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Let the radius of each semicircle be $r$.
In the top diagram, let the side-length of the square be $2x$. By Pythagoras' Theorem,
$x^2 + (2x)^2 = r^2$ and so $5x^2 = r^2$. So this shaded area is $(2x)^2 = 4x^2 = \frac{4r^2}{5}$.

In the bottom diagram, let the side-length of each square be $y$. Then by Pythagoras' Theorem, $y^2 + y^2 = r^2$ and so $2y^2 = r^2$. So this shaded area is $r^2$.

Therefore the ratio of the two shaded areas is $\frac{4}{5} : 1 = 4 : 5$.

This problem is taken from the UKMT Mathematical Challenges.
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