What does it mean to simplify a fraction?

Simplifying a fraction — also called reducing it to lowest terms — means dividing the numerator and denominator by their greatest common divisor (GCD) so that no integer greater than 1 divides both. For example, 12/18 simplifies to 2/3 because GCD(12, 18) = 6: both 12 and 18 divide exactly by 6.

How do you find the GCD of two numbers?

This calculator uses the Euclidean algorithm: repeatedly replace the larger number with the remainder when the larger is divided by the smaller, until the remainder is zero. The last non-zero remainder is the GCD. For 12 and 18: 18 mod 12 = 6, then 12 mod 6 = 0, so GCD = 6. It is fast and exact for any pair of integers.

What is a fraction in lowest terms?

A fraction N/D is in lowest terms (or simplest form) when GCD(N, D) = 1 — the only positive integer that divides both the numerator and denominator evenly is 1. At that point, no further reduction is possible.

Can I simplify a fraction with a negative numerator or denominator?

Yes. The calculator handles negative values: it carries the sign on the numerator and keeps the denominator positive (the standard mathematical convention). For example, 4/(-6) is normalised and simplified to -2/3.

Can I enter a mixed number?

Yes — switch to "Paste fraction" mode and type a mixed number such as 2 3/4. The calculator converts it to an improper fraction (11/4), then checks whether it can be reduced. It also shows the mixed-number form of any improper result.

What if the GCD is 1?

If the GCD of the numerator and denominator is already 1, the fraction is already in its simplest form and the calculator tells you so. No division step is needed.

Simplify Fraction Calculator

Reducing a fraction to its lowest terms is a fundamental arithmetic skill — and one where small errors are easy to make when the numbers are large. This calculator handles the entire process for you: it finds the greatest common divisor (GCD), divides both the numerator and denominator by it, normalises the sign, and presents every intermediate step in plain language.

How it works

A fraction N/D is in simplest form when GCD(N, D) = 1 — the only positive integer dividing both parts is 1. To get there, the tool runs the Euclidean algorithm on the absolute values of the numerator and denominator:

GCD(a, b): while b is not 0, replace (a, b) with (b, a mod b). The last non-zero value is the GCD.

Once the GCD is known, both parts are divided by it:

simplified N = N / GCD(N, D) simplified D = D / GCD(N, D)

If the original denominator was negative, the sign is moved to the numerator so that the denominator is always positive — the conventional representation. The result is displayed as a fraction, optionally as a mixed number (when the numerator is larger than the denominator in absolute value), and as a decimal.

Text diagram of the reduction

  12        12 ÷ 6        2
 ----  -->  --------  =  ---
  18        18 ÷ 6        3

The GCD (6 in this case) is the “common factor” that cancels from both parts simultaneously.

Worked example

Start with 36/48.

Find GCD(36, 48): 48 mod 36 = 12, then 36 mod 12 = 0, so GCD = 12.
Divide: numerator 36 / 12 = 3, denominator 48 / 12 = 4.
Result: 3/4 (lowest terms, since GCD(3, 4) = 1).
Decimal: 3 / 4 = 0.75.

Another example: -15/25.

GCD(15, 25) = 5.
Divide: -15 / 5 = -3, 25 / 5 = 5.
Result: -3/5; decimal: -0.6.

And an improper fraction: 22/8.

GCD(22, 8) = 2.
Divide: 22 / 2 = 11, 8 / 2 = 4.
Result: 11/4; as a mixed number: 2 3/4; decimal: 2.75.

The formula

For integers N and D with D not equal to zero, the simplified fraction is:

N/D = (N / g) / (D / g), where g = GCD(|N|, |D|)

The GCD is found by repeated application of:

GCD(a, b) = GCD(b, a mod b) until b = 0

This algorithm terminates in O(log(min(a, b))) steps, making it extremely fast even for very large numbers.

Why lowest terms matter

Fractions in lowest terms are easier to compare, add, subtract, and reason about. When a recipe says 2/4 cup of flour, you probably want to think of it as 1/2. When a probability comes out as 15/60, expressing it as 1/4 immediately tells you there is a one-in-four chance. Teachers, examiners, and most real-world contexts expect answers in simplest form — leaving 12/18 as the final answer rather than 2/3 is technically equivalent but considered incomplete.