How Fractions Work — Addition, Subtraction, Multiplication and Division
Fractions are one of those topics that many people find confusing precisely because the rules are different for different operations. Addition and subtraction require a common denominator; multiplication and division don't. Here's each operation explained clearly.
The Anatomy of a Fraction
A fraction has two parts: the numerator (top number) and the denominator (bottom number). The denominator tells you how many equal parts the whole is divided into; the numerator tells you how many of those parts you have. ¾ means 3 out of 4 equal parts.
Adding and Subtracting Fractions
You can only add or subtract fractions that have the same denominator. If they don't, you first convert them to a common denominator.
Example: ½ + ⅓. The common denominator is 6. Convert: 3/6 + 2/6 = 5/6.
To find a common denominator, find the Lowest Common Multiple (LCM) of the two denominators, then multiply each fraction's numerator by whatever you multiplied its denominator by.
Multiplying Fractions
This is the simplest operation — multiply the numerators together and the denominators together. No common denominator needed.
Example: ⅔ × ¾ = (2×3) ÷ (3×4) = 6/12 = ½ (simplified).
Always simplify the result by dividing both numerator and denominator by their Greatest Common Divisor (GCD).
Dividing Fractions
Dividing by a fraction is equivalent to multiplying by its reciprocal (flip the second fraction upside down, then multiply).
Example: ⅔ ÷ ¾ = ⅔ × 4/3 = 8/9.
The phrase "keep, change, flip" is the common mnemonic: keep the first fraction, change division to multiplication, flip the second fraction.
Mixed Numbers
Mixed numbers (like 2½) combine a whole number and a fraction. To calculate with them, first convert to an improper fraction: multiply the whole number by the denominator and add the numerator. 2½ = (2×2 + 1)/2 = 5/2. Then do the calculation, then convert back if needed.
Simplifying Fractions
A fraction is in its simplest form when the numerator and denominator share no common factors other than 1. To simplify, divide both by their GCD. 12/18: GCD is 6. 12÷6 = 2, 18÷6 = 3. Simplified: ⅔.