### Calculating equivalent fractions

As we saw above, \(\dfrac{1}{2}\) is equivalent to \(\dfrac{3}{6}\).

There are two steps of length \(\dfrac{1}{2}\) to go from 0 to 1.

There are six steps of length \(\dfrac{1}{6}\) to go from 0 to 1.

There is one step of length \(\dfrac{1}{2}\) to go from 0 to \(\dfrac{1}{2}\).

There are three steps of length \(\dfrac{1}{6}\)to go from 0 to \(\dfrac{1}{2}\).

We can see that the numerator of \(\dfrac{1}{2}\) is multiplied by three and the denominator is multiplied by three.

\begin{align}\dfrac{1}{2}&=\dfrac{1×3}{2×3}\\\\ &=\dfrac{3}{6}\end{align}We can also see that the numerator of \(\dfrac{3}{6}\) is divided by three and the denominator is divided by three.

\begin{align}\dfrac{3}{6}&=\dfrac{3÷3}{6÷3}\\\\ &=\dfrac{1}{2}\end{align}In general, to form equivalent fractions from some given fraction, multiply or divide both the numerator and the denominator of the given fraction by the same (non-zero) whole number.

### Negative fractions and the number line

The process for calculating negative equivalent fractions is the same. The number line below demonstrates that \(–\dfrac{6}{8}=\ –\dfrac{3}{4}, –\dfrac{4}{8}=\ –\dfrac{1}{2}\ \text{and}\ –\dfrac{2}{8}=\ –\dfrac{1}{4}.\)