1.3 Adding and Subtracting Fractions

In this lesson, we will review how to add and subtract fractions with common denominators and different denominators.

Adding and Subtracting Fractions with a Common Denominator

Table 1.3.1
Adding and Subtracting Fractions
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.
Example 1

Find the sum: \frac{1}{8}+\frac{5}{8}.

Table 1.3.2
Steps Solution
Add or subtract the numerators of your fractions. Keep the denominator the same. \frac{1}{8}+\frac{5}{8}

\frac{1+5}{8}

\frac{6}{8}
Simplify. \frac{3}{4}
Practice 1

Find the sum or difference. If necessary, simplify and leave answers as improper fractions.

a) \frac{3}{11}+\frac{6}{11}

 

 

 b) \frac{9}{15}-\frac{4}{15}

 

 

 c) \frac{3}{6}+\frac{1}{6}

 

 
d) \frac{12}{17}-\frac{19}{17}

 

 

 e) \frac{7}{8}+\frac{7}{8}

 

 

 f) \frac{2}{10}-\frac{8}{10}

 

 
Practice 2

Use what you know about adding and subtracting integers to evaluate the following. If necessary, simplify and leave answers as improper fractions.

a) \phantom{\rule{0.2em}{0ex}}\frac{6}{18}--\phantom{\rule{0.2em}{0ex}}\frac{14}{18}

 

 

 

 b) -\phantom{\rule{0.2em}{0ex}}\frac{23}{24}-\phantom{\rule{0.2em}{0ex}}\frac{13}{24}

 

 

 

 c) -\phantom{\rule{0.2em}{0ex}}\frac{6}{8}+\phantom{\rule{0.2em}{0ex}}\frac{2}{8}

 

 

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

 

 

 

 e) \frac{1}{12}+\phantom{\rule{0.2em}{0ex}}\frac{7}{12}\right)-\left(-\phantom{\rule{0.2em}{0ex}}\frac{3}{12}\right)

 

 

 

 f) \frac{4}{5}+\phantom{\rule{0.2em}{0ex}}\frac{2}{5}\right)+\left(-\phantom{\rule{0.2em}{0ex}}\frac{6}{5}\right)

 

 

 

Adding and Subtracting Fractions with Different Denominators

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

Table 1.3.3
To add and subtract fractions that do not have a common denominator:
Step 1: Find the LCD. This will be your common denominator.
Step 2: Rewrite your fractions so that they all have a common denominator.
Step 3: Add or subtract the numerators of your fractions. Keep the denominator the same.
Step 4: If possible, simplify.
Example 2

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

Table 1.3.4
Steps Solution
Find the LCD. This will be your common denominator. multiples of 12: 12, 24, 36, 48, …
multiples of 18: 18, 36, 54, …
The lowest common denominator is 36.
Rewrite your fractions so they all have a common denominator. \frac{7}{12}+\frac{5}{18}

\frac{7}{12} \bullet \frac{3}{3}+\frac{5}{18} \bullet \frac{2}{2}

\frac{21}{36}+\frac{10}{36}
Add or subtract the numerators of your fractions. Keep the denominator the same. \frac{31}{36}
Simplify. 31 is prime and has no factors in common with 36, so this answer is simplified.
Practice 3

Find the sum or difference. If necessary, simplify and leave answers as improper fractions.

a) \frac{5}{12}+\frac{1}{4}

 

 

 

 b) \frac{7}{9}-\frac{2}{3}

 

 

 

 c) \frac{5}{6}+\frac{2}{9}

 

 

 
d) \frac{5}{8}-\frac{4}{5}

 

 

 

 e) \frac{7}{11}+\frac{5}{13}

 

 

 

 f) \frac{13}{8}-2

 

 

 

