Answer
$\frac{1}{3}$
Work Step by Step
The shaded areas form a geometric series:
$(0.5a)^2, ((0.5)^2a)^2, ((0.5)^3a)^2 ,...$
where $a$ is the side length of the largest square. We, have
$a_1=(0.5)^2a^2, r= (0.5)^2$
Thus, the sum of all the areas is
$S=\frac{a_1}{1-r}=\frac{(0.5)^2a^2}{1-(0.5)^2}=\frac{1}{3}a^2$
and the ratio is
$\frac{S}{a^2}=\frac{1}{3}$
