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In the figure shown, if the area of the shaded region is 3 times

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In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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

Kudos for a correct solution.

Attachment:
2015-10-22_0821.png
2015-10-22_0821.png [ 7.04 KiB | Viewed 21522 times ]

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Originally posted by Bunuel on 21 Oct 2015, 21:22.
Last edited by Vyshak on 16 Apr 2016, 09:27, edited 1 time in total.
Edited the topic name
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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 02 May 2016, 09:31
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Bunuel wrote:
Image
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) \(\sqrt{2}\)

Kudos for a correct solution.

Attachment:
2015-10-22_0821.png


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.

Image

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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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 21 Oct 2015, 23:28
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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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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 21 Oct 2015, 23:11
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Bunuel wrote:
Image
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) \(\sqrt{2}\)

Kudos for a correct solution.


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
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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 21 Oct 2015, 23:32
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answer is option c.

ratio of areas of shaded area to smaller circle = 3
so ratio of area of larger and smaller circle = 4
so ratio of radius = ratio of circumference = 2
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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 22 Oct 2015, 00:41
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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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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 23 Oct 2015, 00:50
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Bunuel wrote:
Image
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) \(\sqrt{2}\)

Kudos for a correct solution.

Attachment:
2015-10-22_0821.png


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.
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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 23 Oct 2015, 08:27
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
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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 30 Oct 2015, 01:10
I had a query, From what i got from most of the answers the answer shown is '2' Since R = 2r

Although Isn't the circumference of the larger circle, the outer circumference + the Inner circumference i.e.
R + r => 2r+r => 3r

and therefore 3 should be the answer?

Bunuel wrote:
Image
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) \(\sqrt{2}\)

Kudos for a correct solution.

Attachment:
2015-10-22_0821.png
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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 30 Oct 2015, 11:48
kelvind13 wrote:
I had a query, From what i got from most of the answers the answer shown is '2' Since R = 2r

Although Isn't the circumference of the larger circle, the outer circumference + the Inner circumference i.e.
R + r => 2r+r => 3r

and therefore 3 should be the answer?

Bunuel wrote:
Image
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) \(\sqrt{2}\)

Kudos for a correct solution.

Attachment:
2015-10-22_0821.png


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.
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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 30 Oct 2015, 22:10
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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
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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 02 Dec 2015, 19:23
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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.
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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 16 Apr 2016, 08:43
I have a question about the solution to this problem.

In part of the solution it shows R^2 = 4r^2 then R=2r. How is this accomplished? I would think you cannot take the square root of the the entire (4r^2) as the quare only applies to the r since there is no bracket around the 2r. I.e. (2r)^2.

Is the trick to assume 1R^2 = 4r^2 ? Or do I have rule wrong and brackets are not need around the 4r^2 (i.e. the bracket square applies to both the 4 and the r?


Thanks in advance for any help!
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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 16 Apr 2016, 09:25
grainflow wrote:
I have a question about the solution to this problem.

In part of the solution it shows R^2 = 4r^2 then R=2r. How is this accomplished? I would think you cannot take the square root of the the entire (4r^2) as the quare only applies to the r since there is no bracket around the 2r. I.e. (2r)^2.

Is the trick to assume 1R^2 = 4r^2 ? Or do I have rule wrong and brackets are not need around the 4r^2 (i.e. the bracket square applies to both the 4 and the r?


Thanks in advance for any help!


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

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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 16 Apr 2016, 11:38
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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
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In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 16 Apr 2016, 13:53
eliminating ⫪,
lc area/sc area=4/1=2^2/1^2
lc circ/sc circ=(2)(2)/(1)(2)=2/1
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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 16 Sep 2016, 18:45
Bunuel wrote:
Image
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) \(\sqrt{2}\)

Kudos for a correct solution.

Attachment:
2015-10-22_0821.png



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

Given pi*R^2 - pi*r^2 = 3pi*r^2 ==> R^2 = 4r^2 ==> R = 2r (since radius will always be positive)

Larger Circumference / Smaller Circumference = 2*pi*R / 2*pi*r ==> R/r ==> R/ r/2 ==> 2

So C is correct answer.

We can generalise if radius of one circle is twice that of 2nd circle, area of larger circle will be 4 times that of smaller circle and circumference will be twice that of smaller circle.
similarly if area is 4 times or circumference is twice, then radius of two circles will be in ratio of 1:2
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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 08 Jan 2017, 07:18
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)
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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 09 Jan 2017, 04:10
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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 25 Mar 2017, 10:30
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
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Re: In the figure shown, if the area of the shaded region is 3 times   [#permalink] 25 Mar 2017, 10:30

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