Problem Solving

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

How about reasoning it out without solving formulas?

If the area of the shaded region is 3 times as much as that of the non-shaded region, then the ratio of the area of the big circle to that of the small circle is 4:1.

Area is based off of the square of the radius... so if the big circle has 4 times the area, it will have root 4 = 2 times the radius.

Since circumference is linear, if the big circle has twice the radius, it will also have twice the circumference.

Stuart Kovinsky wrote:How about reasoning it out without solving formulas?

If the area of the shaded region is 3 times as much as that of the non-shaded region, then the ratio of the area of the big circle to that of the small circle is 4:1.

Area is based off of the square of the radius... so if the big circle has 4 times the area, it will have root 4 = 2 times the radius.

Since circumference is linear, if the big circle has twice the radius, it will also have twice the circumference.

A very good point ... and a different way of looking at this !