Expressions: Combining, Dividing & Splitting Fractions
Master every fraction operation tested on the SAT Math section
1. Fundamentals of Algebraic Fractions
What Is an Algebraic Fraction?
An algebraic fraction (also called a rational expression) is a fraction that contains variables in the numerator, the denominator, or both.
Structure:
$$\frac{a}{b} = a \div b$$- Numerator $a$ — the top expression
- Denominator $b$ — the bottom expression
- Rule: $b \neq 0$ always!
Examples of algebraic fractions:
| $\dfrac{x}{3}$ | Variable in numerator only |
| $\dfrac{2x+1}{5}$ | Expression in numerator |
| $\dfrac{x^2-1}{x+3}$ | Variables in both |
Domain Restrictions
| Expression | Restriction | Why? |
|---|---|---|
| $\dfrac{5}{x-3}$ | $x \neq 3$ | $x-3=0 \Rightarrow x=3$ |
| $\dfrac{1}{x^2-4}$ | $x \neq \pm 2$ | $(x-2)(x+2)=0$ |
| $\dfrac{x}{2x+1}$ | $x \neq -\frac{1}{2}$ | $2x+1=0$ |
Equivalent Fractions
Multiply or divide numerator and denominator by the same non-zero expression:
$$\frac{a}{b} = \frac{a \cdot c}{b \cdot c} \quad (c \neq 0)$$Example: $\dfrac{3}{x} = \dfrac{3 \cdot x}{x \cdot x} = \dfrac{3x}{x^2}$
2. Combining Fractions — Addition & Subtraction
2A. Same Denominator (Easy Case)
Rule:
$$\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c} \qquad \frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}$$Keep the denominator. Add or subtract the numerators.
Example 1: $\dfrac{3}{x} + \dfrac{5}{x} = \dfrac{3+5}{x} = \dfrac{8}{x}$
Example 2: $\dfrac{2x}{x+1} + \dfrac{3}{x+1} = \dfrac{2x+3}{x+1}$
Subtraction Caution!
Distribute the negative to ALL terms in the second numerator:
$$\frac{5x-2}{x^2} - \frac{3x+1}{x^2} = \frac{(5x-2)-(3x+1)}{x^2}$$ $$= \frac{5x-2-3x-1}{x^2} = \frac{2x-3}{x^2}$$Common trap: writing $-3x+1$ instead of $-3x-1$
2B. Different Denominators — Finding the LCD
The LCD (Least Common Denominator) is the smallest expression divisible by ALL denominators.
Steps: ① Factor each denominator → ② Include each factor at its highest power → ③ Multiply them together
$\dfrac{x}{4} + \dfrac{x}{6}$
LCM of 4 and 6:
- $4 = 2^2$
- $6 = 2 \cdot 3$
- LCD $= 2^2 \cdot 3 = 12$
$= \dfrac{3x}{12} + \dfrac{2x}{12} = \dfrac{5x}{12}$
$\dfrac{3}{x} + \dfrac{5}{x^2}$
Denominators: $x$ and $x^2$
- Highest power of $x$ is $x^2$
- LCD $= x^2$
$= \dfrac{3x}{x^2} + \dfrac{5}{x^2} = \dfrac{3x+5}{x^2}$
$\dfrac{1}{x+1} + \dfrac{2}{x-1}$
No common factors:
- LCD $= (x+1)(x-1)$
$= \dfrac{x-1}{(x+1)(x-1)} + \dfrac{2(x+1)}{(x+1)(x-1)}$
$= \dfrac{x-1+2x+2}{(x+1)(x-1)} = \dfrac{3x+1}{x^2-1}$
$\dfrac{1}{x^2-4} + \dfrac{1}{x+2}$
Step 1: Factor: $x^2-4 = (x+2)(x-2)$
Step 2: LCD $= (x+2)(x-2)$
Step 3: Convert:
$$= \frac{1}{(x+2)(x-2)} + \frac{(x-2)}{(x+2)(x-2)}$$Step 4: Combine:
$$= \frac{1 + x - 2}{(x+2)(x-2)} = \frac{x-1}{x^2-4}$$$\dfrac{3}{x-2} - \dfrac{2}{x+3} + \dfrac{1}{(x-2)(x+3)}$
LCD $= (x-2)(x+3)$
$$= \frac{3(x+3) - 2(x-2) + 1}{(x-2)(x+3)}$$ $$= \frac{3x+9 - 2x+4 + 1}{(x-2)(x+3)}$$ $$= \frac{x+14}{(x-2)(x+3)}$$LCD Decision Flowchart
Adding a Whole Number and a Fraction
Convert the whole number to a fraction with the same denominator:
$$3 + \frac{2}{x} = \frac{3x}{x} + \frac{2}{x} = \frac{3x+2}{x}$$