Fraction Simplifier (Live Reduce Fractions with Steps Tool)
Reduce and simplify any proper or improper fraction into its simplest form (lowest terms). Features dual step-by-step derivation proofs using both the Greatest Common Divisor (GCD) and prime factorization cancellation.
To simplify a fraction, find the Greatest Common Divisor (GCD) of the numerator and denominator, then divide both numbers by that common factor: Simplified Fraction = (Numerator ÷ GCD) / (Denominator ÷ GCD). For example, 24/36 has a GCD of 12, reducing to 2/3.
The Two Mathematical Methods to Simplify Fractions
A fraction is in its simplest form (lowest terms) when the top numerator and bottom denominator share no common factors other than 1 (meaning they are coprime integers).
- The Greatest Common Divisor (GCD) Method: The standard mathematical technique. Compute the highest common factor that divides both numerator and denominator without remainder (e.g., for $24/36$, the GCD is 12). Dividing both numbers by 12 yields $2/3$ in a single step.
- The Prime Factorization Method: Break both numbers down into their constituent prime factor trees, then cancel identical matching factors in both the numerator and denominator:
24 = 2 × 2 × 2 × 336 = 2 × 2 × 3 × 3
Canceling two $2$s and one $3$ leaves $\mathbf{2/3}$.
Common Fractions Simplification Reference Matrix
| Unreduced Fraction | Greatest Common Divisor (GCD) | Simplified Form (Lowest Terms) | Decimal Form |
|---|---|---|---|
| 12 / 16 | 4 | 3 / 4 | 0.75 |
| 24 / 36 | 12 | 2 / 3 | 0.6667 |
| 45 / 60 | 15 | 3 / 4 | 0.75 |
| 18 / 4 | 2 | 9 / 2 (or 4 1/2) | 4.50 |
| 75 / 100 | 25 | 3 / 4 | 0.75 |
| 35 / 7 | 7 | 5 / 1 (or 5) | 5.00 |
Related Fraction & Mathematical Calculators:
- Mixed Number to Improper Fraction Calculator — Convert mixed numbers to improper fractions with step-by-step reduction proofs.
- Fraction to Percent Calculator — Convert proper fractions, improper fractions, and mixed numbers to percentages and decimals.
- Decimal to Fraction Calculator — Convert terminating and repeating decimals into simplified fractions and mixed numbers.
- Percent to Fraction Calculator — Convert percentages into simplified fractions and mixed numbers in lowest terms.
- Ratio & Proportion Calculator — Solve proportions A:B = C:D and simplify ratios.
Frequently Asked Questions
What does simplifying a fraction mean?
Simplifying a fraction (also known as reducing a fraction) means finding an equivalent fraction where the numerator and denominator are as small as possible by dividing out all shared common factors.
How do you know if a fraction is already in simplest form?
A fraction is in simplest form when the Greatest Common Divisor (GCD) of the numerator and denominator is 1. For example, in 3/4, the only number that divides both 3 and 4 evenly is 1.
How do you simplify an improper fraction like 18/4?
First, divide both 18 and 4 by their GCD of 2 to yield 9/2. Then, convert 9/2 into a mixed number by dividing 9 by 2, which equals 4 with a remainder of 1, producing 4 1/2.
What happens if the numerator is 0?
If the numerator is 0 (e.g., 0/5), the simplified value is simply 0. Zero divided by any non-zero number is always zero.
Is my homework data saved on a server?
No. All calculations run 100% locally inside your browser's runtime memory using JavaScript. No numbers, math homework entries, or fractions are transmitted over an HTTP network or saved in an external database.