The ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring is:-

Disc: moment of inertia about a diameter $= \dfrac{1}{4}MR^2$. By the parallel axis theorem, about a tangent in its plane:
$I_{disc} = \dfrac{1}{4}MR^2 + MR^2 = \dfrac{5}{4}MR^2 = MK_{disc}^2$
Ring: moment of inertia about a diameter $= \dfrac{1}{2}MR^2$. About a tangent in its plane:
$I_{ring} = \dfrac{1}{2}MR^2 + MR^2 = \dfrac{3}{2}MR^2 = MK_{ring}^2$
$\dfrac{K_{disc}^2}{K_{ring}^2} = \dfrac{5/4}{3/2} = \dfrac{5}{6}$
$\Rightarrow \dfrac{K_{disc}}{K_{ring}} = \sqrt{\dfrac{5}{6}}$
So the ratio of the radii of gyration is $\sqrt{5} : \sqrt{6}$.