Fraction Calculator

Add, subtract, multiply or divide two fractions, automatically simplified.

Enter Two Fractions

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Result

Simplified Result
Decimal Equivalent
0
Before Simplifying
0/1

Visual Comparison

1/2
Fraction 1
1/3
Fraction 2
5/6
Result
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How Fraction Arithmetic Works

Each operation on fractions follows its own rule, depending on why the operation works the way it does.

Why addition and subtraction need a common denominator

A fraction's denominator defines the size of the "pieces" being counted, so different denominators can't be combined directly. Both fractions are first converted to a common denominator by cross-multiplying: for a/b and c/d, addition gives (a×d + c×b) / (b×d), and subtraction gives (a×d − c×b) / (b×d).

Why multiplication and division skip that step

Multiplication doesn't need matching denominators, since you're scaling one fraction by another rather than combining same-sized pieces — it multiplies numerators together and denominators together: (a×c) / (b×d). Division multiplies the first fraction by the reciprocal of the second, since dividing by a number is the same as multiplying by its reciprocal: (a×d) / (b×c).

Worked example

Adding 1/2 + 1/3: common denominator is 2×3 = 6, so 1/2 = 3/6 and 1/3 = 2/6. Adding gives 5/6, which is already in lowest terms (GCD of 5 and 6 is 1), so the simplified result is 5/6 ≈ 0.833.

How to calculate with fractions, step by step

  1. Enter the numerator and denominator of your first fraction.
  2. Choose to Add, Subtract, Multiply or Divide.
  3. Enter the numerator and denominator of your second fraction.
  4. Click Calculate to see the result, automatically simplified to its lowest terms.

Why every result is automatically simplified

Every result is simplified by dividing both the numerator and denominator by their Greatest Common Divisor (GCD), found using the Euclidean algorithm — an efficient method of repeated division and remainders — so you always see the fraction in its lowest terms rather than an unreduced form like 10/12 instead of 5/6.

Notes

Frequently Asked Questions

A fraction's denominator defines the size of each "piece" being counted — halves and thirds are different-sized pieces, so you can't directly add the numerators. Converting both fractions to a common denominator (by cross-multiplying) rewrites them in terms of the same-sized piece, so the numerators can then be added or subtracted meaningfully.

Dividing by a number is the same as multiplying by its reciprocal (1 ÷ x = multiplying by 1/x) — this holds for fractions too. Dividing by a/b is the same as multiplying by b/a, which is why fraction division flips the second fraction and multiplies: (a×d) / (b×c) for a/b ÷ c/d.

Multiplication doesn't require the pieces to be the same size, because you're scaling one fraction by another rather than combining pieces of the same size — numerators multiply together and denominators multiply together directly: (a×c) / (b×d), with no common denominator needed first.

The Euclidean algorithm is an efficient method for finding the Greatest Common Divisor (GCD) of two numbers by repeatedly dividing and taking remainders. Once the GCD of a fraction's numerator and denominator is found, dividing both by it produces the same fraction in its simplest, lowest-terms form — this is why every result from this calculator is automatically simplified.

A fraction's denominator represents how many equal pieces something is divided into — dividing into zero pieces is mathematically undefined, since division by zero has no defined result. This calculator flags a zero denominator as invalid and prevents the calculation, rather than showing a nonsensical answer.

Negative fractions are fully supported. Regardless of whether you enter the negative sign on the numerator or denominator, this calculator normalises it onto the numerator internally (so -a/b and a/-b are treated identically), which keeps the arithmetic and simplification logic consistent.
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