From the symmetry of the figure, the two circles must be concentric. Let their centre be $O$. Let the radius of the semicircles be $r$. Then the radius of the outer circle is $2r$ and, by Pythagoras' Theorem, the radius of the inner shaded circle is $\sqrt{r^2+r^2}$, that is $\sqrt{2}r$.

So the radii of the two circles are in the ratio $\sqrt{2}:2$, that is $1:\sqrt{2}$, and hence the ratio of their areas is $1:2$.

*This problem is taken from the UKMT Mathematical Challenges.*