25x-9}{30}
Simplify, if possible. There are no common factors. The fraction is simplified.

Notice we needed an LCD to add \frac{5x}{6}-\phantom{\rule{0.2em}{0ex}}\frac{3}{10}, but not to multiply \frac{5x}{6}\cdot\frac{3}{10}.

Simplify. a) \frac{27a-32}{36} b) \frac{2a}{3}

Show answer

a) \frac{27a-32}{36} b) \frac{2a}{3}

Simplify: a) \frac{4k}{5}-\phantom{\rule{0.2em}{0ex}}\frac{1}{6} b) \frac{4k}{5}\cdot \frac{1}{6}.

Show answer

a) \frac{24k-5}{30} b) \frac{2k}{15}

Use the Order of Operations to Simplify Complex Fractions

We have seen that a complex fraction is a fraction in which the numerator or denominator contains a fraction. The fraction bar indicates division. We simplified the complex fraction \frac{\frac{3}{4}}{\frac{5}{8}} by dividing \frac{3}{4} by \frac{5}{8}.

Now we'll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator.

Simplify: \frac{{\left(\frac{1}{2}\right)}^{2}}{4+{3}^{2}}.

Simplify: \frac{{\left(\frac{1}{3}\right)}^{2}}{{2}^{3}+2}.

Show answer

\frac{1}{90}

Simplify: \frac{1+{4}^{2}}{{\left(\frac{1}{4}\right)}^{2}}.

Show answer

272

  1. Simplify the numerator.
  2. Simplify the denominator.
  3. Divide the numerator by the denominator. Simplify if possible.

Simplify: \frac{\frac{1}{2}+\frac{2}{3}}{\frac{3}{4}-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}}.

Simplify: \frac{\frac{1}{3}+\frac{1}{2}}{\frac{3}{4}-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}}.

Simplify: \frac{\frac{2}{3}-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}}{\frac{1}{4}+\frac{1}{3}}.

Show answer

\frac{2}{7}

Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

Evaluate -\phantom{\rule{0.2em}{0ex}}\frac{5}{6}-y when y=-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}.

Evaluate -\phantom{\rule{0.2em}{0ex}}\frac{1}{2}-y when y=-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}.

Show answer

-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}

Evaluate -\phantom{\rule{0.2em}{0ex}}\frac{3}{8}-y when x=-\phantom{\rule{0.2em}{0ex}}\frac{5}{2}.

Show answer

-\phantom{\rule{0.2em}{0ex}}\frac{17}{8}

Evaluate 3a{b}^{2} when a=-\phantom{\rule{0.2em}{0ex}}\frac{2}{3} and b=-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}.

Show answer

-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}

Evaluate 4{c}^{3}d when c=-\phantom{\rule{0.2em}{0ex}}\frac{1}{2} and d=-\phantom{\rule{0.2em}{0ex}}\frac{4}{3}.

Show answer

\frac{2}{3}

The next example will have only variables, no constants.

Evaluate \frac{p+q}{r} when p=-4,q=-2,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r=8.

Evaluate \frac{a+b}{c} when a=-8,b=-7,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c=6.

Show answer

-\phantom{\rule{0.2em}{0ex}}\frac{5}{2}

Evaluate \frac{x+y}{z} when x=9,y=-18,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}z=-6.

Show answer

\frac{3}{2}

Key Concepts

Glossary

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

Practice Makes Perfect

Add and Subtract Fractions with a Common Denominator

In the following exercises, add.

In the following exercises, subtract.

11. \frac{11}{15}-\phantom{\rule{0.2em}{0ex}}\frac{7}{15} 12. \frac{9}{13}-\phantom{\rule{0.2em}{0ex}}\frac{4}{13}
13. \frac{11}{12}-\phantom{\rule{0.2em}{0ex}}\frac{5}{12} 14. \frac{7}{12}-\phantom{\rule{0.2em}{0ex}}\frac{5}{12}
15. \frac{19}{21}-\phantom{\rule{0.2em}{0ex}}\frac{4}{21} 16. \frac{17}{21}-\phantom{\rule{0.2em}{0ex}}\frac{8}{21}
17. \frac{5y}{8}-\phantom{\rule{0.2em}{0ex}}\frac{7}{8} 18. \frac{11z}{13}-\phantom{\rule{0.2em}{0ex}}\frac{8}{13}
19. -\phantom{\rule{0.2em}{0ex}}\frac{23}{u}-\phantom{\rule{0.2em}{0ex}}\frac{15}{u} 20. -\phantom{\rule{0.2em}{0ex}}\frac{29}{v}-\phantom{\rule{0.2em}{0ex}}\frac{26}{v}
21. -\phantom{\rule{0.2em}{0ex}}\frac{3}{5}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}\right) 22. -\phantom{\rule{0.2em}{0ex}}\frac{3}{7}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{7}\right)
23. -\phantom{\rule{0.2em}{0ex}}\frac{7}{9}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{9}\right) 24. -\phantom{\rule{0.2em}{0ex}}\frac{8}{11}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{11}\right)

Mixed Practice

In the following exercises, simplify.

