jmc

The volume of a cone with radius $r$ and height $h$ is $(1/3)\pi r^{2}h$; the volume of a cylinder with similar dimensions is $\pi r^{2}h$. The cone has the same radius as the cylinder and half the height, so it has $1/3$ the volume of half the cylinder and thus has $1/2\cdot 1/3=1/6$ the volume of the whole cylinder. Hence the desired ratio is $\boxed{\frac{1}{6}}$.