Carcass wrote:

In the figure above, the five circles have points in common as shown. P is the center of the largest circle, Q and R are centers of the medium-sized circles, and Q, P, and R are points on a straight line. What fraction of the largest circular region is shaded?
(A) \(\frac{1}{16}\)
(B) \(\frac{1}{8}\)
(C) \(\frac{3}{16}\)
(D) \(\frac{1}{4}\)
(E) \(\frac{1}{2}\)
Let's start by assigning a "nice" value to the radius of the smallest circle.
Let's say the
blue circle has radius
1
If the
blue circle has radius
1, then the DIAMETER of the
blue circle must be 2.
This means the radius of the
green circle must be
2
If the radius of the
green circle is
2, then the DIAMETER of the
green circle must be 4.
This means the radius of the
red circle must be
4
Area of circle \(= \pi r^2\)So, the area of the
red circle \(= \pi (4^2) = 16\pi\)
And the area of ONE
blue circle \(= \pi (1^2) = \pi\)
So, the area of TWO
blue circles \(= 2\pi\)
What fraction of the largest circular region is shaded?Answer \(= \frac{2\pi}{16\pi} = \frac{1}{8}\)
Answer: B