Notes
similar rectangles solution

Solution to the Similar Rectangles Puzzle

Solution by Similarity

Let the short side of the small red rectangle be $1$ and the long side $a$. Then the blue rectangles and the smaller yellow rectangle have short side $a$ and so long side $a \times a = a^2$. The larger red rectangle therefore has short side $a^2$ and so long side $a \times a^2 = a^3$. This is the short side of the larger yellow rectangle so its long side is $a \times a^3 = a^4$. This is the same as $3$ lots of the short side of the small rectangle and $2$ lots of the long side of the blue, which is $3 + 2 a^2$. Thus:

This rearranges to $a^4 - 2a^2 - 3 = 0$ and this factorises as $(a^2 - 3)(a^2 + 1)$. Since $a$ is a length, and so positive, it must be $\sqrt{3}$.

The total area is $(a + a^3) \times (a^2 + a^4) = a^3(1 + a^2)^2 = 48\sqrt{3}$. The area of the red rectangles is $3 a + a^2 \times a^3 = 12 \sqrt{3}$. The red area therefore comprises $\frac{1}{4}$ of the design.