1. Radius of gyration formula

For a body of mass $M$, radius of gyration $k$ about a given axis is defined by $I = Mk^2$ where $I$ is the moment of inertia about that axis.

2. Ring about an axis through center, perpendicular to its plane

For a ring of radius $R_r$, $I_{\text{ring}} = MR_r^2$ So its radius of gyration is $k_{\text{ring}} = \sqrt{\frac{I}{M}} = \sqrt{\frac{MR_r^2}{M}} = R_r$

3. Solid sphere about an axis through its center

For a solid sphere of radius $R_s$, $I_{\text{sphere}} = \frac{2}{5}MR_s^2$ So its radius of gyration is $k_{\text{sphere}} = \sqrt{\frac{I}{M}} = \sqrt{\frac{\frac{2}{5}MR_s^2}{M}} = R_s\sqrt{\frac{2}{5}}$

4. Given same radii of gyration

According to the question, $k_{\text{ring}} = k_{\text{sphere}}$ Therefore, $R_r = R_s\sqrt{\frac{2}{5}}$ Hence, $\frac{R_r}{R_s} = \sqrt{\frac{2}{5}}$

5. Compare with the given form

Given, $\frac{R_r}{R_s} = \sqrt{\frac{2}{x}}$ So, $\sqrt{\frac{2}{x}} = \sqrt{\frac{2}{5}}$ Squaring both sides, $\frac{2}{x} = \frac{2}{5}$ Thus, $x=5$

6. Final answer

$\boxed{5}$