A ring and a solid sphere rotating about an axis passing through...

The moment of inertia of the ring about an axis perpendicular to the plane of ring is ${I}_{r} = m{k}^{2}$.

The moment of inertia of the sphere is

${I}_{s} = \dfrac{2}{5}m{R}^{2}s$

It is given that

$m{k}^{2} = m{R}^{2}r = \dfrac{2}{5}m{{R}_{s}}^{2} . . . ( i )$

From equation (i)

${R}_{r} = \sqrt{\dfrac{2}{5}}{R}_{s}$

From the question, it can be written that

$\dfrac{{R}_{r}}{{R}_{s}} = \sqrt{\dfrac{2}{x}}$

$⇒ \dfrac{{R}_{r}}{{R}_{s}} = \sqrt{\dfrac{2}{5}} ⇒ x = 5$