Simplify Fraction Calculator

What Does It Mean to Simplify a Fraction?

A fraction is in its simplest form (also called reduced form or lowest terms) when the numerator and denominator share no common factor other than 1. In other words, there is no whole number greater than 1 that divides evenly into both parts.

For example, \(\dfrac{6}{8}\) is not in lowest terms because both 6 and 8 are divisible by 2. Dividing both by 2 gives \(\dfrac{3}{4}\), which is fully simplified because 3 and 4 share no common factors.

The Euclidean Algorithm

The most reliable way to simplify a fraction is to find the Greatest Common Divisor (GCD) of the numerator and denominator, then divide both by it. The GCD is the largest whole number that divides evenly into both values.

The Euclidean algorithm finds the GCD efficiently without needing to list all factors. It works by repeatedly replacing the larger number with the remainder left after dividing:

Example — finding \(\gcd(48, 18)\):

$$48 = 2 \times 18 + 12 \implies \gcd(48,18) = \gcd(18, 12)$$ $$18 = 1 \times 12 + 6 \implies \gcd(18, 12) = \gcd(12, 6)$$ $$12 = 2 \times 6 + 0 \implies \gcd(12, 6) = 6$$

So \(\gcd(48, 18) = 6\), and therefore:

$$\dfrac{48}{18} = \dfrac{48 \div 6}{18 \div 6} = \dfrac{8}{3} = 2\dfrac{2}{3}$$

Alternative Method: Prime Factorization

Another approach is to factor both numerator and denominator into their prime factors, then cancel the common ones:

$$\dfrac{36}{48} = \dfrac{2^2 \times 3^2}{2^4 \times 3} = \dfrac{\cancel{2^2} \times \cancel{3} \times 3}{\cancel{2^2} \times 2^2 \times \cancel{3}} = \dfrac{3}{2^2} = \dfrac{3}{4}$$

The Euclidean algorithm is generally faster for large numbers, while prime factorization gives more insight into the structure of the fraction.

Mixed Numbers

When you enter a mixed number like \(2\,\dfrac{6}{8}\), the fraction part is simplified first, then combined back with the whole number: \(2\,\dfrac{6}{8} = 2\,\dfrac{3}{4}\).

If the fractional part simplifies to a whole number (e.g. \(\dfrac{4}{4} = 1\)), it is added to the whole number part.

When Is a Fraction Already in Lowest Terms?

If \(\gcd(\text{numerator}, \text{denominator}) = 1\), the fraction is already fully simplified. Numbers whose GCD is 1 are called coprime (or relatively prime). For example, \(\dfrac{7}{12}\) is already in lowest terms because \(\gcd(7, 12) = 1\).

Related Calculators

Once you have a simplified fraction, these tools can help you work with it further:

• Fraction Calculator — add, subtract, multiply, or divide two fractions with step-by-step solutions
• Fraction to Decimal Calculator — convert a simplified fraction to its decimal equivalent
• Decimal to Fraction Calculator — convert a decimal back to a fraction

Frequently Asked Questions

Can I simplify a fraction where the numerator is larger than the denominator?

Yes. An improper fraction like \(\dfrac{18}{12}\) simplifies to \(\dfrac{3}{2}\), which can also be written as the mixed number \(1\,\dfrac{1}{2}\). Both forms are shown in the result.

What if the GCD is 1?

If the GCD of the numerator and denominator is 1, the fraction is already in its simplest form and no further reduction is possible.

Does simplifying change the value of the fraction?

No. Dividing both the numerator and the denominator by the same number creates an equivalent fraction — one that represents exactly the same value. \(\dfrac{6}{8}\) and \(\dfrac{3}{4}\) are the same point on a number line.

What about negative fractions?

Enter a minus sign before the fraction: -12/18 simplifies to \(-\dfrac{2}{3}\). The sign is preserved; only the absolute values are used to find the GCD.

Can I enter a fraction with a zero denominator?

No — a zero denominator makes a fraction undefined. The calculator will show an error if you try.