In the diagram, $P Q$ is a diameter of a larger circle, point $R$ is on $P Q$, and smaller semi-circles with diameters $P R$ and $Q R$ are drawn. If $P R=6$ and $Q R=4$, what is the ratio of the area of the shaded region to the area of the unshaded region?
(A) $4: 9$
(B) $2: 3$
(C) $3: 5$
(D) $2: 5$
(E) $1: 2$

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-023.jpg?height=363&width=290&top_left_y=716&top_left_x=1313)
Since point $R$ is on $P Q$, then $P Q=P R+Q R=6+4=10$.

Semi-circles with diameters of 10,6 and 4 have radii of 5,3 and 2, respectively.

A semi-circle with diameter $P Q$ has area $\frac{1}{2} \times \pi \times 5^{2}=\frac{25}{2} \pi$.

A semi-circle with diameter $P R$ has area $\frac{1}{2} \times \pi \times 3^{2}=\frac{9}{2} \pi$.

A semi-circle with diameter $Q R$ has area $\frac{1}{2} \times \pi \times 2^{2}=2 \pi$.

The shaded region consists of a section to the left of $P Q$ and a section to the right of $P Q$.

The area of the section to the left of $P Q$ equals the area of the semi-circle with diameter $P Q$ minus the area of the semi-circle with diameter $P R$.

This area is thus $\frac{25}{2} \pi-\frac{9}{2} \pi=8 \pi$.

The area of the section to the right of $P Q$ equals the area of the semi-circle with diameter $Q R$. This area is thus $2 \pi$.

Therefore, the area of the entire shaded region is $8 \pi+2 \pi=10 \pi$.

Since the large circle has radius 5, its area is $\pi \times 5^{2}=25 \pi$.

Since the area of the shaded region is $10 \pi$, then the area of the unshaded region is equal to $25 \pi-10 \pi=15 \pi$.

The ratio of the area of the shaded region to the area of the unshaded region is $10 \pi: 15 \pi$ which is equivalent to $2: 3$.

ANswer: (B)