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) \(\sqrt{3}\)
(E) \(\sqrt{2}\)



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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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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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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

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

Using larger radius as x and smaller one as y, we get : \(πx^{2} - πy^{2} = 3πy^{2}\) OR \(πx^{2} = 4πy^{2}\)............(1)

We need to find the ratio of the circumference, \(\frac{2πx}{2πy}= ?\) OR \(\frac{x}{y}= ?\)

From (1); \(\frac{x}{y} =\frac{2}{1}\)

Hence answer is C

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

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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Shaded Region Problems from our Special Questions Directory.
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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