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Difficulty:
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Question Stats:
75% (01:49) correct
25%
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wrong
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An artist is painting a white circular region on the top surface of a square blue tile that measures 8 inches on a side. If the circle is inscribed in the square, what fraction of the top surface of the tile will be blue?
A. \(\frac{1}{4}\)
B. \(\frac{1}{2}\)
C. \(\frac{\pi}{4}\)
D. \(\frac{(4 – \pi)}{\pi}\)
E. \(\frac{(4 – \pi)}{4}\)
A. \(\frac{1}{4}\)
B. \(\frac{1}{2}\)
C. \(\frac{\pi}{4}\)
D. \(\frac{(4 – \pi)}{\pi}\)
E. \(\frac{(4 – \pi)}{4}\)
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saiesta
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Area of Square is \(8^2\) = 64
Area of circle is \(π*4^2\) = \(16π\)
Area of Blue region is \(64 - 16π\)
So, Fraction of tile that will be Blue is = \(\frac{64 - 16π}{64}\)
Or, Fraction of tile that will be Blue is = \(\frac{16 (4 - π)}{16*4}\) = \(\frac{4 - π}{4}\)
Hence, Correct answer will be (E) \(\frac{4 - π}{4}\)
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It breaks down like this:
A circle inscribed in a square, means the circle touches all 4 inside edges of the square.
So if the square length is 8", so is the Diameter of the circle, which means the Radius is 4.
Area of a circle is πr^2 or in this case: π16"
So lets compare blue area to total area to get the final answer:
1) Area of Square - Area of the Circle = The Remaining Blue Area
64" - π16" = 64-π16 *Calculating this out at this point would be a mistake...
2) The Remaining Blue Area / Area of the Square = Answer
(64-π16) / 64 =(4-π1)/4 or (4-π)/4
Answer: E
A circle inscribed in a square, means the circle touches all 4 inside edges of the square.
So if the square length is 8", so is the Diameter of the circle, which means the Radius is 4.
Area of a circle is πr^2 or in this case: π16"
So lets compare blue area to total area to get the final answer:
1) Area of Square - Area of the Circle = The Remaining Blue Area
64" - π16" = 64-π16 *Calculating this out at this point would be a mistake...
2) The Remaining Blue Area / Area of the Square = Answer
(64-π16) / 64 =(4-π1)/4 or (4-π)/4
Answer: E
General Discussion
Posts: 3
Hi mromrell,
I can't wrap my head around the last step you are doing.
We have the area of the square = 8*8 = 64
We also have the area of the circle = 4^2*π => 16π
Then the blue area must be = 64-16π
How do we go from this step to (4-π)/4 ?
Hope you can clarify, thanks!
I can't wrap my head around the last step you are doing.
We have the area of the square = 8*8 = 64
We also have the area of the circle = 4^2*π => 16π
Then the blue area must be = 64-16π
How do we go from this step to (4-π)/4 ?
Hope you can clarify, thanks!
Posts: 3
Abhishek009
Hi Abhishek009
Thank you very much for the explanation. I misread the question. I thought that we were being asked for the blue area, but I see now that we HAVE to divide by 64 as we are being asked for the fraction of the blue area to the square.
Circle inscribed in a square=> diamter of circle = side of quare
Fraction of Blue area = (area of Square-Area of Circle)/Area of Square
= (8^2-pi*4^2)/8^2 = (4-pi)/4
Fraction of Blue area = (area of Square-Area of Circle)/Area of Square
= (8^2-pi*4^2)/8^2 = (4-pi)/4
Quote:
HI GMATGuruNY ,EMPOWERgmatRichC , MentorTutoring GMATCoachBen
An artist is painting a white circular region on the top surface of a square blue tile that measures 8 inches on a side. Are we finding the circumference outside the square?
Can you please help me with this problem? I'm unable to comprehend the wordings.
Posts: 1,348
Updated on: 02 Aug 2020, 08:36
Originally posted by GMATGuruNY on 31 Jul 2020, 04:32.
Last edited by GMATGuruNY on 02 Aug 2020, 08:36, edited 1 time in total.
Last edited by GMATGuruNY on 02 Aug 2020, 08:36, edited 1 time in total.
NandishSS
We are asked to calculate the blue area in the figure below:
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NandishSS
The short answer is no, you are not trying to find the circumference of a circle that is larger than the square. How do you know? There are two ways you can tell:
1) The white circle is going to be painted on the top surface of the blue square tile, not covering up all the blue.
2) The question that follows references the circle being inscribed in the square, so the square must be the larger figure. (Otherwise, we would expect the language to be a bit different—e.g., the circle circumscribes the square or the square is inscribed in the circle.)
As much as possible, when you get bogged down in the phrasing of a problem, step back and look for small details, clues such as those I have drawn attention to above, to point you in the right direction.
I hope that helps. Thank you for bringing the question to my attention.
- Andrew
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Expert reply
NandishSS
Hi NandishSS,
When one shape is INscribed in another shape, it means that you are drawing the first shape "inside" of the 2nd shape (with the sides 'touching'). In this question, we will draw a white circle inside of a blue square - and the diameter of the circle will be EQUAL to the side length of the square.
Thus, most of the blue square will be painted white. Since we know that the side length is 8, we can calculate the area of the square and the area of the circle. The difference in those two numbers will be equal to the 'leftover' parts of the square that are still blue. That number, when divided by 64 (re: the area of the square), will tell us what percentage of the square is still blue.
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Expert reply
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saiesta
Solution:
The diameter of the inscribed circular region is the same as the side of the square tile; thus, it’s 8 inches. Therefore, the area of the circular region is π x (8/2)^2 = 16π sq. inches. Since the area of the square is 8^6 = 64 sq. inches, the area of the square tile that is still blue is 64 - 16π sq. inches. Therefore, the fraction of the square tile that is still blue is (64 - 16π) / 64 = (4 - π) / 4.
Answer: E
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Since the circle is inscribed in the square, diameter of circle = side of the square. This implies radius =8/2=4
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