---
tags:
  - prealgebra
  - fractions
---

Two fractions are equivalent if they represent the same value. To begin with we
can represent this visually:

![equiv-fractions.png](equiv-fractions.png)

_Each shaded area is taking up the same proportion of the whole._

The same properties can be represented arithmetically by multiplying the
numerator and denominator at each step by 2. Thus:

$$
\\frac{1 (\cdot 2)}{3 (\cdot 2)} = \frac{2}{6}
$$

Therefore the following rule obtains:

> If you start with a fraction and multiply both its numerator and denominator
> by the same value, the resulting fraction is equivalent to the original
> fraction.

$$
\\frac{a}{b} = \frac{a \cdot x}{b \cdot x}
$$

This process works in reverse when we invert the operator and use division:

$$
\\frac{2 (/ 2)}{6 (/ 2)} = \frac{1}{3}
$$

Thus:

> If you start with a fraction and divide both its numerator and denominator by
> the same value, the resulting fraction is equivalent to the original fraction.

$$
\\frac{a}{b} = \frac{a / x}{b / x}
$$