Adding and Subtracting Fractions With Different Denominators, Step by Step
Adding or subtracting fractions with different denominators requires first converting both to a common denominator (a shared multiple of both original denominators), then adding or subtracting only the numerators — the denominators themselves are never added or subtracted directly, which is a common mistake when the underlying rule isn't fully understood.
This is one of the most fundamental fraction rules, and walking through a worked example makes the "why" behind the common-denominator step concrete.
A worked example
Adding 1/4 and 1/6: the least common denominator of 4 and 6 is 12. Converting each fraction: 1/4 becomes 3/12 (multiplying numerator and denominator by 3), and 1/6 becomes 2/12 (multiplying by 2). Adding the numerators now that denominators match: 3/12 + 2/12 = 5/12.
Why a common denominator is necessary
A fraction's denominator defines the size of each piece being counted — 1/4 and 1/6 are pieces of different sizes, so their numerators (the count of pieces) can't be meaningfully combined until both fractions are expressed in terms of the same-sized piece, which is exactly what finding a common denominator accomplishes.
The common mistake this rule prevents
Adding numerators and denominators directly (1/4 + 1/6 = 2/10, incorrectly) is a common error that ignores the different piece sizes entirely — this shortcut isn't mathematically valid and produces a wrong answer, which is exactly why the common-denominator step can't be skipped.