Equivalent Fractions

Intuition and core definition

Equivalent fractions name the same number using different-sized unit parts. Multiplying or dividing numerator and denominator by the same nonzero number preserves value because it multiplies the fraction by $k/k=1$. Thus $\frac23=\frac{2\cdot4}{3\cdot4}=\frac8{12}$.

Notation, language, and conditions

In $\frac ab$, $a$ is the numerator and $b\ne0$ is the denominator. The equivalence $\frac ab=\frac cd$ can be checked by cross-products $ad=bc$, provided both denominators are nonzero. A fraction is in simplest form when numerator and denominator share no positive integer factor greater than $1$.

Why this idea matters

Equivalent fractions preserve one point on the number line while changing the number and size of the pieces used to name it.

A dependable method

1. Identify the known numerator or denominator and its scale factor to the target.
2. Multiply or divide both numerator and denominator by that same nonzero factor.
3. If no scale factor is obvious, use the equality $ad=bc$ to solve for the missing entry.
4. Reduce by the greatest common factor when simplest form is requested.
5. Check with cross-products, a diagram, or decimal values.

Worked example

Common mistakes