The rule
a/b = (a × k) ÷ (b × k) for any non-zero k
Why it works
Multiplying the numerator and denominator by the same number does not change a fraction's value, because you are multiplying by a disguised 1. Every fraction therefore has infinitely many equivalents, and exactly one in lowest terms.
How to do it by hand
- Reduce the fraction to lowest terms first
- Multiply both parts by 2, then by 3, then by 4
- Each result is a different way of writing the same value
- Only one of them, the irreducible one, has no common factor
What is worth knowing
The irreducible form is the canonical representative of the whole infinite family, which is what makes it useful: two fractions are equal exactly when they reduce to the same thing. That test is faster and more reliable than cross-multiplying, especially with larger numbers. Equivalent fractions are also what you construct every time you add fractions with different denominators — finding a common denominator means replacing each with an equivalent that happens to share one.