Simplify Fractions Calculator

Enter a numerator and denominator to reduce the fraction to lowest terms (A/GCD over B/GCD), plus its decimal and mixed-number equivalents.

Quick Facts

Simplify formula
a/b = (a÷g) / (b÷g)
Where g = GCD(a, b), the greatest common divisor of the numerator and denominator.
Lowest terms test
GCD(a, b) = 1
A fraction is fully reduced when its numerator and denominator share no common factor besides 1.
Sign convention
Denominator kept positive
Any negative sign is moved to (or kept on) the numerator.

Your Results

Calculated
Simplified Fraction
-
Reduced to lowest terms
Greatest Common Divisor
-
GCD used to reduce
Decimal Equivalent
-
Numerator ÷ denominator
Mixed Number
-
Whole number + remaining fraction

Ready

Enter a numerator and denominator, then press Calculate.

How the Simplify Fractions Calculator works

Simplifying (or "reducing") a fraction means rewriting it with the smallest possible whole-number numerator and denominator while keeping the same value. This is done by dividing both the numerator and denominator by their greatest common divisor (GCD) — the largest whole number that divides both evenly. If a = 12 and b = 18, then g = GCD(12, 18) = 6, so the fraction reduces to 12/6 over 18/6 = 2/3.

Formula and method

For a fraction a/b, the simplified form is (a ÷ g) / (b ÷ g), where g = GCD(|a|, |b|). This calculator finds g using the Euclidean algorithm: repeatedly replace the larger of the two numbers with the remainder of dividing it by the smaller number, until the remainder reaches 0 — the last nonzero remainder is the GCD. For example, GCD(12, 18): 18 mod 12 = 6, then 12 mod 6 = 0, so GCD = 6. Once the fraction is reduced, the calculator also reports the decimal equivalent (numerator ÷ denominator) and, for improper fractions (numerator larger than denominator), the equivalent mixed number.

Common sources of error

  • Stopping at a common factor instead of the GCD: dividing 12/18 by 2 gives 6/9, which is not fully reduced — you must divide by the full GCD (6) to reach lowest terms.
  • Sign handling: a negative fraction like -6/-9 should reduce to 2/3 (both signs cancel), while 6/-9 should reduce to -2/3 (denominator kept positive).
  • Zero denominator: a fraction with a denominator of 0 is undefined and cannot be simplified or converted to a decimal.

Checking your result

After simplifying, verify that the numerator and denominator of the reduced fraction share no common factor other than 1 (their GCD should equal 1) — if it does not, you have not fully reduced the fraction. You can also cross-check by confirming that the original and simplified fractions produce the same decimal value.

Applications

Simplified fractions are easier to compare, add, subtract, and communicate than unreduced ones. They show up constantly in cooking measurements, construction and woodworking (reading a tape measure in reduced eighths or sixteenths of an inch), probability calculations, and basic algebra, where an unsimplified fraction like 8/12 is harder to work with than its reduced form, 2/3.

Frequently Asked Questions

How do you simplify a fraction?
Find the greatest common divisor (GCD) of the numerator and denominator, then divide both by that number. For example, 12/18 has a GCD of 6, so it simplifies to 2/3.
How is the GCD calculated?
The Euclidean algorithm finds the GCD by repeated division: divide the larger number by the smaller, replace the larger number with the remainder, and repeat until the remainder is 0. The last nonzero remainder is the GCD.
How do I know if a fraction is already in lowest terms?
A fraction is in lowest terms (fully reduced) when the GCD of its numerator and denominator is 1, meaning they share no common factor other than 1.
How do you convert an improper fraction to a mixed number?
Divide the numerator by the denominator. The whole-number quotient becomes the whole part, and the remainder over the original denominator becomes the fractional part, which should then be simplified using the GCD.

Related calculators