Fraction arithmetic follows four short rules, and the only real work is simplifying at the end. This calculator does both — exact answers via integer math (no floating-point rounding), with the simplification step spelled out.
Adding and subtracting: the common denominator
Fractions add only when they speak the same units. 1⁄3 + 1⁄4 becomes 4⁄12 + 3⁄12 = 7⁄12 — twelfths being the common language of thirds and quarters. Cross-multiplication (ad + bc over bd) automates the rewrite; simplification then shrinks whatever it produced.
Multiplying and dividing: no common ground needed
Multiplication goes straight across: 2⁄3 × 3⁄4 = 6⁄12 = 1⁄2. Division is multiplication in disguise — flip the second fraction: 1⁄2 ÷ 3⁄4 = 1⁄2 × 4⁄3 = 2⁄3. The "keep, change, flip" mnemonic survives generations of classrooms because it's the entire rule.
Simplifying: divide by the GCF
A result like 18⁄24 shrinks by the greatest common factor of top and bottom — here 6 — to 3⁄4. The steps line shows exactly which factor was divided out. Under the hood the calculator uses arbitrary-precision integers, so even huge intermediate products stay exact — a promise decimals can't make, since 1⁄3 has no finite decimal at all (see the fraction ⇄ decimal converter for that story).
Mixed numbers and negatives
Top-heavy results also appear as mixed numbers: 7⁄4 = 1 3⁄4. Negative fractions keep their sign on the numerator by convention. For ratios and proportions — fractions' close cousins — the ratio calculator solves A:B = C:? directly.
Exact integer arithmetic; decimals shown to 6 places.