Solution:

The centers of the three new circles form a triangle. The diameter of the new circles is $R - r$, so the side length of the triangle is $R - r$. Call the center of the concentric circles $O$, two vertices of the triangle $A$ and $B$, and $AB$'s midpoint $D$. $OA$ is the average of $R$ and $r$, namely $\frac{R + r}{2}$. Using the law of sines on triangle $DAO$, we get $\frac{\sin(30)}{AD} = \frac{\sin(90)}{AO} \Rightarrow R = 3r$, so $\frac{R}{r} = 3$.