The Three Circles

Let the radii of the smaller circles be $2r$ and the bigger circle be $3r$ . The shaded area is formed by 2 overlapping sectors $60^\circ$ each. Their area is $\frac{πr^{2}}{6}$ each, and $\frac{πr^{2}}{3}$ for both.

But we counted a region twice. An equilateral triangle to be exact. So we draw the equilateral triangle inside the shaded region. The area of the triangle is $\frac{r^{2}\sqrt{3}}{4}$ .

Subtracting the area of the triangle from the area of the two sectors gives us the area which is $r^{2}(\frac{π}{3} - \frac{\sqrt{3}}{4})$

The area of the biggest circle is $\frac{9πr^{2}}{4}$ .

Dividing the two areas and multiplying by 100, we get 8.6889 or approximately

$\boxed{8.69}$

Since the centers of the smaller circles are separated by a distance 1/3 the diameter of the big circle, the radii of the smaller circles are also 1/3 the diameter of the big circle. This also means that each of the smaller circles passes through the center of the other smaller circle. This means that the area of the shaded region is that of 2 sectors of the smaller circles subtended by angles of pi/3 radians, minus the area of the equilateral triangle they share in common that is formed by their centers and the upper point of intersection. If the radius of the smaller circles is R, then the area of the shaded region is

2 * (1/6) * pi * R^2 - (1/2) * R * (1/2) * sqrt(3 )* R =

[(pi/3) - (1/4) * sqrt(3)] * R^2.

Now the area of the big circle will be pi [(3/2) R]^2 = (9/4) pi R^2, giving us a fractional area of [(pi/3) - (1/4) sqrt(3)] / [(9/4) pi] = 0.0869 to 3 significant figures, or 8.69%.