Add or Subtract Fractions with Different Denominators

In the following exercises, add or subtract.

33. \frac{1}{2}+\frac{1}{7} 34. \frac{1}{3}+\frac{1}{8}
35. \frac{1}{3}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{1}{9}\right) 36. \frac{1}{4}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{1}{8}\right)
37. \frac{7}{12}+\frac{5}{8} 38. \frac{5}{12}+\frac{3}{8}
39. \frac{7}{12}-\phantom{\rule{0.2em}{0ex}}\frac{9}{16} 40. \frac{7}{16}-\phantom{\rule{0.2em}{0ex}}\frac{5}{12}
41. \frac{2}{3}-\phantom{\rule{0.2em}{0ex}}\frac{3}{8} 42. \frac{5}{6}-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}
43. -\phantom{\rule{0.2em}{0ex}}\frac{11}{30}+\frac{27}{40} 44. -\phantom{\rule{0.2em}{0ex}}\frac{9}{20}+\frac{17}{30}
45. -\phantom{\rule{0.2em}{0ex}}\frac{13}{30}+\frac{25}{42} 46. -\phantom{\rule{0.2em}{0ex}}\frac{23}{30}+\frac{5}{48}
47. -\phantom{\rule{0.2em}{0ex}}\frac{39}{56}-\phantom{\rule{0.2em}{0ex}}\frac{22}{35} 48. -\phantom{\rule{0.2em}{0ex}}\frac{33}{49}-\phantom{\rule{0.2em}{0ex}}\frac{18}{35}
49. -\phantom{\rule{0.2em}{0ex}}\frac{2}{3}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\right) 50. -\phantom{\rule{0.2em}{0ex}}\frac{3}{4}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}\right)
51. 1+\frac{7}{8} 52. 1-\phantom{\rule{0.2em}{0ex}}\frac{3}{10}
53. \frac{x}{3}+\frac{1}{4} 54. \frac{y}{2}+\frac{2}{3}
55. \frac{y}{4}-\phantom{\rule{0.2em}{0ex}}\frac{3}{5} 56. \frac{x}{5}-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}

Mixed Practice

In the following exercises, simplify.

57. a) \frac{2}{3}+\frac{1}{6} b) \frac{2}{3}\div \frac{1}{6} 58. a) -\phantom{\rule{0.2em}{0ex}}\frac{2}{5}-\phantom{\rule{0.2em}{0ex}}\frac{1}{8} b) -\phantom{\rule{0.2em}{0ex}}\frac{2}{5}\cdot \frac{1}{8}
59. a) \frac{5n}{6}\div \frac{8}{15} b) \frac{5n}{6}-\phantom{\rule{0.2em}{0ex}}\frac{8}{15} 60. a) \frac{3a}{8}\div \frac{7}{12} b) \frac{3a}{8}-\phantom{\rule{0.2em}{0ex}}\frac{7}{12}
61. -\phantom{\rule{0.2em}{0ex}}\frac{3}{8}\div \left(-\phantom{\rule{0.2em}{0ex}}\frac{3}{10}\right) 62. -\phantom{\rule{0.2em}{0ex}}\frac{5}{12}\div \left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{9}\right)
63. -\phantom{\rule{0.2em}{0ex}}\frac{3}{8}+\frac{5}{12} 64. -\phantom{\rule{0.2em}{0ex}}\frac{1}{8}+\frac{7}{12}
65. \frac{5}{6}-\phantom{\rule{0.2em}{0ex}}\frac{1}{9} 66. \frac{5}{9}-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}
67. -\phantom{\rule{0.2em}{0ex}}\frac{7}{15}-\phantom{\rule{0.2em}{0ex}}\frac{y}{4} 68. -\phantom{\rule{0.2em}{0ex}}\frac{3}{8}-\phantom{\rule{0.2em}{0ex}}\frac{x}{11}
69. \frac{11}{12a}\cdot \frac{9a}{16} 70. \frac{10y}{13}\cdot \frac{8}{15y}

