Adding Fractions
Adding fractions requires a common denominator; then the numerators are added while the denominator stays the same.
Formula
\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}; \quad \frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}
Definition
To add fractions, the denominators must be the same: if they already match, just add the numerators and keep the denominator; if not, first find a common denominator (ideally the LCD) before adding. Formally, for $a/b + c/d$: if $b = d$, the sum is $(a+c)/b$; if $b \neq d$, rewrite both fractions over the LCD, add the resulting numerators, and simplify to lowest terms. In the language of equivalence classes, addition in $\mathbb{Q}$ is defined by $[a/b] + [c/d] = [(ad+bc)/(bd)]$, and proving this is well-defined, that equivalent representatives always give equivalent sums, confirms $\mathbb{Q}$ is a group under addition.
Example
With the same denominator, $2/7 + 3/7 = 5/7$; with different denominators, $1/3 + 1/4$ becomes $4/12 + 3/12 = 7/12$ using $12$ as the common denominator. For $3/8 + 5/12$, the LCD is $24$: $3/8 = 9/24$ and $5/12 = 10/24$, giving $19/24$ (already lowest terms). To verify the formula is well-defined regardless of representative: since $1/2 \sim 2/4$, computing $(1 \times 3 + 2 \times 5)/(2 \times 5) = 13/10$ and $(2 \times 3 + 4 \times 5)/(4 \times 5) = 26/20$ gives equivalent results, confirmed by $13 \times 20 = 260 = 26 \times 10$.
Key Insight
Adding fractions with different denominators is like adding apples and oranges: you first convert everything to the same "unit" (the common denominator), then simply add the counts. The general formula $a/b + c/d = (ad+bc)/(bd)$ always works without finding the LCD, but often gives a fraction that needs more reduction, so using the LCD is more efficient since the result is already over the smallest possible denominator. The abstract definition of addition in $\mathbb{Q}$ via equivalence classes is an instance of how algebraic structures are built from quotient constructions, and verifying well-definedness is the algebraic price of working with equivalence classes rather than canonical representatives.