Adding and Subtracting Fractions.

A fraction $\frac{a}{b}$ means "$a$ parts out of $b$ equal pieces." You can only add fractions when they're cut into the same size pieces — that means the denominators must match.

Same denominator — easy case

When the bottoms already match, just add the tops: $\frac{3}{8} + \frac{2}{8} \;=\; \frac{5}{8}$

Different denominators — find a common one

You need a number that both denominators divide into evenly. The smallest one is the Least Common Denominator (LCD).

Quickest method: multiply each fraction by the form of $1$ that gets you there.

$\frac{1}{3} + \frac{1}{4}$ LCD is $12$. Multiply $\frac{1}{3}$ by $\frac{4}{4}$, and $\frac{1}{4}$ by $\frac{3}{3}$: $\frac{1}{3} \cdot \frac{4}{4} + \frac{1}{4} \cdot \frac{3}{3} \;=\; \frac{4}{12} + \frac{3}{12} \;=\; \frac{7}{12}$

Subtraction works the same way

$\frac{5}{6} - \frac{1}{4}$ LCD is $12$. $\frac{5}{6} = \frac{10}{12}$ and $\frac{1}{4} = \frac{3}{12}$. $\frac{10}{12} - \frac{3}{12} \;=\; \frac{7}{12}$

Always reduce at the end

$\frac{6}{8}$ should be written as $\frac{3}{4}$ (divide top and bottom by 2). A fraction is reduced when top and bottom share no common factor.