Fraction Simplifier

Understanding the fraction simplifier — concept and principle

Simplifying a fraction means rewriting it in lowest terms: the same value, but with a numerator and denominator that share no common factor other than 1. Students use it to check homework, cooks and builders use it to turn measurements like 6/8 inch into 3/4 inch, and anyone comparing fractions finds it easier when both are reduced.

How it is calculated

• GCD: find G, the greatest common divisor of the numerator a and denominator b.
• Simplified: (a ÷ G) / (b ÷ G).
• Decimal: a ÷ b.
• Mixed number: shown only when |a| ≥ |b|. The whole part is the integer quotient of a ÷ b, and the remainder over b is reduced by G. For proper fractions the field shows "—".

The principle

Multiplying or dividing the top and bottom of a fraction by the same non-zero number does not change its value, because you are really multiplying by G/G = 1. Dividing by the greatest common divisor removes all shared factors in a single step, so the result is guaranteed to be in lowest terms. Dividing by a smaller common factor, such as 2, also gives an equivalent fraction, but you may need to repeat the step. The GCD itself is found with the Euclidean algorithm: replace the larger number by the remainder of dividing it by the smaller, and repeat until the remainder is zero.

Worked example

1. Inputs: numerator 42, denominator 56.
2. Euclidean algorithm: 56 = 1 × 42 + 14, then 42 = 3 × 14 + 0, so GCD = 14.
3. Divide both terms: 42 ÷ 14 = 3 and 56 ÷ 14 = 4, giving 3/4.
4. Decimal: 42 ÷ 56 = 0.75.
5. Because 42 is smaller than 56, this is a proper fraction and the mixed-number field shows —.

How to read the result

If the GCD shown is 1, the fraction was already in lowest terms. A large GCD means the original numbers shared many factors. The decimal is useful for comparing sizes, while the reduced fraction is exact; for example 1/3 is exact but 0.333… is not.

Common mistakes

• Cancelling digits instead of factors, such as striking the 6 in 16/64. It happens to give 1/4 here, but the method is wrong.
• Cancelling terms in a sum: (2 + 3)/2 is not 3.
• Stopping after dividing by 2 when a larger common factor remains.
• Forgetting that a denominator of 0 is undefined.

Frequently asked questions

How do I simplify a negative fraction?

Reduce the absolute values and keep one minus sign, usually on the numerator: −42/56 becomes −3/4.

Is an improper fraction acceptable as a final answer?

In algebra, yes, improper fractions such as 7/4 are standard. In everyday measurement, the mixed number 1 3/4 is often easier to read.

How can I tell whether two fractions are equal?

Reduce both to lowest terms or cross-multiply: a/b = c/d exactly when a × d = b × c.

Sources: Euclid, Elements, Book VII, Propositions 1–2 (Euclidean algorithm) · D. E. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, Section 4.5.2