A charge $Q$ is divided into two parts of $q$ and $Q-q$. If the Coulomb repulsion between them when they are separated, is to be maximum, the ratio of $\frac{Q}{q}$ should be:

Let parts are separated at a distance of 'd'

So repulsion force,

$F=\frac{kq(Q-q)}{d^2}$

here, Q and d are constant so for maximum force differentiate the force w.r.t q,

$\frac{dF}{dq}=\frac{k}{d^2}(Q-2q)$

For maximum force, $\frac{dF}{dq}=0$

                   $\frac{k}{d^2}(Q-2q)=0$

                           $Q-2q=0$

                                   $\frac{Q}{q}=2$