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Solution

Consider the following diagram.

If we let the radius of the circle be $r$ then the area of the circle is given by $\pi r^2$ and the area of the outside square will be $(2r)^2 = 4r^2$. It should also be clear that the red square is exactly half the area of the outside square, so its area will be $2r^2$.

$\therefore$ (Area of Circle) / (Area of Large Square) $= \dfrac{\pi r^2}{4r^2} = \dfrac{\pi}{4} \approx$ 78.5%

   (Area of Small Square) / (Area of Circle) $= \dfrac{2r^2}{\pi r^2} = \dfrac{2}{\pi} \approx$ 63.7%

But to compare these ratios without the aid of a calculator we write both over the same denominator:

$$\frac{\pi}{4} = \frac{\pi^2}{4\pi}$$$$\frac{2}{\pi} = \frac{8}{4\pi}$$

As $\pi > 3, \pi^2 > 9,$   so it follows that $\dfrac{\pi}{4} > \dfrac{2}{\pi}$.