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M25-23

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Bunuel
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M25-23 [#permalink] New post   16 Sep 2014, 01:23
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The diagram above shows a circle inscribed in a square. The area of the shaded region is approximately what percent of the area of the square?


A. 14%
B. 18%
C. 22%
D. 28%
E. 30%
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C
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Bunuel
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Re M25-23 [#permalink] New post   16 Sep 2014, 01:23
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Official Solution:




The diagram above shows a circle inscribed in a square. The area of the shaded region is approximately what percent of the area of the square?


A. 14%
B. 18%
C. 22%
D. 28%
E. 30%


Consider the side of the square to be 20 centimeters. Notice that the diameter of the circle equals to the side of the square, hence the radius equals to \(\frac{20}{2}=10\) centimeters.

Next, the area of the square is \(20^2=400\) square centimeters and the area of the circle is \(\pi r^2 = 100 \pi \approx 314\) square centimeters. Since the area of the shaded region equals to the difference of those areas then it's \(400-314=86\) square centimeters. So the area of the shaded region is approximately \(\frac{86}{400}*100 \approx 22%\) of the area of the square.


Answer: C
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Re: M25-23 [#permalink] New post   11 Oct 2014, 06:46
assigning numbers, assume the radius = 2. Therefore the circle area = 4pi. With a radius = 2, each side of the square = 4, so the area is 16. Now, the area of the sq - area of the circle is what you want. 16 - ~12 = ~4, then ~4 / 16 = slightly less than 1/4, so Ans. C
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Re: M25-23 [#permalink] New post   24 Oct 2014, 07:14
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Bunuel wrote:
Official Solution:




The diagram above shows a circle inscribed in a square. The area of the shaded region is approximately what percent of the area of the square?


A. 14%
B. 18%
C. 22%
D. 28%
E. 30%


Consider the side of the square to be 20 centimeters. Notice that the diameter of the circle equals to the side of the square, hence the radius equals to \(\frac{20}{2}=10\) centimeters.

Next, the area of the square is \(20^2=400\) square centimeters and the area of the square is \(\pi r^2 = 100 \pi \approx 314\) square centimeters. Since the area of the shaded region equals to the difference of those areas then it's \(400-314=86\) square centimeters. So the area of the shaded region is approximately \(\frac{86}{400}*100 \approx 22%\) of the area of the square.


Answer: C


It appears that there is a typo.(indicated in RED color).
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Bunuel
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Re: M25-23 [#permalink] New post   24 Oct 2014, 08:15
Expert Reply
arunspanda wrote:
Bunuel wrote:
Official Solution:




The diagram above shows a circle inscribed in a square. The area of the shaded region is approximately what percent of the area of the square?


A. 14%
B. 18%
C. 22%
D. 28%
E. 30%


Consider the side of the square to be 20 centimeters. Notice that the diameter of the circle equals to the side of the square, hence the radius equals to \(\frac{20}{2}=10\) centimeters.

Next, the area of the square is \(20^2=400\) square centimeters and the area of the square is \(\pi r^2 = 100 \pi \approx 314\) square centimeters. Since the area of the shaded region equals to the difference of those areas then it's \(400-314=86\) square centimeters. So the area of the shaded region is approximately \(\frac{86}{400}*100 \approx 22%\) of the area of the square.


Answer: C


It appears that there is a typo.(indicated in RED color).


Edited. Thank you.
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Re: M25-23 [#permalink] New post   28 Jun 2018, 09:28
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Lets pick up a smart number.

Let the side of the square to be 1 inch. side of square = diameter of circle.

Area of square is \(1^2\) = 1 and area of circle is \(\pi r^2 = \frac{1}{4} \pi \approx 0.78\)

Since the area of the shaded region equals to area of square minus the area of the circle we have \(1-0.78=0.22\) square inch.

So the area of the shaded region is approximately \(\frac{0.22}{1}*100 \approx 22 percent\) of the area of the square.

Correct Answer: C
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Re: M25-23 [#permalink] New post   29 Jun 2018, 01:46
I did not have to solve it. Looking at the picture (and knowing it is drawn to scale), I could see what percentage it would be.
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Re: M25-23 [#permalink] New post   06 Jul 2018, 13:21
Area of square = s^2
Area of circle = pi * r^2

r = 1/2 s
2r = s

Compare pi*r^2 with 4*r^2...
pi = 3.14
4 = 4.00
4.00-3.14 = 0.86
1/4 = 25% (=1.0)
20% of 4 = 0.8
Answer has to be someplace between 25% and 20%, so pick 22%.
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Re: M25-23 [#permalink] New post   28 Oct 2020, 17:09
Bunuel, say one wants to solve it without substituting numbers, i.e.:

Area of square s^2; area of circle pie*r^2 , is there anyway of doing so?
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Re: M25-23 [#permalink] New post   28 Oct 2020, 23:14
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DanielEMZ wrote:
Bunuel, say one wants to solve it without substituting numbers, i.e.:

Area of square s^2; area of circle pie*r^2 , is there anyway of doing so?


Just substitute 20 with s in the solution,

Consider the side of the square to be s centimeters. Notice that the diameter of the circle equals to the side of the square, hence the radius equals to \(\frac{s}{2}\) centimeters.

Next, the area of the square is \(s^2\) square centimeters and the area of the circle is \(\pi (s/2)^2\) square centimeters. Since the area of the shaded region equals to the difference of those areas then it's \(s^2-\pi (s/2)^2=(1 - π/4) s^2 \approx 0.21s^2\) square centimeters. So the area of the shaded region is approximately \(\frac{0.21s^2}{s^2}*100 \approx 21\%\) of the area of the square.


Answer: C
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Re: M25-23 [#permalink] New post   29 Oct 2020, 06:00
Bunuel wrote:


The diagram above shows a circle inscribed in a square. The area of the shaded region is approximately what percent of the area of the square?


A. 14%
B. 18%
C. 22%
D. 28%
E. 30%


let each side of square be 4 are of square 16
for circle the radius is 2 and area is 4pi
area of shaded region 16-4pi
% of area shaded region to that of square ; 16-4pi/16 ; ~ 22%
option C
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Re: M25-23 [#permalink]
29 Oct 2020, 06:00
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