In the figure below, the outside square contains infinitely many squares, each of them with the same center and sides parallel to the outside square. The ratio of the side length of a square to the side length of the next larger square is $k$, where $0 < k < 1.$ The spaces between squares are alternately shaded as shown in the figure (which is not necessarily drawn to scale). The area of the shaded portion of the figure is $64\%$ of the area of the original square. What is $k$? $\textbf{(A) } \frac{3}{5} \qquad$ $\textbf{(B) } \frac{16}{25} \qquad$ $\textbf{(C) } \frac{2}{3} \qquad$ $\textbf{(D) } \frac{3}{4} \qquad$ $\textbf{(E) } \frac{4}{5}$
Solution
Let the outer square have side $1$. The squares have areas $1,\ k^2,\ k^4,\ \dots$, so the successive regions between them have areas $1-k^2,\ k^2-k^4,\ k^4-k^6,\dots$ Starting from the outermost one, every other region is shaded: $1-k^2,\ k^4-k^6,\ k^8-k^{10},\dots$.
These form a geometric series with first term $1-k^2$ and ratio $k^4$, so the shaded area is $$\frac{1-k^2}{1-k^4}=\frac1{1+k^2}.$$
Setting $\dfrac1{1+k^2}=\dfrac{64}{100}$ gives $k^2=\dfrac9{16}$, so $k=\dfrac34$.
Infinitely many concentric circles are drawn, such that the ratio of the radius of each circle to the radius of the next larger circle is $k$, where $0<k<1$. The regions between consecutive circles are shaded alternately, starting with the outermost ring shaded. (The innermost disk is not part of any ring.) The shaded area is $\frac{25}{34}$ of the area of the largest disk. What is $k$?
