Answer
The moment of inertia will be equal if the solid sphere is rotating around an axis that is a distance of $\sqrt{\frac{4}{15}}~R$ from the center.
Work Step by Step
solid sphere: $I_1 = \frac{2}{5}MR^2$
hollow sphere: $I_2 = \frac{2}{3}MR^2$
Let $d$ be the distance from the center of the solid sphere to the axis of rotation. We can use the parallel axis theorem to solve this question.
$\frac{2}{5}MR^2 + Md^2 = \frac{2}{3}MR^2$
$Md^2 = \frac{2}{3}MR^2 - \frac{2}{5}MR^2$
$Md^2 = \frac{4}{15}MR^2$
$d^2 = \frac{4}{15}R^2$
$d = \sqrt{\frac{4}{15}}~R$
The moment of inertia will be equal if the solid sphere is rotating around an axis that is a distance of $\sqrt{\frac{4}{15}}~R$ from the center.
