AMC 12 2025a Problem 5

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 inner 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).

unitsize(1cm);

int n = 25;
real s = 6;
real ratio = 0.767; //definitely not on purpose
real a = s;

for (int i = 0; i < n; ++i) {
real b = a * ratio;

// Draw current square
draw(box((-a/2,-a/2),(a/2,a/2)));

if (i % 2 == 1) {
fill(box((-a/2,-a/2),(a/2,a/2)), gray(1));
} else {
fill(box((-a/2,-a/2),(a/2,a/2)), lightred);
}

a = b;
}

draw(box((-a/2,-a/2),(a/2,a/2)));

The area of the shaded portion of the figure is $64\%$ of the area of the original square. What is $k$?

$\textbf{(A) } \frac{3}{5} \textbf{(B) } \frac{16}{25} \textbf{(C) } \frac{2}{3} \textbf{(D) } \frac{3}{4} \textbf{(E) } \frac{4}{5}$