In the diagram, the four circles have a common centre, and have radii of $1,2,3$, and 4 . The ratio of the area of the shaded regions to the area of the largest circle is
(A) $5: 8$
(B) $1: 4$
(C) $7: 16$
(D) $1: 2$
(E) $3: 8$

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-124.jpg?height=231&width=247&top_left_y=340&top_left_x=1362)
Since the largest circle has a radius of 4 , its area is $\pi\left(4^{2}\right)=16 \pi$.

We must calculate the area of each of the shaded regions.

The innermost shaded region is a circle of radius 1 , and so it has area $\pi\left(1^{2}\right)=\pi$.

The outermost shaded region is the region inside a circle of radius 3 and outside a circle of radius 2. Therefore its area is the difference between the areas of these two circles, or $\pi\left(3^{2}\right)-\pi\left(2^{2}\right)=5 \pi$.

Therefore, the total area of the shaded regions is $\pi+5 \pi=6 \pi$, and the required ratio is $6 \pi: 16 \pi=6: 16=3: 8$.

ANSWER: (E)