Reduce a Fraction

Simplify fractions using the greatest common divisor

Fraction Simplifier

What is calculated?

This tool reduces a fraction to its simplest form. The greatest common divisor (gcd) of numerator and denominator is computed and both are divided by it. Optionally include a whole number part.

Enter fraction
Example: 12/15 = 4/5
Reduced fraction
The fraction will be reduced to its simplest form

Fraction Reduction Info

Properties

Reduction rules:

Find gcd Divide numerator & denominator Fully simplify Coprime

Note: A fraction is fully reduced when numerator and denominator are coprime (gcd = 1).

Examples
Simple: 6/8 = 3/4
Large: 24/36 = 2/3
Mixed: 2 12/15 = 2 4/5
Already reduced: 3/5 = 3/5

Formulas & Rules for Fraction Reduction

Principle
\[\frac{a}{b} = \frac{a \div d}{b \div d}\] d = gcd(a,b)
Greatest common divisor
\[\text{gcd}(a,b)\] Euclidean algorithm
Fully reduced
\[\text{gcd}(a,b) = 1\] Numerator and denominator coprime
Value preservation
\[\frac{a}{b} = \frac{a'}{b'}\] Equality preserved
Mixed numbers
\[n\frac{a}{b} = n\frac{a'}{b'}\] Only fractional part reduced
Euclidean algorithm
\[\text{gcd}(a,b) = \text{gcd}(b, a \bmod b)\] Recursive computation
Negative fractions
\[\frac{-a}{b} = \frac{a}{-b} = -\frac{a}{b}\] Sign preserved
Prime factorization
\[a = p_1^{a_1} \cdot p_2^{a_2} \cdots\] Alternative method

Step-by-step Example

Example: Reduce 12/15
Step 1: Find gcd of 12 and 15
\[\text{gcd}(12, 15)\] Divisors of 12: 1, 2, 3, 4, 6, 12
Divisors of 15: 1, 3, 5, 15
Common divisors: 1, 3
\[\text{gcd}(12, 15) = 3\]

The greatest common divisor is 3.

Step 2: Divide numerator and denominator by gcd
\[\frac{12}{15} = \frac{12 \div 3}{15 \div 3} = \frac{4}{5}\]
Step 3: Verify
\[\text{gcd}(4, 5) = 1\] The fraction is fully reduced.
Step 4: Result
\[\frac{12}{15} = \color{blue}{\frac{4}{5}}\]
More examples
Simple reduction:
\[\frac{6}{8} = \frac{3}{4}\] (gcd = 2)
Large numbers:
\[\frac{24}{36} = \frac{2}{3}\] (gcd = 12)
Already reduced:
\[\frac{7}{11} = \frac{7}{11}\] (gcd = 1)
Euclidean algorithm
For gcd(12, 15):
15 = 1 × 12 + 3
12 = 4 × 3 + 0
→ gcd = 3
Negative mixed number
\[-2\frac{2}{3}\] is interpreted as \[-(2\frac{2}{3})\]
The sign applies to the entire value.
Reduction steps
1. Compute gcd
2. Divide numerator
3. Divide denominator
4. Verify

The greatest common divisor determines how much the fraction can be reduced.

Applications of fraction reduction

Reduction is important in many mathematical and practical areas:

Simplifying calculations
  • Simpler numbers for computation
  • Lower chance of mistakes
  • Faster mental arithmetic
  • Clearer representation
Ratios & proportions
  • Mixing ratios
  • Recipe proportions
  • Scalings
  • Probabilities
Mathematical education
  • Understand equivalence
  • Number theory basics
  • Algorithmic understanding
  • Foundations of fraction arithmetic
Technical applications
  • Gear ratios
  • Frequency ratios
  • Material mixtures
  • CAD constructions

Mathematical context

Description

Fraction reduction is a fundamental concept in number theory and algebra. It relies on the greatest common divisor and the idea of equivalence of rational numbers. Reducing fractions yields simpler yet equivalent representations. The Euclidean algorithm for computing the gcd is one of the oldest known algorithms and demonstrates the elegant link between practical computation and theoretical mathematics.

Summary

Fraction reduction is more than an arithmetic rule — it's a tool for simplification and deeper insight into the structure of rational numbers. It teaches divisibility, primes and equivalence classes. From practical applications in craft and engineering to abstract mathematical theories, fraction reduction highlights the beauty and utility of mathematical thinking.

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