Fraction Calculator

About this calculator

A free fraction calculator that works the way fractions actually work: in exact integer arithmetic, never floating-point approximations. Add, subtract, multiply, or divide any two fractions and the result comes back fully simplified — 1/3 + 1/4 is exactly 7/12, not a rounded decimal forced back into fraction form — alongside its mixed-number decomposition and a six-decimal display value. Two companion views convert any fraction to a decimal (and say whether that decimal terminates or repeats) and reduce any fraction to simplest form with the greatest common divisor. Everything calculates in your browser on the numbers you enter; nothing is uploaded or stored. Negative fractions are handled with the sign carried on the numerator, results past one are decomposed into mixed numbers automatically, and impossible inputs — a zero denominator, or dividing by a fraction equal to zero — are called out instead of returning a bogus number.

Adding fractions, the common-denominator walk-through

Fractions can only be added or subtracted once they describe pieces of the same size, which is all a common denominator is. For 1/3 + 1/4, scale each fraction by the other's denominator: 1/3 becomes 4/12 and 1/4 becomes 3/12. Both now count twelfths, so the numerators add directly: 4/12 + 3/12 = 7/12. The calculator uses the product of the denominators as the common denominator and then reduces the result, which always lands on the same answer as the least-common-denominator method taught in school.

When a sum passes one, the mixed-number line decomposes it: 5/6 + 3/4 over the common denominator 12 is 10/12 + 9/12 = 19/12, reported as the improper fraction 19/12, the mixed number 1 7/12, and the decimal ≈1.583333. Subtraction is the same walk with a minus sign: 3/4 − 1/6 = 9/12 − 2/12 = 7/12.

Why multiplying and dividing skip the common denominator

Multiplication needs no setup at all: multiply the numerators, multiply the denominators, reduce. 2/3 × 3/4 = 6/12, which simplifies to 1/2. The reduction step is what keeps answers readable — without it, chained multiplications snowball into fractions that are technically correct and practically useless.

Division uses the invert-and-multiply rule: dividing by a fraction is the same as multiplying by its reciprocal. 1/2 ÷ 3/4 flips the second fraction to 4/3, giving 1/2 × 4/3 = 4/6 = 2/3. The rule covers whole numbers too, since any whole number sits over a denominator of one: 3/4 ÷ 2 is 3/4 × 1/2 = 3/8. The one impossible case is dividing by a fraction equal to zero, which the calculator flags instead of answering.

Simplifying with the greatest common divisor

Every result is reduced by dividing the numerator and denominator by their greatest common divisor (GCD), found with Euclid's algorithm: repeatedly replace the larger number with the remainder of dividing it by the smaller, and the last non-zero remainder is the GCD. For 48/36: 48 mod 36 = 12, then 36 mod 12 = 0, so the GCD is 12 and 48/36 reduces to 4/3 — the mixed number 1 1/3.

Euclid's algorithm finds the whole GCD in one pass, unlike the schoolbook habit of dividing by small primes one at a time and hoping nothing was missed — stop that process early and 48/36 stalls at 8/6 instead of reaching 4/3. Signs are normalized in the same step: the denominator is kept positive and the sign rides on the numerator, so 4/−6 reduces to −2/3.

Mixed numbers and improper fractions

An improper fraction like 7/4 and the mixed number 1 3/4 are the same value in two costumes: divide 7 by 4 to get whole part 1 with remainder 3. The calculator shows both forms for every result, because improper fractions are easier to compute with and mixed numbers are easier to read.

Going the other way, multiply the whole part by the denominator and add the numerator: 2 1/3 = (2 × 3 + 1)/3 = 7/3. For negative mixed numbers the sign belongs to the whole quantity, not just the whole part: −1 3/4 means −(1 + 3/4) = −7/4.

Terminating or repeating? Read the denominator

Whether a fraction's decimal form ends or repeats forever is decided entirely by the reduced denominator's prime factors. If they are only 2s and 5s — the primes of 10 — the decimal terminates: 3/8 has denominator 2³ and ends at exactly 0.375, and 9/20 has denominator 2² × 5 and ends at 0.45.

Any other prime factor forces repetition. 1/3 is 0.333… forever, and 7/12 repeats because 12 carries a factor of 3 — its decimal reads ≈0.583333 only after rounding. The fraction-to-decimal view applies this test automatically and labels every conversion as terminating or repeating.

By variant

• Fraction to Decimal Converter
• Simplify Fractions Calculator

Questions

Is the fraction calculator free?

Yes. It is free, needs no account, and calculates in your browser; nothing you enter is uploaded or stored.

Are the results exact or approximations?

Exact. All arithmetic happens on integer numerators and denominators and every result is reduced with the greatest common divisor — 1/3 + 1/4 is exactly 7/12. Only the optional decimal line is rounded, to six places, and repeating decimals are marked as such.

Does it simplify answers automatically?

Yes. Every result is reduced to simplest form, so 2/4 + 2/4 returns 1, not 16/16, and the mixed-number form is shown whenever the result passes one.

Can I enter negative fractions?

Yes — put the minus sign on the numerator. −1/2 + 1/3 = −1/6, with the sign always reported on the numerator and the denominator kept positive.