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

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

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New post 03 Apr 2016, 03:06
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A
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Question Stats:

89% (01:26) correct 11% (01:36) wrong based on 90 sessions

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Attachment:
circle.jpg
circle.jpg [ 3.97 KiB | Viewed 2684 times ]

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

A. 1/2
B. 2/3
C. 3/4
D. 4/5
E. 5/6


* A solution will be posted in two days.

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

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New post 03 Apr 2016, 03:33
MathRevolution wrote:
Attachment:
circle.jpg

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

A. 1/2
B. 2/3
C. 3/4
D. 4/5
E. 5/6


* A solution will be posted in two days.


Hi,
you will have to mention that the two circles meet at one point..
Area of larger circle =\(pi * r^2\)

for smaller circle, the diameter = r, so radius = r/2
Area of smaller circle = \(pi*(\frac{r}{2})^2\)

area of shaded region = \(pi * r^2\) - \(pi*(\frac{r}{2})^2\) = \(pi*(\frac{3}{4})*r^2\)

ratio= \((pi*(\frac{3}{4})*r^2)/(pi * r^2)\) = \(\frac{3}{4}\)
C
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Re: A smaller circle is inscribed in a larger circle shown as above figure  [#permalink]

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New post 03 Apr 2016, 04:10
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MathRevolution wrote:
Attachment:
The attachment circle.jpg is no longer available

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

A. 1/2
B. 2/3
C. 3/4
D. 4/5
E. 5/6


* A solution will be posted in two days.


IMHO the best and easiest approach will be plugging in some values as below -

Attachment:
Circle 1.png
Circle 1.png [ 6.65 KiB | Viewed 2630 times ]


Area of the bigger circle will be π\((4)^2\) = 16π

Area of the smaller circle will be π\((2)^2\) = 4π

So, Area of the shaded region will be 16π - 4π = 12π


We are required to find -
Quote:
the ratio of the region shaded to the larger circle’s area?


The required ratio is 12π / 16π = 3/4
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Re: A smaller circle is inscribed in a larger circle shown as above figure  [#permalink]

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New post 03 Apr 2016, 06:22
MathRevolution wrote:
Attachment:
circle.jpg

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

A. 1/2
B. 2/3
C. 3/4
D. 4/5
E. 5/6


* A solution will be posted in two days.


From Figure we see that

radius of bigger circle = Diameter of smaller circle

Let, Radius of Bigger circle = 2 = Diameter of smaller circle

i.e. Radius of Smaller circle = 2/2 = 1

Area of Bigger circle = Pi 2^2 = 4 Pi
Area of Smaller circle = Pi 1^2 = Pi

Area of Shaded Region = Big circle - Small Circle = 4 Pi - Pi = 3Pi

Shaded Area/Area of Big circle = 3Pi/4 Pi = 3/4

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

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New post 03 Apr 2016, 12:55
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Re: A smaller circle is inscribed in a larger circle shown as above figure  [#permalink]

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New post 05 Apr 2016, 04:57
As we know that the smaller circle will have radius half to that of main circle
So area of smaller circle = 1/4* area of larger circle
Now as per question it asks us the ratio of shaded region to that of whole
So shaded region = area of larger - area of smaller = 1- 1/4 = 3/4

So answer is C

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

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New post 05 Apr 2016, 05:58
A smaller circle is inscribed in a larger circle shown as above figure. If the smaller circle passes through a center of the larger circle,what is the ratio of the region shaded to the larger circle’s area?

A. 1/2
B. 2/3
C. 3/4
D. 4/5
E. 5/6


-> As the ratio of the radius is 1:2, the ratio of the area is 1:4.
If the area of the small circle is a, the shaded area is 3a.
Thus, 3a/(a+3a)=3/4 and the answer is C.
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Re: A smaller circle is inscribed in a larger circle shown as above figure  [#permalink]

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New post 27 Jul 2017, 19:36
I have a small doubt in this question. If you consider the radius of the smaller circle to be 3 and the radius of the larger circle to be 5 then the area of the smaller is pi and the larger area is 25pi . Therefore the shaded part is 25pi-pi=16pi
16pi/25pi

So here i am missing out on the fact that the smaller circle passes through the center of the larger circle. Is that so? and in what circumstance does this method hold valid. Thank you
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Re: A smaller circle is inscribed in a larger circle shown as above figure   [#permalink] 27 Jul 2017, 19:36
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A smaller circle is inscribed in a larger circle shown as above figure

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