Adding & Subtracting Fractions
Adding and subtracting fractions requires one crucial step: the fractions must have the same denominator, called a common denominator.
Why Common Denominators?
Think of denominators as telling you the size of the pieces. You can't directly add or subtract pieces of different sizes (like adding $\frac{1}{2}$ and $\frac{1}{4}$) any more than you can directly add 1 apple and 1 orange and say you have "2 apple-oranges".
Getting Common Denominators
To make the denominators the same, you need to find the Least Common Denominator (LCD). This is the smallest number that both original denominators divide into evenly (their Least Common Multiple).
How to find the LCD (e.g., for $\frac{1}{4}$ and $\frac{5}{6}$):
- List multiples of 4: 4, 8, 12, 16, ...
- List multiples of 6: 6, 12, 18, 24, ...
- The smallest number in both lists is 12. So, the LCD is 12.
Once you find the LCD, rewrite each fraction as an equivalent fraction with that denominator.
The Steps for Adding/Subtracting Fractions
- Convert (if needed): Change any mixed numbers to improper fractions.
- Find LCD: Determine the Least Common Denominator for all fractions involved.
- Rewrite Fractions: Create equivalent fractions using the LCD. Remember to multiply the numerator by the same factor you multiplied the denominator by.
- Add/Subtract Numerators: Perform the addition or subtraction on the top numbers (numerators). Keep the common denominator.
- Simplify: Reduce the resulting fraction to its lowest terms. Convert back to a mixed number if appropriate.
Reminder: Sign Rules
- Adding a negative is the same as subtracting: $5 + (-3) = 5 - 3 = 2$
- Subtracting a negative is the same as adding: $7 - (-2) = 7 + 2 = 9$
- When adding/subtracting with different signs, think about number lines or "owing" money.
- When signs are the same for addition (or after converting subtraction), add the values and keep the sign: $-4 - 5 = -9$
Examples with Step-by-Step Solutions
Example 1: Calculate $\frac{2}{5} + \frac{1}{3}$
Find LCD and rewrite fractions:
$$ \begin{align*} & \frac{2}{5} + \frac{1}{3} \quad & \text{(Denominators 5 and 3. LCD = 15)} \\ = & \boxed{\frac{2 \times 3}{5 \times 3}} + \boxed{\frac{1 \times 5}{3 \times 5}} \quad & \text{(Rewrite with LCD)} \\ = & \frac{6}{15} + \frac{5}{15} \\ = & \frac{\boxed{6 + 5}}{15} \quad & \text{(Add Numerators)} \\ = & \frac{11}{15} \end{align*} $$Result: $\frac{11}{15}$ (already simplified)
Example 2: Calculate $-\frac{1}{4} - \frac{5}{6}$
Find LCD and watch the signs:
$$ \begin{align*} & -\frac{1}{4} - \frac{5}{6} \quad & \text{(Denominators 4 and 6. LCD = 12)} \\ = & \boxed{-\frac{1 \times 3}{4 \times 3}} - \boxed{\frac{5 \times 2}{6 \times 2}} \quad & \text{(Rewrite with LCD)} \\ = & -\frac{3}{12} - \frac{10}{12} \\ = & \frac{\boxed{-3 - 10}}{12} \quad & \text{(Subtract Numerators: -3 take away 10)} \\ = & \frac{-13}{12} \end{align*} $$Result: $-\frac{13}{12}$ or $-1\frac{1}{12}$
Example 3: Calculate $2\frac{1}{2} - \left(-\frac{3}{4}\right)$
Convert mixed number and simplify signs first:
$$ \begin{align*} & \boxed{2\frac{1}{2}} - \left(-\frac{3}{4}\right) \quad & \text{(Convert mixed to improper: } \frac{5}{2}) \\ = & \frac{5}{2} - \left(-\frac{3}{4}\right) \\ = & \boxed{\frac{5}{2} + \frac{3}{4}} \quad & \text{(Subtracting a negative is adding)} \\ & \quad & \text{(Denominators 2 and 4. LCD = 4)} \\ = & \boxed{\frac{5 \times 2}{2 \times 2}} + \frac{3}{4} \quad & \text{(Rewrite first fraction with LCD)} \\ = & \frac{10}{4} + \frac{3}{4} \\ = & \frac{\boxed{10 + 3}}{4} \quad & \text{(Add Numerators)} \\ = & \frac{13}{4} \end{align*} $$Result: $\frac{13}{4}$ or $3\frac{1}{4}$