🔢 Fraction Simplifier – Reduce Any Fraction to Lowest Terms
The Fraction Simplifier reduces any integer fraction to its lowest terms using exact integer arithmetic and the Euclidean algorithm. Whether you're working with proper fractions, improper fractions, signed fractions, or mixed numbers, this tool delivers an exact simplified result along with a clear, step-by-step explanation.
What Does "Lowest Terms" Mean?
A fraction is in lowest terms (also called simplest form or reduced form) when the numerator and denominator share no common factor other than 1. For example, 84/126 simplifies to 2/3 because both 84 and 126 are divisible by their Greatest Common Divisor (GCD) of 42. Dividing both by 42 gives 2/3, which cannot be reduced further.
How the Euclidean Algorithm Works
The GCD is found using the Euclidean algorithm, one of the oldest and most efficient algorithms in mathematics:
gcd(84, 126)
84 = 126 × 0 + 84
126 = 84 × 1 + 42
84 = 42 × 2 + 0
→ GCD = 42Once the GCD is known, the simplified fraction is obtained by dividing both numerator and denominator by it:
numerator = 84 ÷ 42 = 2
denominator = 126 ÷ 42 = 3
Result: 2/3Supported Fraction Types
Proper Fractions
Numerator is smaller than the denominator — e.g. 12/18 → 2/3.
Improper Fractions
Numerator is larger than the denominator — e.g. 42/18 → 7/3, shown as mixed number 2 1/3.
Signed Fractions
Negative numerators or denominators are handled correctly — e.g. −18/−24 → 3/4. The sign is always moved to the numerator.
Mixed Numbers
Enter a whole part plus a fractional remainder in Mixed-Number mode — e.g. whole part 3, fraction 6/8 → converts to 30/8 → 15/4 = 3 3/4.
Sign Normalization
The tool always normalizes the sign so the denominator is positive. If you enter 4/−6, the result is displayed as −2/3, not 2/−3. This matches standard mathematical convention and avoids ambiguity when reading or copying the result.
Mixed Number Conversion
When the simplified fraction is improper (|numerator| ≥ denominator), the tool automatically computes the mixed-number form using:
wholePart = trunc(|numerator| / denominator)
remainder = |numerator| mod denominator
Mixed form: wholePart remainder/denominatorExample: 7/3 → 2 1/3 because 7 ÷ 3 = 2 remainder 1.
Optional Representations
Beyond the simplified fraction, the tool optionally shows:
- Decimal equivalent — the numeric value rounded to 6 significant digits (e.g.
2/3 ≈ 0.666667). - Percent equivalent — useful in probability and statistics contexts (e.g.
2/3 ≈ 66.6667%).
Common Use Cases
- Homework and exam prep — verify that your manual simplification is correct with an exact check and explanation.
- Algebra clean-up — reduce intermediate results before substituting fractions into equations.
- Cooking and measurement scaling — convert a messy fraction like
12/16cup to the cleaner equivalent3/4cup. - Probability — express event probabilities in their simplest form, e.g.
6/9 → 2/3. - Finance — simplify share ratios, rate fractions, or any numeric proportion before presenting to stakeholders.
Limitations and Notes
The tool uses JavaScript's native number type, which handles integers up to ±9,007,199,254,740,991 (Number.MAX_SAFE_INTEGER) exactly. For values beyond this range, results may lose precision. All inputs must be plain integers — decimal strings, scientific notation, and non-numeric characters are rejected.