Equivalent fractions

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

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.

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