How many times the circumference of the smaller circle?

[AAPL]

In the figure shown, if the area of the shaded region is 3 times the area of the smaller circular region, then the circumference of the larger circle is how many times the circumference of the smaller circle?

A. 4
B. 3
C. 2
$$D.\sqrt{3}$$
$$E.\sqrt{2}$$

The OA is C.

I don't have clear this PS question but I have an idea about how to solve it,

I can say that the area of the larger circle is,
$$A_{Lc}=\pi\cdot R^2$$
And the area of the smaller circle is,
$$A_{Sc}=\pi\cdot r^2$$
Also I know that the area of the shaded region is 3 times the area of the smaller circle, then
$$\pi\cdot R^2-\pi\cdot r^2=3\cdot\pi\cdot r^2$$
Now, I can get the difference between the smaller and larger radius, right?
$$\pi\cdot R^2=4\cdot\pi\cdot r^2\ then\ R^2=4\cdot r^2\ finally\ \frac{R}{r}=2$$
That's mean that the circumference of the larger circle is twice than the circumference of the smaller circle, right?

I appreciate if any expert explain it for me. Thank you so much.

[DrMaths]

In the figure shown, if the area of the shaded region is 3 times the area of the smaller circular region, then the circumference of the larger circle is how many times the circumference of the smaller circle?

A. 4
B. 3
C. 2
$$D.\sqrt{3}$$
$$E.\sqrt{2}$$

For speed, ignoring the proportionality constant pi gives:
W = Whole circle area = w^2, where w = radius of W
I = Inner circle area, i^2, where i = radius of I
S = Shaded area = 3I = 3i^2

W = I + S
w^2 = i^2 + 3i^2
w^2 = 4i^2
Square root both sides to get...
w = 2i
So w/i = 2
ANSWER = C

We could have included pi, but it would have all been divided by pi later, cancelling out - so it saved time ignoring it at the start.

[GMATGuruNY]

In the figure shown, if the area of the shaded region is 3 times the area of the smaller circular region, then the circumference of the larger circle is how many times the circumference of the smaller circle?

A. 4
B. 3
C. 2
$$D.\sqrt{3}$$
$$E.\sqrt{2}$$

Let the area of the smaller circle = π, implying that the shaded region = 3π and that the area of the larger circle = smaller circle + shaded = π + 3π = 4π.

Smaller circle:
Since the smaller circle has an area of π, we get:
πr² = π
r² = 1
r = 1.
Circumference = 2Ï€r = 2*Ï€*1 = 2Ï€.

Larger circle:
Since the larger circle has an area of 4Ï€, we get:
πr² =4π
r² = 4
r = 2.
Circumference = 2Ï€r = 2*Ï€*2 = 4Ï€.

The circumference of the larger circle (4Ï€) is two times the circumference of the smaller circle (2Ï€).

The correct answer is C.

[Scott@TargetTestPrep]

In the figure shown, if the area of the shaded region is 3 times the area of the smaller circular region, then the circumference of the larger circle is how many times the circumference of the smaller circle?

A. 4
B. 3
C. 2
$$D.\sqrt{3}$$
$$E.\sqrt{2}$$

If we let A = the radius of the larger circle and B = the radius of the smaller circle, then we can create the equation:

(A^2 - B^2)Ï€ = area of shaded region

Area of the smaller circle = πB^2; thus:

(A^2 - B^2)Ï€ = 3Ï€B^2

A^2 - B^2 = 3B^2

A^2 = 4B^2

A = 2B

Since the radius of the larger circle can be expressed as 2B, the circumference of the larger circle is 4BÏ€, and the circumference of the smaller circle is 2BÏ€, so the circumference of the larger circle is twice that of the smaller circle.

Answer: C