Find the gravitational force of attraction between the ring and sphere as shown in the diagram, where the plane of the ring is perpendicular to the line joining the centres. If $R\sqrt8$ is the distance between the centres of a ring (of mass $m$) and a sphere (of mass $M$) where both have equal radius $R$.
The gravitational force from the sphere on any "chunk" $\Delta m$ of the ring is indeed $G M \Delta m / (3 R)^2$ (note the parentheses). However, this does not mean that the net force on the whole ring is just $G M m / (3 R)^2$. This is because the forces on different parts of the rings point in different directions, and so some of their components cancel out.
It is not too hard to see, from symmetry, that only the horizontal components of the force on each "chunk" of the ring are important. By using simple trigonometry, you can calculate the horizontal components of the force on each "chunk" of the ring, and you'll find that you get the desired result.
