Given that the radii of gyration for the ring and the solid sphere are equal, we have:

$$ K_1 = K_2 $$

For the ring, the moment of inertia is:

$$ I_{ring} = mR_1^2 = mK_1^2 $$

Thus, the radius of gyration for the ring is:

$$ K_1 = R_1 $$

For the solid sphere, the moment of inertia is:

$$ I_{sphere} = \frac{2}{5}m'R_2^2 = m'K_2^2 $$

Hence, the radius of gyration for the solid sphere is:

$$ K_2 = \sqrt{\frac{2}{5}}R_2 $$

Since the radii of gyration are equal:

$$ R_1 = \sqrt{\frac{2}{5}}R_2 $$

Therefore, the ratio of the radius of the ring to that of the sphere is:

$$ \frac{R_1}{R_2} = \sqrt{\frac{2}{5}} $$

So, the value of $$x$$ is:

$$ x = 5 $$