Training Fractions Adding and Subtracting Fractions
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Adding and Subtracting Fractions

20 min Fractions

Adding and Subtracting Fractions

Adding and subtracting fractions is one of the most important skills in elementary mathematics. The central idea is simple — you can only add or subtract fractions that share the same denominator, just as you can only add like terms in algebra.

When the denominators differ, you must first find a common denominator, convert each fraction to an equivalent form, and then combine the numerators. The least common denominator (LCD) keeps the numbers as small as possible and makes the arithmetic cleaner.

This lesson walks through the process step by step with plenty of examples, including mixed numbers and problems that require borrowing.

Same Denominator

Rule

$$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} \qquad\quad \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$$

Example 1

$\dfrac{2}{7} + \dfrac{3}{7}$

$$\frac{2 + 3}{7} = \frac{5}{7}$$

Different Denominators

Procedure
  1. Find the LCD (least common denominator).
  2. Rewrite each fraction with the LCD.
  3. Add or subtract the numerators.
  4. Simplify if possible.
Example 2

$\dfrac{1}{4} + \dfrac{2}{3}$

LCD of 4 and 3 is 12.

$$\frac{3}{12} + \frac{8}{12} = \frac{11}{12}$$

Example 3

$\dfrac{5}{6} - \dfrac{1}{4}$

LCD = 12. $\;\dfrac{10}{12} - \dfrac{3}{12} = \dfrac{7}{12}$

Example 4

$\dfrac{3}{8} + \dfrac{5}{12}$

LCD = 24. $\;\dfrac{9}{24} + \dfrac{10}{24} = \dfrac{19}{24}$

Mixed Numbers

Example 5

$5\dfrac{1}{4} - 2\dfrac{2}{3}$

LCD = 12. $\;5\dfrac{3}{12} - 2\dfrac{8}{12}$.

Since $\dfrac{3}{12} < \dfrac{8}{12}$, borrow: $4\dfrac{15}{12} - 2\dfrac{8}{12} = 2\dfrac{7}{12}$.

Practice Problems

1. $\dfrac{3}{8} + \dfrac{1}{8}$
2. $\dfrac{7}{10} - \dfrac{3}{10}$
3. $\dfrac{2}{5} + \dfrac{1}{3}$
4. $\dfrac{5}{6} - \dfrac{1}{2}$
5. $\dfrac{3}{4} + \dfrac{5}{6}$
6. $\dfrac{7}{8} - \dfrac{2}{3}$
7. $4\dfrac{1}{3} + 2\dfrac{1}{2}$
8. $6\dfrac{1}{4} - 3\dfrac{3}{4}$
9. $\dfrac{1}{6} + \dfrac{3}{8} + \dfrac{1}{4}$
10. $8\dfrac{1}{5} - 4\dfrac{3}{10}$
11. $\dfrac{5}{9} + \dfrac{1}{6}$
12. $7\dfrac{1}{3} - 2\dfrac{5}{6}$
13. $\dfrac{2}{3} + \dfrac{3}{4} + \dfrac{5}{6}$
14. $10 - 4\dfrac{2}{5}$
15. $3\dfrac{5}{8} + 4\dfrac{7}{8}$
Show Answer Key

1. $\dfrac{1}{2}$

2. $\dfrac{2}{5}$

3. $\dfrac{11}{15}$

4. $\dfrac{1}{3}$

5. $1\dfrac{7}{12}$

6. $\dfrac{5}{24}$

7. $6\dfrac{5}{6}$

8. $2\dfrac{1}{2}$

9. $\dfrac{19}{24}$

10. $3\dfrac{9}{10}$

11. $\dfrac{13}{18}$

12. $4\dfrac{1}{2}$

13. $2\dfrac{1}{4}$

14. $5\dfrac{3}{5}$

15. $8\dfrac{1}{2}$

Fraction Basics and Equivalent Fractions Multiplying and Dividing Fractions