In this section, we will learn how to add and subtract fractions.
Fractions can be added or subtracted only when they have the same denominators.
Adding and Subtracting Fractions with Same Denominators:
$\dfrac{a}{b}+ \dfrac{c}{b} = \dfrac{a+ c}{b}$
$\dfrac{a}{b}- \dfrac{c}{b} = \dfrac{a- c}{b}$
For example: $\dfrac{1}{5}+\dfrac{3}{5}=\dfrac{4}{5}$ , and $\dfrac{1}{5}-\dfrac{3}{5}=\dfrac{-2}{5}$ .
Adding or subtracting fractions with different denominators requires us to first find the least common denominator (LCD).
Adding and Subtracting Fractions with Different Denominators:
Step 1. Finding the Least Common Denominator (LCD)
- Make a list of (enough) multiples of each denominator.
- Identify the lowest common multiple. If you can’t see one, then your lists need to be expanded.
Step 2. Rewrite both fractions (by multiplying both numerator and denominator by the appropriate same number) to get the LCD as denominator.
Step 3. Add or subtract then simplify if needed.
Example: Find the LCD and then add and simplify $\dfrac{3}{12}+ \dfrac{5}{8}$
Step 1. Finding the LCD of 12 and 8
$8: 8, 16, 24, 36,…$
$12: 12, 24, 36,…$
LCD: 24
Step 2. Rewrite each fraction using the LCD
$\dfrac{3}{12}=\dfrac{3\cdot 2}{12\cdot 2}=\dfrac{6}{24}$
$\dfrac{5}{8}=\dfrac{5\cdot 3}{8\cdot 3}=\dfrac{15}{24}$
Step 3. Add or subtract then simplify if needed.
$\dfrac{6}{24}+\dfrac{15}{24}=\dfrac{21}{24}=\dfrac{3\cdot7}{3\cdot8}=\dfrac{7}{8}$
The above three steps can be combined as:
$\dfrac{3}{12}+\dfrac{5}{8}=\dfrac{3\cdot 2}{12\cdot 2}+\dfrac{5\cdot 3}{8\cdot 3}=\dfrac{6}{24}+\dfrac{15}{24}=\dfrac{21}{24}=\dfrac{7}{8}$
Example: Find the LCD and then subtract and simplify $\dfrac{1}{9}- \dfrac{3}{5}$
Step 1. Finding the LCD of 9 and 5
$9: 9, 18, 27, 36, 45, 54, …$
$5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, …$
LCD: 45
Step 2. Rewrite each fraction using the LCD
$\dfrac{1}{9}=\dfrac{1\cdot 5}{9\cdot 5}=\dfrac{5}{45}$
$\dfrac{3}{5}=\dfrac{3\cdot 9}{5\cdot 9}=\dfrac{27}{45}$
Step 3. Add or subtract then simplify if needed.
$\dfrac{5}{45}-\dfrac{27}{45}=\dfrac{-22}{45}$
The above three steps can be combined as:
$\dfrac{1}{9}-\dfrac{3}{5}=\dfrac{1\cdot 5}{9\cdot 5}-\dfrac{3\cdot 9}{5\cdot 5}=\dfrac{5}{45}-\dfrac{27}{45}=\dfrac{-22}{45}$
Practice Problems:
1. Add or subtract the following fractions:
(a) $\dfrac{3}{5}-\dfrac{5}{6}$
(b) $-\dfrac{7}{11}+\dfrac{2}{33}$
(c) $\dfrac{1}{3}-\dfrac{1}{2}$
(b) $-\dfrac{1}{2}+2$ (note that $2=2/1$)
Answer Key: (a) (b) (c) (d)
For more detailed explanation, please read: Arithmetic|Algebra Chapter 2.
