To divide two Rational Numbers there is a simple rule:

$\frac{a}{b}~/~\frac{c}{d}~=~\frac{a}{b}~\times~\frac{d}{c}~=~\frac{ad}{bc}$

As to why we do this, that's more complicated. Here's a way of thinking about it.

We can think of a rational number as a stretching by the numerator, and shrinking by the denominator.

We can think of a rational number as a stretching by the numerator, and shrinking by the denominator. We think of dividing as undoing a multiplication. If we multiply by $\frac{c}{d}$ then we stretch by $c$ and shrink by $d.$ To undo that we must shrink by $c$ and stretch by $d.$ In other words, undoing a multiply by $\frac{c}{d}$ is a shrink by $c$ and stretch by $d,$ which we write as a multiply by $\frac{d}{c}.$

Thus dividing by $\frac{c}{d}$ is the same as multiplying by $\frac{d}{c}.$

Hence the rule.

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