Consider an equilibrium reaction
The rate of the forward reaction, $r_f$ is equal to the rate of the backward reaction, $r_r$
$r_f=k_1[A],r_r=k_2[B]$
where $k_1$ and $k_2$ are the rate constants of the forward and reverse reactions respectively.
$⟹k_1[A]=k_2[B]$
$\frac{k_1}{k_2}=\frac{[B]}{[A]}=kf,kf=\frac{[B]}{[A]}$y the same approach, $k_r=\frac{[A]}{[B]}$
where $k_f$ and $k_r$ are the equilibrium constants of the forward and reverse reactions respectively.
$$\begin{array}{rl}{k}_f& =\frac{[B]}{[A]\times [B1/[B]}=\frac{[B]}{[A]/[B]\times [B]}\\ & =\frac{1}{[A//B]}=\frac{1}{k_r}\end{array}$$
i.e. $k_f$ is the multiplicative inver-
se of $k_r$