First off, there is no "known ratio" since the volume of a sphere is a function of one variable and the volume of a (right-circular) cylinder is a function of two.There is a known ratio of sphere volume to cylinder volume.
Edit: just like cone volume.
But even if you constrain things so as to get your "known ratio" of 4/3, you can't use it for this problem because it is derived from the formulas for the volume of a sphere and the volume of a cylinder, which you are not allowed to use either directly or indirectly. You aren't even allowed to derive those formulas from first principles. So unless you can establish how to get that 4/3 ratio via some means that is independent of the volume formulas, you can't use it.