The rule
simplified = numerator ÷ gcd, denominator ÷ gcd
Why it works
Simplifying does not change a fraction's value, only how it is written. Dividing both the numerator and the denominator by the same number leaves the ratio untouched, and dividing by the greatest common divisor does it in one step.
How to do it by hand
- Find the greatest common divisor of the numerator and the denominator
- Divide both by it
- The result is in lowest terms: no further common factor exists
- If the denominator becomes 1, the fraction was a whole number
What is worth knowing
Using the greatest common divisor is what makes this a single step rather than a hunt. Dividing by any common factor works, but only the greatest one finishes the job at once — divide 18/24 by 2 and you get 9/12, which still simplifies. Euclid's algorithm finds the GCD quickly by repeated division with remainder, and it is over two thousand years old and still the method computers use. A fraction is irreducible exactly when its numerator and denominator are coprime, which is the same thing said in the language of number theory.