Fraction Simplifier
By Talcart · Last updated July 10, 2026
Fraction Simplifier
A fraction simplifier reduces any fraction to its lowest terms in one step: enter 12/18 and it returns 2/3, because the greatest common factor of 12 and 18 is 6 and dividing both parts by 6 leaves nothing more to cancel. It works on proper, improper, and negative fractions, and shows the GCF used at each stage.
Key facts
- The probability that two randomly chosen integers are already coprime - needing no simplification - is 6/pi^2, about 60.8%.
- By Lame's theorem (1844), the Euclidean algorithm never needs more than five times as many steps as the smaller number has decimal digits; the worst case occurs at consecutive Fibonacci numbers.
- Anomalous cancellation is a famous trap: 16/64 = 1/4 looks like you can "cancel the 6s", but the shortcut is a coincidence that fails for almost every other fraction.
What is the Fraction Simplifier?
A fraction is in lowest terms - also called simplest form or an irreducible fraction - when its numerator and denominator share no common factor greater than 1. Simplifying never changes a fraction's value, only its representation: 12/18, 6/9, and 2/3 all mark the same point on the number line, but 2/3 is the canonical form. Reduced fractions are easier to compare, combine, and grade, which is why textbooks and exam mark schemes expect final answers in lowest terms.
How does the Fraction Simplifier work?
The simplifier computes the greatest common factor of the numerator and denominator with the Euclidean algorithm - repeatedly replacing the larger number with the remainder of the division until the remainder is 0 - then divides both parts by that GCF. For 12/18: gcf(12, 18) = 6, so 12/6 = 2 and 18/6 = 3, giving 2/3. Because the Euclidean algorithm needs only O(log n) steps, even fractions with very large terms reduce instantly. A negative sign is normalized onto the numerator, and improper results can also be shown as mixed numbers.
What is the Fraction Simplifier formula?
- gcf – greatest common factor
Worked simplification examples
| Fraction | GCF | Simplified |
|---|---|---|
| 12/18 | 6 | 2/3 |
| 8/12 | 4 | 2/3 |
| 24/36 | 12 | 2/3 |
| 45/60 | 15 | 3/4 |
| 50/100 | 50 | 1/2 |
| 21/49 | 7 | 3/7 |
| 16/64 | 16 | 1/4 |
| 100/250 | 50 | 2/5 |
How do you use the Fraction Simplifier?
- Enter the fraction.
- Read the simplified form.
Worked example
| Scenario | 12/18 |
| Calculation | gcf(12,18)=6; 12/6 ÷ 18/6 |
| Result | 2/3. |
Common use cases
Tips & best practices
If GCF = 1, the fraction is already simplified.
Frequently asked questions
12/18 simplifies to 2/3. The greatest common factor of 12 and 18 is 6, so divide both parts by 6: 12 / 6 = 2 and 18 / 6 = 3. You can also reduce in stages - divide by 2 to get 6/9, then by 3 to get 2/3 - but using the GCF finishes the job in one step.
A fraction is fully simplified when the greatest common factor of its numerator and denominator is 1, meaning the two numbers are coprime. For example, 2/3 is fully reduced because gcf(2, 3) = 1, while 4/6 is not because both are divisible by 2. If any prime divides both the top and the bottom, more cancelling remains.
Use the Euclidean algorithm to find the GCF, then divide once. For 462/1071: 1071 = 2 x 462 + 147, then 462 = 3 x 147 + 21, then 147 = 7 x 21 + 0, so the GCF is 21 and 462/1071 = 22/51. Hunting for common factors by trial division is much slower for large numbers.
Yes - simplification works identically whether the numerator is smaller or larger than the denominator. For example, 10/4 reduces to 5/2 because gcf(10, 4) = 2. You can then optionally rewrite 5/2 as the mixed number 2 1/2. Reduce first, convert second: the arithmetic is easier with smaller numbers.
Simplify the absolute values normally and keep a single negative sign, conventionally on the numerator. So -12/18 reduces to -2/3, and 12/-18 is also written -2/3. If both the numerator and denominator are negative, the signs cancel and the result is positive: -12/-18 = 2/3.
No. Simplifying divides the numerator and denominator by the same nonzero number, which is equivalent to multiplying the fraction by 1. That is why 12/18 = 2/3 exactly: both equal 0.6666... as decimals. Only the representation gets shorter; the quantity, and any calculation built on it, is unchanged.