Homework 

  1. Find the sum or difference. Simplify and leave your answers as improper fractions where necessary.

    a) \frac{4}{9} + \frac{5}{9}

     

    		 b) \frac{7}{12} - \frac{3}{12}

     

    		 c) \frac{2}{5} + 1

     
    d) \frac{8}{14} - \frac{10}{14}

     

    		 e) \frac{3}{7} + \frac{3}{7}

     

    		 f) \frac{6}{15} - \frac{4}{15}

     
    g) \frac{5}{11} + \frac{6}{11}

     

    		 h) \frac{9}{20} - \frac{5}{20}

     

    		 i) \frac{7}{10} + \frac{2}{10}

     
    j) \frac{11}{13} - \frac{8}{13}

     

    		 k) \frac{4}{6} + \frac{2}{6}

     

    		 l) \frac{13}{18} - \frac{7}{18}

     
    m) \frac{34}{56} + \frac{28}{56}

     

    		 n) \frac{81}{120} - \frac{45}{120}

     

    		 o) \frac{123}{150} + \frac{77}{150}

     
    p) \frac{98}{145} - \frac{52}{145}

     

     

    		 q) \frac{67}{84} + \frac{52}{84}

     

     

    		 r) \frac{144}{225} - \frac{88}{225}

     

     

  2. Add or subtract. Simplify and leave your answers as improper fractions where necessary.

    a) \frac{8}{15} - \left( -\phantom{\rule{0.2em}{0ex}} \frac{4}{15} \right)

     

     

    		 b) \frac{7}{10} + \left( - \phantom{\rule{0.2em}{0ex}} \frac{9}{10} \right)

     

     

    		 c) -\phantom{\rule{0.2em}{0ex}} \frac{5}{9} + \phantom{\rule{0.2em}{0ex}} \frac{3}{9}

     

     
    d) -\phantom{\rule{0.2em}{0ex}} \frac{11}{18} - \phantom{\rule{0.2em}{0ex}} \frac{5}{18}

     

     

    		 e) \frac{2}{7} + \left( - \phantom{\rule{0.2em}{0ex}} \frac{4}{7} \right)

     

     

    		 f) \frac{3}{10} + \left( - \phantom{\rule{0.2em}{0ex}} \frac{4}{10} \right)

     

     
    g) \phantom{\rule{0.2em}{0ex}} \frac{7}{12} - \phantom{\rule{0.2em}{0ex}} \frac{9}{12}

     

     

    		 h) \frac{5}{8} + \left( - \phantom{\rule{0.2em}{0ex}} \frac{3}{8} \right)

     

     

    		 i) -\phantom{\rule{0.2em}{0ex}} \frac{4}{6} + \phantom{\rule{0.2em}{0ex}} \frac{1}{6}

     

     
    j) \frac{9}{14} + \left( - \phantom{\rule{0.2em}{0ex}} \frac{11}{14} \right)

     

     

     

    		 k) \frac{2}{5} + \left( - \phantom{\rule{0.2em}{0ex}} \frac{3}{5} \right) + \phantom{\rule{0.2em}{0ex}} \frac{4}{5}

     

     

     

    		 l) \phantom{\rule{0.2em}{0ex}} \frac{7}{9} - \left( - \phantom{\rule{0.2em}{0ex}} \frac{5}{9} \right) + \frac{3}{9}

     

     

     
    m) -\phantom{\rule{0.2em}{0ex}} \frac{2}{3} + \phantom{\rule{0.2em}{0ex}} \frac{4}{3} - \frac{1}{3}

     

     

     

    		 n) \frac{8}{13} + \left( - \phantom{\rule{0.2em}{0ex}} \frac{6}{13} \right)

     

     

     

    		 o) \frac{6}{7} - \left( -\frac{3}{7} \right) + \frac{2}{7}

     

     

     
    p) \frac{10}{15} + \left( -\frac{3}{15} \right) - \frac{5}{15}

     

     

     

    		 q) \frac{5}{6} + \left( - \phantom{\rule{0.2em}{0ex}} \frac{2}{6} \right)

     

     

     

    		 r) \frac{3}{11} - \left( - \phantom{\rule{0.2em}{0ex}} \frac{1}{11} \right)

     

     

     

  3. Add or subtract. Simplify and leave your answers as improper fractions where necessary.

    a) \frac{3}{8} + \frac{5}{16}

     

     

     

    		 b) \frac{7}{12} - \frac{3}{4}

     

     

     

    		 c) \frac{2}{5} + \frac{1}{3}

     

     

     
    d) \frac{7}{10} - \frac{2}{5}

     

     

     

    		 e) \frac{9}{14} + \frac{1}{7}

     

     

     

    		 f) \frac{5}{6} -4

     

     

     
    g) \frac{8}{15} + \frac{1}{5}

     

     

     

    		 h) \frac{11}{18} - \frac{3}{8}

     

