For each of the three circles shown above, the diameter of the smaller circle is equal to the radius of the larger. If the area of the largest circle is 52π square inches greater than the area of the shaded region, then what is the area of the shaded region in square inches?
A. 4π
B. 6π
C. 12π
D. 16π
E. 26π

(C) Let’s denote the radius of the smallest circle by x inches. Then the radius of the black circle is 2x inches and the radius of the largest circle is 4x inches.

Now we use the formula πr² to find the areas of the circles. The areas of the circles are πx², π(2x)², and π(4x)² square inches respectively. The area of the shaded region is the area of the black circle minus the area of the smallest circle. It equals π(2x)² – πx² = 3πx² square inches.

We know that the area of the largest circle is 52π square inches greater than the area of the shaded region. Let’s write it down as the following equation:
π(4x)² = 52π + 3πx²
divide the both sides by π
16x² = 52 + 3x²
13x² = 52
x² = 4
x = 2
We know that the area of the shaded region is 3πx² = 12π square inches. The correct answer s C.
----------
The question does not say the larger circle passes through the centre of the largest circle. So, the radius of the largest circle cannot be said to be 4x.

You have apparently missed the word "each" in the question statement. It says Since a diameter is twice a radius, then the radii of the circles are x, 2x and (2 × 2x) respectively.