2. Adding & Subtracting Fractions

You can only add or subtract fractions when the pieces are the same size — that is, when the denominators match. You can't add 1 quarter and 1 third directly any more than you can add 1 apple and 1 orange without a common name ('fruit').

Same denominator? Just add the tops and keep the bottom:
\[ \frac{2}{5} + \frac{1}{5} = \frac{3}{5}. \]
Different denominators? First rewrite them with a common denominator (a number both bottoms divide into), then add.

Example: \(\frac{1}{2} + \frac{1}{3}\). The smallest common denominator is 6.
\[ \frac{1}{2} = \frac{3}{6}, \qquad \frac{1}{3} = \frac{2}{6}, \qquad \frac{3}{6} + \frac{2}{6} = \frac{5}{6}. \]

⚠️ Common mistake: never add the denominators. \(\frac{1}{2} + \frac{1}{3}\) is not \(\frac{2}{5}\). The bottom tells you the size of the piece, not a quantity to add.