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# A smaller circle is inscribed in a larger circle shown as above figure

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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GPA: 3.82
A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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11 Jul 2017, 00:41
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58% (01:43) correct 42% (01:41) wrong based on 59 sessions

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7.11.png [ 4.08 KiB | Viewed 1041 times ]

A smaller circle is inscribed in a larger circle shown as above figure. If the smaller circle passes through the center of the larger circle, what is the ratio of the area of the region shaded to the area of the larger circle?

A. $$\frac{1}{2}$$
B. $$\frac{2}{3}$$
C. $$\frac{3}{4}$$
D. $$\frac{4}{5}$$
E. $$\frac{5}{6}$$
[Reveal] Spoiler: OA

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Math Expert
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Posts: 5660
Re: A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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11 Jul 2017, 19:20
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MathRevolution wrote:
Attachment:
7.11.png

A smaller circle is inscribed in a larger circle shown as above figure. If the smaller circle passes through the center of the larger circle, what is the ratio of the area of the region shaded to the area of the larger circle?

A. $$\frac{1}{2}$$
B. $$\frac{2}{3}$$
C. $$\frac{3}{4}$$
D. $$\frac{4}{5}$$
E. $$\frac{5}{6}$$

Hi,

The figure is incorrect and does not represent a sketch that is intended by Q.

Also there can be various smaller circles that can satisfy the Q so the Q must be meaning :-
smaller circle passing through the centre and touching the circumference at just one point.

Solution:-
This circle will have the DIAMETER equal to the RADIUS of larger circle...
Area of larger circle =πr^2...
Area of smaller circle=$$π(\frac{r}{2)}^2=π\frac{r^2}{4}$$..
Thus area of shaded region =$$πr^2-π\frac{r^2}{4}=\frac{3πr^2}{4}$$...
Thus ratio =3/4

C
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Re: A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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11 Jul 2017, 23:05
chetan2u wrote:
MathRevolution wrote:
Attachment:
7.11.png

A smaller circle is inscribed in a larger circle shown as above figure. If the smaller circle passes through the center of the larger circle, what is the ratio of the area of the region shaded to the area of the larger circle?

A. $$\frac{1}{2}$$
B. $$\frac{2}{3}$$
C. $$\frac{3}{4}$$
D. $$\frac{4}{5}$$
E. $$\frac{5}{6}$$

Hi,

The figure is incorrect and does not represent a sketch that is intended by Q.

Also there can be various smaller circles that can satisfy the Q so the Q must be meaning :-
smaller circle passing through the centre and touching the circumference at just one point.

Solution:-
This circle will have the DIAMETER equal to the RADIUS of larger circle...
Area of larger circle =πr^2...
Area of smaller circle=$$π(\frac{r}{2)}^2=π\frac{r^2}{4}$$..
Thus area of shaded region =$$πr^2-π\frac{r^2}{4}=\frac{3πr^2}{4}$$...
Thus ratio =3/4

C

How diameter of small circle equals to radius of larger circle??
VP
Joined: 22 May 2016
Posts: 1346
A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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12 Jul 2017, 08:07
2
KUDOS
chetan2u wrote:
MathRevolution wrote:

A smaller circle is inscribed in a larger circle shown as above figure. If the smaller circle passes through the center of the larger circle, what is the ratio of the area of the region shaded to the area of the larger circle?

A. $$\frac{1}{2}$$
B. $$\frac{2}{3}$$
C. $$\frac{3}{4}$$
D. $$\frac{4}{5}$$
E. $$\frac{5}{6}$$

Hi,

The figure is incorrect and does not represent a sketch that is intended by Q.

Also there can be various smaller circles that can satisfy the Q so the Q must be meaning :-
smaller circle passing through the centre and touching the circumference at just one point.

Solution:-
This circle will have the DIAMETER equal to the RADIUS of larger circle...
Area of larger circle =πr^2...
Area of smaller circle=$$π(\frac{r}{2)}^2=π\frac{r^2}{4}$$..
Thus area of shaded region =$$πr^2-π\frac{r^2}{4}=\frac{3πr^2}{4}$$...
Thus ratio =3/4

C

How diameter of small circle equals to radius of larger circle??

chetan2u , thanks for clarifying. I thought the same, but waited for an expert . . .

Adityagmatclub , see my figure and solution immediately below where smaller circle passes through center of larger circle and smaller circle touches larger circle's circumference at just one point.
Attachment:

circle--in-circle.png [ 2.34 KiB | Viewed 897 times ]

Similar to chetan2u 's analysis: Disregard the figure. Rely instead on the phrase "is inscribed in," see my figure above, where the smaller circle is completely inside the larger circle, which is what I understand "inscribed" to mean.

Smaller circle passes through larger circle's center and touches larger circle's circumference at just one point.

Shaded area = Area of large circle - area of smaller circle.

Let radius of larger circle be 4.

Radius of larger circle is diameter of small circle, and d = 2r, so radius of small circle is: d = 4 = 2r means r = 2.

Area of large circle = $$\pi*r^2$$ = $$\pi*4^2$$ = 16$$\pi$$
Area of small circle (r is 2) = 4$$\pi$$

Large - small = 16$$\pi$$ - 4$$\pi$$ = 12$$\pi$$ --> is the area of the shaded portion

Ratio of area of shaded portion to area of larger circle is $$\frac{12\pi}{16\pi}$$ = $$\frac{3}{4}$$

Hope it helps.
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Posts: 121
Re: A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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12 Jul 2017, 08:25
chetan2u wrote:
smaller circle passing through the centre and touching the circumference at just one point.
C

Good catch, I was also confused after reading and only by assuming that was able to resolve the case
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 4908
GPA: 3.82
Re: A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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13 Jul 2017, 01:06
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==> Square of the ratio of the length=ratio of the area. The diameter of the smaller circle is equal to the radius of the larger circle, so the ratio of the length becomes 1:2. Then, the ratio of the area becomes 1^2:2^2=1:4. Thus, if the area of the smaller circle=k, the area of the larger circle=4k. Therefore, the area of the region shaded:the area of the larger circle=(4k-k):4k=3:4=¾.

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Manager
Joined: 12 Feb 2017
Posts: 71
Re: A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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03 Sep 2017, 00:25
Please correct the figure, I got it wrong just because of faulty figure.
Intern
Joined: 24 Jun 2017
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GMAT 1: 750 Q49 V44
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Re: A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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03 Sep 2017, 10:04
This question is a complete waste of time with current attachment

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Re: A smaller circle is inscribed in a larger circle shown as above figure   [#permalink] 03 Sep 2017, 10:04
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