     

     

    		 i) \frac{5}{9} + \frac{2}{7}

     

     

     
    j) \frac{6}{13} - \frac{1}{26}

     

     

     

    		 k) \frac{3}{4} + \frac{2}{3}

     

     

     

    		 l) \frac{8}{9} - \frac{4}{5}

     

     

     
    m) \frac{5}{8} + \frac{3}{16}

     

     

     

    		 n) \frac{7}{10} - \frac{5}{12}

     

     

     

    		 o) \frac{13}{24} + \frac{7}{16}

     

     

     
    p) \frac{11}{15} - \frac{1}{3}

     

     

     

    		 q) \frac{5}{6} + \frac{2}{9}

     

     

     

    		 r) \frac{3}{7} - \frac{2}{5}

     

     

     
    s) \frac{9}{16} + \frac{1}{4}

     

     

     

    		 t) \frac{5}{8} - \frac{3}{7}

     

     

     

    		 u) \frac{2}{9} + \frac{5}{6}

     

     

     
    v) \frac{3}{11} - \frac{7}{22}

     

     

     

     

    		 w) \frac{1}{2} + \frac{3}{5}

     

     

     

     

    		 x) \frac{5}{12} - \frac{7}{18}

     

     

     

     

  4. What is \frac{3}{8} + \frac{5}{12} equal to?

    a) \frac{19}{24}

    b) \frac{8}{15}

    c) \frac{17}{10}

    d) \frac{5}{4}

  5. What is -\frac{1}{6} - \frac{2}{3} equal to?

    a) \frac{1}{4}

    b) \frac{-3}{2}

    c) \frac{9}{5}

    d) \frac{-5}{6}

  6. Which of the following is required when adding or subtracting fractions with unlike denominators?

    a) You need to make the denominators the same by finding a common denominator.
    b) You only need to find a common numerator.
    c) You add the numerators and denominators directly.
    d) You subtract the numerators and denominators directly.

  7. What should you do when you add two fractions with the same denominator?

    a) You add both the numerators and denominators.
    b) You add the numerators and keep the denominator the same.
    c) You subtract both the numerators and denominators.
    d) You multiply both the numerators and denominators.

  8. When adding or subtracting fractions with unlike denominators, what is the purpose of finding the least common denominator (LCD)?

    a) To make the fractions easier to simplify
    b) To make the numerators equal
    c) To ensure that both fractions have the same denominator, which is necessary for adding or subtracting the fractions
    d) To convert the fractions to decimals

Answers

1.

a) 1 b) \frac{1}{3} c) \frac{7}{5} d) -\frac{1}{7} e) \frac{6}{7} f) \frac{2}{15}
g) 1 h) \frac{1}{5} i) \frac{9}{10} j) \frac{3}{13} k) 1 l) \frac{1}{3}
m) \frac{31}{28} n) \frac{3}{10} o) \frac{4}{3} p) \frac{46}{145} q) \frac{119}{84} r) \frac{56}{225}

2.

a) \frac{4}{5} b) -\frac{1}{5} c) -\frac{2}{9} d) -\frac{8}{9} e) -\frac{2}{7} f) -\frac{1}{10}
g) -\frac{1}{6} h) \frac{1}{4} i) -\frac{1}{2} j) -\frac{1}{7} k) \frac{3}{5} l) \frac{5}{3}
m) \frac{1}{3} n) \frac{2}{13} o) \frac{11}{7} p) \frac{2}{15} q) \frac{1}{2} r) \frac{4}{11}

3.

a) \frac{11}{16} b) \frac{-1}{6} c) \frac{11}{15} d) \frac{3}{10} e) \frac{11}{14} f) -\frac{19}{6}
g) \frac{11}{15} h) \frac{17}{72} i) \frac{53}{63} j) \frac{11}{26} k) \frac{17}{12} l) \frac{4}{45}
m) \frac{13}{16} n) \frac{17}{60} o) \frac{47}{48} p) \frac{2}{5} q) \frac{19}{18} r) \frac{1}{35}
s) \frac{13}{16} t) \frac{11}{56} u) \frac{19}{18} v) -\frac{1}{22} w) \frac{11}{10} x) \frac{1}{36}
4. a 5. d 6. a 7. b 8. c

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resource:

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

License

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