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Fraction Simplifier Calculator

Reduce any fraction or mixed number to lowest terms with the steps shown three ways — Euclid’s GCF, prime factorization or dividing by common factors one at a time — plus the mixed number, decimal (repeating digits marked), percent and equivalent fractions.

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Denominator (bottom)

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How it works

simplified = (numerator ÷ GCF) / (denominator ÷ GCF), where GCF is the greatest common factor of top and bottom · Euclid: GCF(a, b) = GCF(b, a mod b) until the remainder is 0 · mixed number w n/d = (w × d + n)/d

A fraction is in lowest terms when its numerator and denominator share no common factor other than 1. To get there, find the greatest common factor (GCF) of the two numbers and divide both by it in a single step. Dividing the top and bottom by the same amount never changes a fraction’s value, only how it is written, and the result is the fraction’s unique simplest form. Choose how the steps find the GCF. Euclid’s algorithm repeatedly divides and keeps the remainder, which is fast even for very large numbers. Prime factorization breaks both numbers into primes and cancels the ones they share. Common factors divides by one shared factor at a time, the way it is often done by hand. Switch on the mixed-number box to simplify something like 2 6/8. Decimals in the numerator or denominator are cleared first by multiplying by a power of 10. The answer is also shown as a mixed number, a decimal with any repeating digits in brackets, and a percent, with equivalent fractions and the reciprocal.

Worked example

18/24: Euclid’s algorithm gives 24 = 1 × 18 + 6, then 18 = 3 × 6 + 0, so the GCF is 6. Dividing both by 6 gives 3/4, which is 0.75 or 75%. With prime factors, 18 = 2 × 3 × 3 and 24 = 2 × 2 × 2 × 3 share 2 × 3 = 6, giving the same 3/4. One factor at a time, 18/24 → 9/12 (÷ 2) → 3/4 (÷ 3). An improper fraction such as 42/12 simplifies to 7/2 = 3 1/2. A decimal fraction such as 1.5/2.25 becomes 150/225 = 2/3 = 0.(6).

Sources: Irreducible fraction — Wikipedia · Euclidean algorithm — Wikipedia

Frequently asked questions

What does it mean for a fraction to be in lowest terms?

It means the top and bottom have no common factor except 1, so it can’t be reduced any further. 3/4 is in lowest terms; 6/8 has the same value but is not reduced.

Why divide by the greatest common factor instead of any factor?

You can cancel any shared factor, but the greatest one finishes the job in a single step. Dividing 18/24 by 2 gives 9/12, which can still be reduced; dividing by 6 lands on 3/4 immediately. The “Common factors” steps show the slower route.

How does the Euclidean algorithm find the GCF?

Divide the larger number by the smaller and keep the remainder, then divide the smaller number by that remainder, and repeat. When the remainder reaches 0, the last divisor is the GCF. For 18 and 24: 24 = 1 × 18 + 6, then 18 = 3 × 6 + 0, so the GCF is 6.

How do I simplify a mixed number?

Simplify the fraction part and keep the whole number: 2 6/8 = 2 3/4. Switch on “Enter a mixed number” to see it converted to 22/8 and reduced to 11/4 = 2 3/4.

What happens to a negative fraction?

The sign is kept on the numerator and the reduction is done on the absolute values, so −18/24 simplifies to −3/4. A negative denominator is moved up top for the standard form.