Answer is 2.

Just plug the equations in assuming the bigger circle is of radius R and the smaller circle is of radius r. you will get the answer.

Calista.

Sorry StarDust845! I cann't be agree with u. According to me,
Let R = radius of large circle
r = radius of small circle
Now 3 * pi * r^2 = pi * (R - r)^2
So (R - r) = sqrt(3) * r

Now circumference of big circle : circum of small circle = 2 * pi * (R - r) / 2 * pi * r = (R - r) / r = sqrt(3).
So IMO D

agree with amitava D

Area of Shaded Region = pi* (R^2 - r^2)
Area of Smaller circle = pi*r^2

Hence (R^2 - r^2)/r^2 = 3

R/r = sqrt(2)

camitava wrote:Sorry StarDust845! I cann't be agree with u. According to me,
Let R = radius of large circle
r = radius of small circle
Now 3 * pi * r^2 = pi * (R - r)^2
So (R - r) = sqrt(3) * r

Now circumference of big circle : circum of small circle = 2 * pi * (R - r) / 2 * pi * r = (R - r) / r = sqrt(3).
So IMO D

I am afraid the equiation should be 3*pi*r^2=pi * (R^2-r^2), answer should be 2.

Sorry guys! It's my mistake - I admit ...

i think answer is 3.

3 pi r^2 = pi(R-r)^2
R-r =sqrt 3 r
R = 1.7r + r
R/r =1.7 +1
= 2.7 or approx 3

Let R be the radius of bigger circle and r be the radius of the smaller one.
Then
*Note Pi is denoted by #
((#R^2)-(#r^2)/(#r^2))=3/1
=> #R^2=4#r^2
=> R=2r

Ratio of Circumference of the two circles is the ratio of the radius
Hence R/r=2 from above.
Correct answer is 2.

Auzbee wrote:Let R be the radius of bigger circle and r be the radius of the smaller one.
Then
*Note Pi is denoted by #
((#R^2)-(#r^2)/(#r^2))=3/1
=> #R^2=4#r^2
=> R=2r

Ratio of Circumference of the two circles is the ratio of the radius
Hence R/r=2 from above.
Correct answer is 2.

how do you get from
"((#R^2)-(#r^2)/(#r^2))=3/1"
to
"=> #R^2=4#r^2"

???

thanks.

how do you get from
"((#R^2)-(#r^2)/(#r^2))=3/1"
to
"=> #R^2=4#r^2"

Solve it like this:
=> (a-b)/b=3/1
=> a-b=3b
=> a=4b

where a=#R^2 and b=#r^2.

I guess the answer should be C.

I am with Auzbee on this. I also think that the answer is 2. Does anyone actually have the official answer:

How I came to the answer, with some help from Auzbee !

Let R = big circle radius
Let r = small circle radius

Big circle area = ∏R²
Small circle area = ∏r²

Shaded area = Big circle area - Small circle area
Shaded area = ∏R² - ∏r²

The question tells us that the shaded area is 3 time the small circle area.

Shaded area = 3 x ∏r²
∏R² - ∏r² = 3(∏r²)
∏R² = 3(∏r²)+(∏r²)
∏R² = 4(∏r²)

Lets get rid of the squares ... so we have to apply √ to both sides of the equation:

√∏R² = √4(∏r²)
∏R = 2∏r
∏R/∏r = 2∏r/∏r
∏R/∏r = 2
Remove ∏ from both numerator and denominator from left hand side:
R/r = 2

So the radius of the big circle (R) is twice as big as the small circle radius (r).

Therefore the circumference will also be twice as big, so the answer is C.

Thanks.
II

I should read all the posts before posting mine. Camitava corrected his solution earlier.. so deleting my comments and posting only the solution.

<deleting>


Given: PI (R^2 - r^2) = 3PIr^2 which implies (R/r)^2 = 4 and hence R/r = 2

That is what we want. Hence the ratio is 2 that is C in the given choices.

Calista.

i agree with 2,
amit dont you think that area of shaded circle is
pi*R^2- pi*r^2 and not pi*(R-r)^2