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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) \(\sqrt{2}\)


Attachment:
2015-10-22_0821.png
2015-10-22_0821.png [ 7.04 KiB | Viewed 31197 times ]

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Originally posted by Bunuel on 21 Oct 2015, 21:22.
Last edited by Bunuel on 15 Sep 2019, 20:55, edited 2 times in total.
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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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4
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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2
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 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 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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1
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, 11:38
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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
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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 2017, 20:02
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.
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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 06 Jul 2017, 13:45
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
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In the figure shown, if the area of the shaded region is 3 t  [#permalink]

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New post 06 Dec 2017, 12:09
monirjewel 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. \(\sqrt{2}\)
Attachment:
1111220830.jpg

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

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New post 01 Feb 2018, 13:19
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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 = 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:

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

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New post 28 Feb 2018, 00:03
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. \(\sqrt{2}\)

The area of the shaded region is \(area_{shaded}=\pi{R^2}-\pi{r^2}\) and the area of the smaller circle is \(area_{small}=\pi{r^2}\). Given: \(\pi{R^2}-\pi{r^2}=3\pi{r^2}\) --> \(R^2=4r^2\) --> \(R=2r\);

Now, the ratio of the circumference of the larger circle to the that of the smaller circle is \(\frac{C}{c}=\frac{2\pi{R}}{2\pi{r}}=\frac{{2r}}{{r}}=2\).

Answer: C.




Hi Bunuel,

Can you share problems that are similar to this?

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

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New post 28 Feb 2018, 00:15
1
shivamtibrewala wrote:
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. \(\sqrt{2}\)

The area of the shaded region is \(area_{shaded}=\pi{R^2}-\pi{r^2}\) and the area of the smaller circle is \(area_{small}=\pi{r^2}\). Given: \(\pi{R^2}-\pi{r^2}=3\pi{r^2}\) --> \(R^2=4r^2\) --> \(R=2r\);

Now, the ratio of the circumference of the larger circle to the that of the smaller circle is \(\frac{C}{c}=\frac{2\pi{R}}{2\pi{r}}=\frac{{2r}}{{r}}=2\).

Answer: C.




Hi Bunuel,

Can you share problems that are similar to this?

Thanks.


Shaded Region Problems from our Special Questions Directory.
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Re: In the figure shown, if the area of the shaded region is 3 t  [#permalink]

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New post 01 Mar 2018, 18:35
monirjewel 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. \(\sqrt{2}\)

Attachment:
1111220830.jpg
Attachment:
Untitled2.png


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

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New post 18 Mar 2018, 23:14
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)
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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 27 Mar 2018, 11:00
Assume smaller circle has the radius of 4. The shaded region has an area 3 times the smaller circle.
Smaller = 16pi
Shaded = 48pi
Total circle area = smaller + shadeded = 64pi

64pi = pi*r^2(area of circle)
r= 8 (radius of total circle

circumference - 2*pi*r

Circumference or bigger circle = 2*pi*8 = 16pi
Circuference of smaller circle = 2*pi*4= 8pi

Larger circle circumference is double smaller circle or 2 times 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 29 Apr 2018, 03:59
reto wrote:
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.


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 :)
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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 29 Apr 2018, 12:15
1
dave13 wrote:
reto wrote:
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.


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 :)


Hey dave13

To understand the part about the circle having an area 4x, you will have to
take a look at the question which states that "area of the shaded region is 3
times the area of the smaller circular region"

We are assuming the smaller circle to have area x. Since the shaded region
has 3 times the area, making the area of the shaded region 3x. Now, the area
of the circle which contains the small circle and the shaded region will have an
area of x+3x = 4x

Once that part is clear, hope you understood the flaw in your reasoning!
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Re: In the figure shown, if the area of the shaded region is 3 times   [#permalink] 29 Apr 2018, 12:15

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

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