How this instrument works
This page performs any of the four basic operations on two fractions, selected with a simple operation number, and returns the result already simplified. Adding or subtracting first finds a common denominator by cross-multiplying (n₁d₂ ± n₂d₁ over d₁d₂), multiplying combines numerators and denominators directly (n₁n₂ over d₁d₂), and dividing multiplies by the second fraction flipped (n₁d₂ over d₁n₂) — four different combination rules, unified here under one operation selector.
Whatever the operation, the raw result is simplified automatically by dividing both the resulting numerator and denominator by their greatest common divisor — the same reduction technique used to put any single fraction into lowest terms, applied here to the combined result rather than to an input fraction directly.
Multiplication and division are structurally simpler than addition and subtraction, since they never actually need a common denominator — multiplying numerators together and denominators together, or flipping and multiplying for division, works directly regardless of whether the two original denominators match.
- Enter the operation into the Operation field: 1 for add, 2 for subtract, 3 for multiply, 4 for divide.
- Enter the first fraction's numerator and denominator into the Fraction 1 fields.
- Enter the second fraction's numerator and denominator into the Fraction 2 fields.
- Read Result numerator and Result denominator: the sheet performs the operation and simplifies the answer automatically.
Worked example — adding 1⁄2 and 1⁄3
Adding 1⁄2 and 1⁄3: (1×3+1×2)⁄(2×3) = 5⁄6, already in lowest terms — a common denominator found by cross-multiplying, with the numerators combined according to the chosen operation.
Multiplying 2⁄3 by 3⁄4 instead: (2×3)⁄(3×4) = 6⁄12, which simplifies down to 1⁄2 once the greatest common divisor, 6, is divided out of both parts.
Questions
How do you add two fractions with different denominators?
Cross-multiply to find a common denominator: (n₁d₂ + n₂d₁) ⁄ (d₁d₂). This works for any two fractions regardless of whether their original denominators already match.
Why doesn't multiplying fractions need a common denominator?
Because multiplication combines numerators together and denominators together directly, with no need for the two parts to share any common structure first — unlike addition and subtraction, which require matching denominators before the numerators can be combined.
How does dividing fractions work?
Multiply by the second fraction flipped (its reciprocal): dividing by n₂⁄d₂ is the same as multiplying by d₂⁄n₂. This 'invert and multiply' rule turns every division into an equivalent multiplication.
Is the result always simplified automatically?
Yes — the sheet always reduces the final numerator and denominator by their greatest common divisor, so the answer comes back already in lowest terms, regardless of which operation was performed.
What if a denominator is entered as zero?
The result is undefined, since a fraction with a zero denominator doesn't represent a real number at all — every denominator must be nonzero for the calculation to produce a meaningful answer.