Say $$c = \frac{a}{b}$$ and $$f = \frac{b}{a}$$
where $a, b , c, d, e, f \in \mathbb{R}$
$c = $ quotient
$f = $ quotient of the reciprocal of a fraction
Is there any relationship between $c$ and $f$? Is it possible to find the quotient of the reciprocal of a fraction from the quotient of the fraction?
If we multiply $c$ by $f$, we get $$ cf = \left(\frac{a}{b} \right) \left(\frac{b}{a} \right) = \frac{ab}{ba} = \frac{ab}{ab} $$ Assuming that $b\neq 0$ and $a\neq 0$, this can be simplified to $$ cf = 1 $$ And therefore we have a relation between the two. This can be rewritten as $$ c = \frac{1}{f} \qquad \text{or, equivalently,} \qquad f = \frac{1}{c} $$
