Fraction Calculator

Fraction arithmetic by hand requires finding common denominators, simplifying results, and converting between mixed and improper forms. It's the math you supposedly mastered in fifth grade and immediately stopped using, which is why most adults silently dread the moment a recipe calls for 2/3 + 3/4.

Enter two fractions (improper or mixed) and an operator, and the calculator returns the result simplified to lowest terms in both improper and mixed forms, with the decimal equivalent. Step-by-step working is shown so you can see how the answer was derived — useful for homework help where the process matters as much as the answer.

Handles negatives, mixed numbers, zero denominators (gracefully, with an error), and arbitrary integer inputs. The simplification uses the Euclidean algorithm for greatest common divisor, which scales fine to large numerators and denominators without overflow concerns at typical input sizes.

The operations chain together using a small set of rules that most people learned and forgot. Addition and subtraction require a common denominator: a/b ± c/d = (ad ± bc) / bd. Multiplication is simpler: a/b × c/d = ac/bd. Division flips the second fraction and multiplies: a/b ÷ c/d = ad/bc. Mixed numbers convert to improper fractions for arithmetic, then back to mixed form for the final answer. The simplification step uses the Euclidean algorithm to find GCD: divide larger by smaller, take the remainder, repeat until remainder is zero, the last nonzero divisor is the GCD. This is the same algorithm Euclid documented in 300 BCE, and it's still the fastest practical method for reasonable input sizes.

Worked example: 2 ¾ + 1 ⅔. Convert: 2 ¾ = 11/4, 1 ⅔ = 5/3. Common denominator 12: 11/4 = 33/12, 5/3 = 20/12. Sum 53/12. Convert back: 53 / 12 = 4 remainder 5, so 4 5/12. Decimal equivalent ≈ 4.4167. The calculator shows each step, which matters when the goal is learning the procedure (homework help) rather than just the answer (baking math).

Where fraction arithmetic gets tricky is repeating decimals and irrational numbers. 1/3 = 0.3333... infinitely; the calculator can show the rational form precisely or the decimal approximation. For irrational results like √2 or π that arise in geometric problems, fractional arithmetic is the wrong tool — those require symbolic math systems (Mathematica, SymPy, WolframAlpha) that handle algebraic numbers exactly. The fraction calculator covers rational arithmetic, which is what 95% of practical fraction problems are.

A common edge case the calculator handles correctly: subtraction that produces a negative result. 1/4 − 1/2 = −1/4. Some manual implementations return absolute values or fail; the math is signed and the result is negative. Similarly, division by zero returns an error rather than crashing or returning infinity-as-fraction. These are the kinds of edge cases that distinguish a usable tool from a fragile one. The Euclidean algorithm itself doesn't care about signs — GCD is taken on absolute values, with the sign tracked separately.