A sphere has a diameter of $$500\sqrt{3}$$ cm. A biggest cube is fitted in it. Now a biggest sphere is fitted within this cube. Again a biggest cube is fitted in the smaller sphere. The ratio of the valume of bigger cube to the valume of smaller cube is

Let us say that the biggest sphere (S1) has a diameter of $$500\sqrt{3}$$, which will be the diagonal of the cube (C1) fitted inside it. Hence, the side of the cube is of 500 cm.

Now, the side of C1 (500 cm) will be the diameter of another sphere (S2) fitted in it. 

In S2, a cube (C2) is fitted which shall have a diagonal equal to the diameter of S2. 

This means, the side of C2 is $$\ \frac{\ 500}{\sqrt{\ 3}}$$ cm. (as  $$a\sqrt{3}$$ = diameter).

Now, the ratio of volume of C1 to volume of C2 will be $$\ \frac{\ \left(500\sqrt{\ 3}\right)^3}{\ \ \ \left(\frac{\ 500}{\sqrt{\ 3}}\right)^3}$$, which is $$3\sqrt{\ 3}:\ 1$$