Simplify fractions to lowest terms
Core idea
Divide numerator and denominator by their greatest common divisor (GCD) to get an equivalent fraction in simplest form.
How it works
- Compute GCD(|a|,|b|) with the Euclidean algorithm.
- Return (a/G, b/G) with a single overall sign and positive denominator.
- If |a| ≥ |b|, you can also express the result as a mixed number for readability.
Sanity checks
- GCD(|num|,|den|)=1 in the final fraction.
- Denominator > 0; sign only on the numerator.
- Original value equals simplified value when converted to decimal.
Shortcuts
- Both even → divide by 2; digit‑sum multiple of 3 → divide by 3.
- If one divides the other, that’s the GCD.
- Use prime factors only when numbers are small; otherwise use Euclid.
Pitfalls
- Stopping after dividing by a small factor when more reduction remains.
- Leaving a negative denominator.
- Confusing “simplify” with “find equivalents” (scaling instead of reducing).
Micro‑examples
- 12/18 → GCD=6 → 2/3
- −8/20 → GCD=4 → −2/5
- 45/15 → GCD=15 → 3/1 = 3
Mini‑FAQ
- Is simplest form unique? Yes, when denominator is positive.
- Why reduce first? It simplifies later operations and comparison.
- Decimal tie‑in? Equal decimals confirm equivalence (subject to display rounding).
Action tip
Enter the fraction and use the steps shown to confirm the GCD and each division—keep the simplified result for further work.