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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R= Radius of larger circle
r= radius of smaller circle
Area of shaded region = Pi* R^2- Pi*r^2 = 3 *Pi*r^2
=>R=2r
Ratio of circumference of larger circle to smaller circle= (2 *pi*R)/(2*pi*r)
=2
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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Plug in some values for the the whole circle. I took r = 4. Which means, the whole circle has area of 16*pi which equals 4x of the area. So the shaded region is 3x, hence 12pi and the small region is 4pi. Therefore the small circle has radius 2 and diameter 4pi.

The large circle has diameter 8pi because i picked initially 4 as the radius (2*r = 8).

Hence the large circumference is double the small one. Answer C.

Answer C.

Important detail to note and not to get into trap is the following: the shaded area excludes the area of the smaller circle so we should not forget it in the calculation. Let the longer radi be "r" and the shorter radi be "t". Then:

(1) Shaded area relates to smaller circle area as Pi r^2 - Pi t^2 = 3 Pi t^2

(2) Boil down to r^2 = 4 t^2

(3) Finally \(\frac{r}{t}\)=\(\frac{2}{1}\)

(4) Hence circumference 2 Pi r relates to 2 Pi t as 2/1.

NOTE: If you miss out that the shaded area does not include the area of the smaller circle, you willr each answer D

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.

Let the radius of the outer circle be R
and that of the inner circle be r

To find \(x\):
\(2\pi\)R = \(x\) 2\(\pi\)r
=> \(\frac{R}{r} = x\)

Now,
Area of shaded region = \(\pi\)\(R^2\) - \(\pi\)\(r^2\) => \(\pi\)(\(R^2\)-\(r^2\))
Area of smaller circle = \(\pi\)\(r^2\)

Given,
=> \(\pi\)(\(R^2\)-\(r^2\)) = 3 (\(\pi\)\(r^2\))
=> \(R^2\) - \(r^2\) = 3\(r^2\)
=> \(R^2\) = 4\(r^2\)
=> \(\frac{R^2}{r^2} = 4\)
=> \(\frac{R}{r} = 2\)

Answer: 2

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.

Algebraically
Let a = Small circle's area
Let s = Shaded region's area
Let A = Large circle's area

\(s = 3a\)
\(A = s + a\)
\(A = 3a + a\)
\(A = 4a\)

\(a = \pi r^2\)
\(A = \pi R^2\)

From above:
\(A = 4a\)
\(\pi R^2 = 4\pi r^2\)
\(R^2 = 4r^2\)
\(\sqrt{R^2}=\sqrt{4r^2}\)
\(R = 2r\)*

Circumference, Large and Small
C = \(2\pi R\), and c = \(2\pi r\)
\(R = 2r\), so
\(\frac{C}{c}=\frac{2\pi (2r)}{2\pi r}= \frac{2r}{r}=2\)

The large circle's circumference is two times the small circle's circumference.

Answer C

Numbers and algebra

Large circle's area: Small circle's area?

Small circle's area = x
Shaded region's area = 3x
Large circle's area = 3x + x = 4x

Large circle circumference/ small circle's circumference?

Let Small circle's radius \(r = 1\)
Area of Small: \(\pi r^2 = \pi\)
Area of Large = (4x) = \((4*\pi) = 4\pi\)

Radius, R, of Large circle:
\(4\pi =\pi R^2\)
\(R = 2\)

Circumference, Small: \(2 \pi r = 2 \pi\)
Circumference, Large: \(2 \pi R = 4 \pi\)

\(\frac{Large}{Small}=\frac{4\pi}{2\pi} = 2\)

The large circle's circumference is two times the small circle's circumference.

Answer C

**OR
\(A = 3a + a\)
\(A - 3a = a\) AND \(A = \pi R^2\)

\(\pi R^2 - 3\pi r^2=\pi r^2\)
\(\pi (R^2 - 3r^2)=\pi r^2\)
\(R^2 - 3r^2 = r^2\)
\(R^2 = 4r^2\) Take square roots:
\(R = 2r\)
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Hi All,

This question can be solved by TESTing VALUES.

We're told that the area of the shaded region is 3 TIMES the area of the central circle...

Area of center = 1pi
Area of shaded region = 3(1pi) = 3pi
Area of FULL CIRCLE = 1pi + 3pi = 4pi

With those values....
Radius of center = 1
Radius of FULL CIRCLE = 2

The question asks how many times the circumference of the full circle is to the smaller circle...

Circumference of small circle = 2pi
Circumference of full circle = 4pi

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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Hi Bunuel,

Can you share problems that are similar to this?

Thanks.

Shaded Region Problems from our Special Questions Directory.
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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)π = area of shaded region

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

(A^2 - B^2)π = 3πB^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 4Bπ, and the circumference of the smaller circle is 2Bπ, so the circumference of the larger circle is twice that of the smaller circle.

Answer: C
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Given: In the figure shown, if the area of the shaded region is 3 times the area of the smaller circular region.

Asked: The circumference of the larger circle is how many times the circumference of the smaller circle?

Let the radius of larger circle be R and radius of smaller circle be r

Area of shaded region = \(\pi (R^2 - r^2)\)
Area of smaller circle = \(\pi r^2\)

\(\pi (R^2 - r^2) = 3\pi r^2\)
R^2 = 4 r^2
R = 2r

2\pi R = 2 * 2\pi r
The circumference of the larger circle is 2 times the circumference of the smaller circle.

IMO C
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Let, The radius of the larger circle \(= R\), and the radius of the smaller circle \(= r\)

So, the Area of the larger circle \(= πR^2\), and the are of the smaller circle \(= πr^2\)

Given that,
\(πR^2-πr^2=3πr^2\)

\(⇒πR^2=4πr^2\)

\(⇒R^2=4r^2\)

\(⇒R=\sqrt{4r^2}\)

\(⇒R=2r\)

The circumference of the larger circle \(= 2πR\), and the circumference of the smaller circle \(=2πr\)

The comparison: \(\frac{2πR}{2πr}=\frac{2π*2r}{2πr}=2\)

The answer is \(C\)
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Very simple way to solve this problem

Assume : Smaller circle area is 9
Then : Shaded region area is 3 times the area of smaller region therefore -> 27
Therefore : Whole circle area is 36 ( Adding the smaller and the shaded region)

Now radius of whole circle is 6 (Since area is 36)
and radius of smaller circle is 3 (Since area is 9)

Therefore the circumference of the bigger circle is 2* the circumference of the smaller circle.
2*6*pi -> bigger circle & 2*3*pi smaller circle

I like your way of solving, and shaped it.

Let, Smaller circle area is 9,

Then the Shaded region area is \(= 9*3=27\) [Three times the area of the smaller region]

Area of the larger circle (whole circle) \(= 9+27=36\)

∴ The are of the larger circle \(πR^2=36, \ or \ πr=6\), or The perimeter \(2πR=12\)

The are of the smaller circle \(πr^2=9, \ or \ πr=3\), or The perimeter \(2πr=6\)

\( \frac{The \ perimeter \ of \ the \ larger \ circle \ 2πR =12}{ The \ perimeter \ of \ the \ smaller \ circle \ 2πr =6}=2\)

The answer is \(C\)
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I will go with the plug-inners here:

The area of the larger circle is 4 times that of the smaller circle.

If radius of smaller circle = x = 1 and radius of larger circle = y = 2, then

y^2 = 4(x^2) (times Pi on both sides)

The circumferences will be 4Pi and 2Pi, respectively.