## Yin Yang Circle

Diameter $ACE$ is divided at $C$ in the ratio $2:3$ The two semicircles, $ABC$ and $CDE$, divide the circular region into an upper (shaded) region and a lower region. Find the ratio of the area of the upper region to that of the lower region.

Source: Julia Robinson Mathematics Festival

SOLUTION

Suppose without loss of generality that diameter $AC=2$ units and diameter $CE=3$ units.
Area of upper region = area of semicircle $CDE$ + area of semicircle $AFE\,-$ area of semicircle $ABC$
$=\frac{1}{2}\pi\!\left (\frac{3}{2}\right )^2+\frac{1}{2}\pi\!\left (\frac{5}{2}\right )^2-\frac{1}{2}\pi\!\left (\frac{2}{2}\right )^2$
$=\frac{1}{2}\pi\!\left (\frac{9}{4}+\frac{25}{4}-\frac{4}{4}\right )$
$=\frac{1}{2}\pi\!\left (\frac{30}{4}\right )$
$=\frac{15\pi}{4}$
Area of lower region = area of semicircle $ABC$ + area of semicircle $AGE-$  area of semicircle $CDE$
$=\frac{1}{2}\pi\!\left (\frac{2}{2}\right )^2+\frac{1}{2}\pi\!\left (\frac{5}{2}\right )^2-\frac{1}{2}\pi\!\left (\frac{3}{2}\right )^2$
$=\frac{1}{2}\pi\!\left (\frac{4}{4}+\frac{25}{4}-\frac{9}{4}\right )$
$=\frac{1}{2}\pi\!\left (\frac{20}{4}\right )$
$=\frac{10\pi}{4}$
Ratio of area of upper region to area of lower region
$\frac{15\pi}{4}\div \frac{10\pi}{4}$
$=\frac{15\pi}{4}\times \frac{4}{10\pi}$
$=\frac{3}{2}$

Answer: $\frac{3}{2}$.