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) 3√3
(E) 2√2

My 2 cents.
Plug in.

So the area of shaded region needs to be 3 times the area of smaller circular region.
Let's say smaller circular region has r=2, so the area would be 4 pi.
As the area of shaded region needs to be 3 times, then it would be 12 pi.
As the area of shaded region can be found by larger - smaller circles, then we have x - 12pi = 4pi.
Then the x would be 16 pi, which means that its radius is 4.

Therefore, the circumference of big circle is 8 pi and that of smaller circle is 4 pi.
C is the correct answer.

shaded region is 3 three times the area of inner circle.

so, total circle area is 4 times the area of inner circle.

big circle area/small circle area = 4/1

(if two triangles are similiar, if the length of the sides are in ratio, k : 1, the area must be k^2 : 1)
we can extend this concept here, since areas are in ratio 4:1, lengths(radius, diameter, circumference) must be in ratio 2:1

(C)

Hi pushpitkc,

i am trying to understand the above solution

if radius = 4 and the whole circle area is 16*pi how can it be equal to 4x ??

also how can shaded region area be 3x ? somehow cant wrap my mind around this this

is it " hence 12pi and the small region is 4pi" 12 pi is it area and what about 4pi

morever, when i read question stem " 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?"

i undestand it like this
pi*r^2 = area of circle
2pi *r = circimference of circle

let area of larger non shaded region is 6

then if if the area of the shaded region is 3 times the area of the smaller circular region, then area of smaller is 2, because 2 goes into 6 three times... this is just a first idea solution that came to my mind

have a great weekend

This is how I solved. Assume the radius of smaller circle r to be 2. then the area = PI (r)^2= Pi (2)^2 = 4 PI
Its also told that area of shaded region is 3 times the smaller circle = 12 PI
Now smaller circle + Shaded area = area of larger circle = 4Pi + 12Pi = 16 Pi
from this we can get the radius of Larger circle = Pi (R)^2 = 16 pi
cancelling pi from both sides = (R)^2 = 16 therefore R=4
Now that we know R we can find out Circumference of Larger circle = 2 PI R = 8 PI

Similarly we know the radius of smaller circle = 2 then Circumference of smaller circle = 4Pi
So to answer the main question that the circumference of the larger circle is how many times the circumference of the smaller circle?
= 8Pi / 4pi = 2 times = C