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A circle is inscribed right in the middle of a semicircle with a diameter of \(\pi\) as shown below. What is the ratio of the area of the semicircle to the area not covered by the inscribed circle?

Image


A. 4:1
B. 3:2
C. 3:1
D. 4:3
E. 2:1
[Reveal] Spoiler: OA

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A circle is inscribed right in the middle of a semicircle with a diameter of \(\pi\) as shown below. What is the ratio of the area of the semicircle to the area not covered by the inscribed circle?

Image


A. 4:1
B. 3:2
C. 3:1
D. 4:3
E. 2:1


Image

Since the radius of the big circle (\(\frac{\pi}{2}\)) is twice the radius of the inscribed circle (\(\frac{\pi}{4}\)) then its area is 4 times greater than the area of the inscribed circle (because in the area formula the radius is squared. For example the area of a circle with radius of 2 is \(4\pi\), which is 4 times greater than the radius of a circle with the radius of 1, which is \(\pi\)).

Thus the area of the semicircle is \(\frac{4}{2}=2\) times greater than the area of the inscribed circle: so, the area of the semicircle is 2 units, the area of the inscribed circle is 1 unit, and the area of the semicircle not covered by the inscribed circle is also 1 unit. Ratio: \(\frac{2}{1}\).


Answer: E
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Re D01-33 [#permalink]

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I think this question is good and helpful.
"Pi" is not marked on the diagram ( atleast in my browser - safari 8.0.3). Please correct it

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Re: D01-33 [#permalink]

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New post 05 Feb 2015, 20:24
Bunuel wrote:
A circle is inscribed right in the middle of a semicircle with a diameter of \(\pi\) as shown below. What is the ratio of the area of the semicircle to the area not covered by the inscribed circle?

Image


A. 4:1
B. 3:2
C. 3:1
D. 4:3
E. 2:1



hi..
ans E... it can be found just by looking at the figure ..
As ratio of the areas would just depend on square of the radius.. we get area of the outer circle to be 4 times of smaller circle as radius of smaller is half of that of larger circle.
So the ratio will come down to 2:1 as we are comparing half of outer circle with full inner circle
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I think this question is good and helpful.

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Re: D01-33 [#permalink]

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New post 06 Sep 2015, 22:19
Bunuel wrote:
A circle is inscribed right in the middle of a semicircle with a diameter of \(\pi\) as shown below. What is the ratio of the area of the semicircle to the area not covered by the inscribed circle?

Image


A. 4:1
B. 3:2
C. 3:1
D. 4:3
E. 2:1



As Randude has already mentioned, the question is a good one but missing the "pi" in the figure. Admin, please correct the figure. Thank you.

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Re: D01-33 [#permalink]

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New post 03 Oct 2015, 06:29
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Bunuel wrote:
Thus the area of the semicircle is \(\frac{4}{2}=2\) times greater than the area of the inscribed circle: so, the area of the semicircle is 2 units, the area of the inscribed circle is 1 unit, and the area of the semicircle not covered by the inscribed circle is also 1 unit. Ratio: \(\frac{2}{1}\).


Answer: E


why is the area of the semicircle NOT covered by the inscribed circle also 1?!
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New post 03 Oct 2015, 06:36
reto wrote:
Bunuel wrote:
Thus the area of the semicircle is \(\frac{4}{2}=2\) times greater than the area of the inscribed circle: so, the area of the semicircle is 2 units, the area of the inscribed circle is 1 unit, and the area of the semicircle not covered by the inscribed circle is also 1 unit. Ratio: \(\frac{2}{1}\).


Answer: E


why is the area of the semicircle NOT covered by the inscribed circle also 1?!


If the area of the semicircle is 2 and the area of the inscribed circle is 1, then how much is the area of the semicircle not covered by the inscribed circle? 2 - 1 = 1.
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Re: D01-33 [#permalink]

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I just plugged into the area formula. It takes a bit longer though.

Big half circle area = (π*(π/2)*(π/2) / 2 = π/8
Little circle area = π/16

Big Half circle - Little circle = area not covered
2/16-1/16 = 1/16

Ratio
1/8:1/16 = 2:1

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New post 17 Apr 2016, 11:41
I think this is a high-quality question and I agree with explanation.

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New post 14 Jun 2016, 04:10
I don't agree with the explanation. 4:3 is the answer. plz verify

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Re: D01-33 [#permalink]

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New post 17 Jun 2016, 00:02
harshanitr

You have taken area of semicircle as (pi) * (r^2) . You need to divide by 2 since it half the area of a circle.
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New post 17 Jun 2016, 10:16
I think this is a high-quality question and I don't agree with the explanation. The explanation states that the area of the circle with bigger radius is four times 'greater' than that with the smaller radius. I think it should be four times and not four times 'greater' as that would affect the answer.

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New post 27 Jul 2016, 03:44
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. How do we know the that the radius of smaller circle is half of the semicircle?

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New post 27 Jul 2016, 03:50
gmatclubb wrote:
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. How do we know the that the radius of smaller circle is half of the semicircle?


Small circle is centered in the semicircle, thus diameter of smaller circle is equal to the radius of the semicircle.
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Re: D01-33 [#permalink]

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New post 09 Sep 2016, 10:36
Radius of semicircle is pi/2
Radius of circle is pi/4.

Then area of semicircle is pi cube/8. And area of circle is pi cube/32.

The question is ratio between area of semicircle to area not covered by circle.

Area not covered is pi cube/8 minus pi cube/32.

That is 3 pi cube/32.

Atlast, ratio = (pi cube/8) / (3 pi cube/32)
= 4/3

Then answer should be 4:3 !?

Whether my approach to this is wrong? Correct me, if is it so..

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Re: D01-33 [#permalink]

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New post 16 Sep 2016, 22:48
Hi Ansari1

There is a small error in your calculations.
And area of circle is pi cube/32.

This will actually be: pi cube/16

Area not covered is pi cube/8 minus pi cube/16.

That is pi cube/16.

ratio = (pi cube/8) / (pi cube/16)
= 2/1

Hope it's clear.

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Re: D01-33 [#permalink]

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New post 09 Jan 2017, 03:28
But the question says "What is the ratio of the area of the semicircle to the area not covered by the inscribed circle?" so answer should be 4:3

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Re: D01-33 [#permalink]

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New post 09 Jan 2017, 03:32
bansi99 wrote:
But the question says "What is the ratio of the area of the semicircle to the area not covered by the inscribed circle?" so answer should be 4:3


The answer should be and is 2:1. Please check the discussion above and post your own solution to help you in finding where you went wrong there.
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Re: D01-33   [#permalink] 09 Jan 2017, 03:32

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