Adding Fractions
Fractions & DecimalsAdding 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 bottom numbers (denominators) must be the same. If they are already the same, just add the top numbers (numerators) and keep the same denominator. If they are different, first find a common denominator.
Example
Same denominator: $2/7 + 3/7 = 5/7$. Different denominators: $1/3 + 1/4$. Use $12$ as the common denominator: $4/12 + 3/12 = 7/12$.
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 you can simply add the counts.
Definition
To add fractions $a/b + c/d$: if $b = d$, sum $= (a+c)/b$. If $b \neq d$, rewrite using the LCD ($= \text{LCM}(b, d)$): $a/b = (a \times \text{LCD}/b)/\text{LCD}$ and $c/d = (c \times \text{LCD}/d)/\text{LCD}$; then add numerators. Always simplify the result to lowest terms.
Example
$3/8 + 5/12$. LCD $= 24$. $3/8 = 9/24$; $5/12 = 10/24$. Sum $= 19/24$ (already in lowest terms since $\gcd(19,24)=1$). For mixed numbers: $2$ and $1/3 + 1$ and $3/4 = 7/3 + 7/4 = 28/12 + 21/12 = 49/12 = 4$ and $1/12$.
Key Insight
The formula $a/b + c/d = (ad+bc)/(bd)$ always works without finding the LCD, but often gives a fraction that needs more reduction. Using the LCD is more efficient because the result is already over the smallest possible denominator.
Definition
Addition in $\mathbb{Q}$ is defined by $[a/b] + [c/d] = [(ad+bc)/(bd)]$, where brackets denote equivalence classes. Well-definedness must be verified: if $a/b \sim a'/b'$ and $c/d \sim c'/d'$, then $(ad+bc)/(bd) \sim (a'd'+b'c')/(b'd')$. This follows from the definition of $\sim$ and basic algebra, confirming $\mathbb{Q}$ is a group under addition.
Example
Verify well-definedness: $1/2 \sim 2/4$. $(1 \times 3 + 2 \times 5)/(2 \times 5) = 13/10$ and $(2 \times 3 + 4 \times 5)/(4 \times 5) = 26/20$. Check: $13 \times 20 = 260 = 26 \times 10$. The cross-multiplication equality confirms the two sums are equivalent.
Key Insight
The abstract definition of addition in $\mathbb{Q}$ via equivalence classes is an instance of how algebraic structures are built from quotient constructions. The verification of well-definedness is the algebraic price of working with equivalence classes rather than canonical representatives.