We are given the diagram of a shaded portion of a circular ring. Let’s sketch and label the diagram. As seen below, we can also use a specific formula for area of the shaded region in a ring.



To determine the area of the shaded ring we can use the formula, where a = radius of the smaller circle and b = radius of the larger circle:

Area of shaded ring = π(b^2 – a^2)

In this particular problem we are given that the area of the shaded ring is 3 times the area of the smaller circular region. We know that the area of the smaller region is πa^2, so we can create the following equation:

π(b^2 – a^2) = 3πa^2

b^2 – a^2 = 3a^2

b^2 = 4a^2

b = 2a

Since the radius of the larger circle is twice the radius of the smaller circle, the circumference of the larger circle is also twice the circumference of the smaller circle.

Answer: C
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if the radius of larger circle is r and that of the smaller circle is a then (pi*r*r-pi*a*a)/pi*a*a=3 which gives us -> r:a=2:1
as circumference ratio is 2*pi*r:2*pi*a, the ratio is also 2:1 or option C

Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer.



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
(E) √2


Let the radius of larger circle be B, and that of smaller be A.
Then the area of the shaded region is pi*B^2 - pi*A^2 = pi*(B^2 – A^2).
So by the assumption pi*(B^2 – A^2)=3 * pi*A^2. --> pi*B^2= 4*pi*A^2 ---> B^2=4*A^2 ---> B=2A.

So the answer is (C)(since circumference is propotional to the radius, the circumference of larger circle is 2 times that of smaller circle).
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Let the radius of the larger circle be R and the smaller one be r. By problem. The area of shaded region is thrice the area of the smaller circle. i.e. pi(R^2-r^2)=3pi (r^2).

Solving we get
R=2r..----(1)

Circumference of smaller circle is 2pi (r) and that of the larger circle is 2pi(R)...
Substituting (1)
Circum of smaller cirlce is 2pi(r) and that of larger circle is {2pi(2r)=4pi(r)}.

Circum of larger circle/Circum of smaller circle= 4pi(r)/2pi(r)=2

Option C

Your assumption is valid when it says "outer ring" and not the "larger circle". Circles by definition have only one circumference. You make an interesting point however - GMAT might sneak in the word "ring" for a tough problem.

suppose the radius of the small circle, not shaded, is r1
and the radius of the big circle is r2.
now, the area of the shaded region is 3 times the area of the not shaded region:

(pi*r2 - pi*r1)/pi*r1 = 3/1
simplify by pi

(r2-r1)/r1=3/1
now cross multiply:
3r1 = r2-r1
add r1:
2r1 = r2
Ok, so the radius of the big circle is twice the radius of the small circle:

Circumference is 2*pi*r

now, write everything as:
C of big one:
2*pi*2r1
C of the small one:
2*pi*r1

divide C big / C small, and the answer is 2.

Merging the topics. Please refer to the discussions above and follow the rules for posting.

Rules for posting: rules-for-posting-please-read-this-before-posting-133935.html

eliminating ⫪,
lc area/sc area=4/1=2^2/1^2
lc circ/sc circ=(2)(2)/(1)(2)=2/1

Hi, I have a question. Can I just take smart number?

I put the small radius = 1 => Area = 1pi => The shaded area = 3pi => The big radius must be 2 (4pi-1pi=3pi for the shaded area)

So 2/1 = 2 Answer C (Because the circumferences formula simplify themselves)

First, let's write down what we have:
r-small radius
l-large radius

(l^2-r^2)/r^2=3
we need to find l/r
from the equation above we can calculate that 4r^2=l^2
l=2r
2r/r=2
Answer is C