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

71. \frac{{2}^{3}+{4}^{2}}{{\left(\frac{2}{3}\right)}^{2}} 72. \frac{{3}^{3}-{3}^{2}}{{\left(\frac{3}{4}\right)}^{2}}
73. \frac{{\left(\frac{3}{5}\right)}^{2}}{{\left(\frac{3}{7}\right)}^{2}} 74. \frac{{\left(\frac{3}{4}\right)}^{2}}{{\left(\frac{5}{8}\right)}^{2}}
75. \frac{2}{\frac{1}{3}+\frac{1}{5}} 76. \frac{5}{\frac{1}{4}+\frac{1}{3}}
77. \frac{\frac{7}{8}-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}}{\frac{1}{2}+\frac{3}{8}} 78. \frac{\frac{3}{4}-\phantom{\rule{0.2em}{0ex}}\frac{3}{5}}{\frac{1}{4}+\frac{2}{5}}
79. \frac{1}{2}+\frac{2}{3}\cdot \frac{5}{12} 80. \frac{1}{3}+\frac{2}{5}\cdot \frac{3}{4}
81. 1-\phantom{\rule{0.2em}{0ex}}\frac{3}{5}\div \frac{1}{10} 82. 1-\phantom{\rule{0.2em}{0ex}}\frac{5}{6}\div \frac{1}{12}
83. \frac{2}{3}+\frac{1}{6}+\frac{3}{4} 84. \frac{2}{3}+\frac{1}{4}+\frac{3}{5}
85. \frac{3}{8}-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}+\frac{3}{4} 86. \frac{2}{5}+\frac{5}{8}-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}
87. 12\left(\frac{9}{20}-\phantom{\rule{0.2em}{0ex}}\frac{4}{15}\right) 88. 8\left(\frac{15}{16}-\phantom{\rule{0.2em}{0ex}}\frac{5}{6}\right)
89. \frac{\frac{5}{8}+\frac{1}{6}}{\frac{19}{24}} 90. \frac{\frac{1}{6}+\frac{3}{10}}{\frac{14}{30}}
91. \left(\frac{5}{9}+\frac{1}{6}\right)\div \left(\frac{2}{3}-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}\right) 92. \left(\frac{3}{4}+\frac{1}{6}\right)\div \left(\frac{5}{8}-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}\right)

Evaluate Variable Expressions with Fractions

In the following exercises, evaluate.

Everyday Math

Writing Exercises

105. Why do you need a common denominator to add or subtract fractions? Explain. 106. How do you find the LCD of 2 fractions?

Answers

1. \frac{11}{13} 3. \frac{x+3}{4} 5. -\phantom{\rule{0.2em}{0ex}}\frac{5}{8}
7. \frac{7}{17} 9. -\phantom{\rule{0.2em}{0ex}}\frac{16}{13} 11. \frac{4}{15}
13. \frac{1}{2} 15. \frac{5}{7} 17. \frac{5y-7}{8}
19. -\phantom{\rule{0.2em}{0ex}}\frac{38}{u} 21. \frac{1}{5} 23. -\phantom{\rule{0.2em}{0ex}}\frac{2}{9}
25. -\phantom{\rule{0.2em}{0ex}}\frac{1}{4} 27. \frac{n-4}{5} 29. -\phantom{\rule{0.2em}{0ex}}\frac{5}{24}
31. \frac{2}{9} 33. \frac{9}{14} 35. \frac{4}{9}
37. \frac{29}{24} 39. \frac{1}{48} 41. \frac{7}{24}
43. \frac{37}{120} 45. \frac{17}{105} 47. -\phantom{\rule{0.2em}{0ex}}\frac{53}{40}
49. \frac{1}{12} 51. \frac{15}{8} 53. \frac{4x+3}{12}
55. \frac{4y-12}{20} 57. a)\frac{5}{6}b) 4 59. a)\frac{25n}{16}b)\frac{25n-16}{30}
61. \frac{5}{4} 63. \frac{1}{24} 65. \frac{13}{18}
67. \frac{-28-15y}{60} 69. \frac{33}{64} 71. 54
73. \frac{49}{25} 75. \frac{15}{4} 77. \frac{5}{21}
79. \frac{7}{9} 81. -5 83. \frac{19}{12}
85. \frac{23}{24} 87. \frac{11}{5} 89. 1
91. \frac{13}{3} 93. a) -\phantom{\rule{0.2em}{0ex}}\frac{1}{2} b) -1 95. a) \frac{1}{5} b) -1
97. a) \frac{1}{5} b) \frac{6}{5} 99. -\phantom{\rule{0.2em}{0ex}}\frac{1}{9} 101. -\phantom{\rule{0.2em}{0ex}}\frac{5}{11}
103. \frac{7}{8} yard 105. Answers may vary

Attributions

This chapter has been adapted from "Add and Subtract Fractions" in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

jordanhicessell.blogspot.com

Source: https://opentextbc.ca/introalgebra/chapter/add-and-subtract-fractions/

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