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 Post subject: GMAT Probability
PostPosted: Mon Sep 20, 2010 9:06 am 
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A wooden cube whose edge length is 10 inches is composed of smaller cubes with edge lengths of one inch. The outside surface of the large cube is painted red and then it is split up into its smaller cubes. If one cube is randomly selected from the small cubes, what is the probability that the cube will have AT LEAST one red face?
A. 36.0%
B. 48.8%
C. 50.0%
D. 52.5%
E. 60%

(B) The large cube is composed of 10 small cubes in each dimension, so it is composed of 1000 small cubes in all because the volume of a cube is:
Volume = length × width × height = 10 × 10 × 10 = 1000.

The cubes on the surface will have at least one face painted, but the inside cubes will not. It is difficult to directly solve for the number of small cubes along the surface, so we will calculate the number of cubes on the inside (not painted), and subtract this number from the total number of cubes, to get the number of cubes that are painted.

The number of small cubes that are on the inside (not painted) is:
8 × 8 × 8 = 512.
This is because the first and last cube in each dimension will be painted, meaning that there are rows of 8 unpainted cubes in each dimension.

Then, the number of painted cubes is:
1000 – 512 = 488.
Therefore, the probability of selecting a cube with at least one face painted is:
488/1000 = 48.8%.

The correct answer is choice (B).

Note: If you chose choice (A), you were probably working with a 2-dimensional figure instead of a cube (3-dimensional).
---------
I can't imagine this problem in 3D. Could you help, please?


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 Post subject: Re: math: 3D geometry, probability.
PostPosted: Tue Sep 21, 2010 5:42 am 
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At the beginning we have a 10 × 10 cube which sides are painted red:


We want to count number of unit cubes that are NOT painted (it's easier). For that we take off outer layer of painted unit cubes:


As the result we get inner 8 × 8 cube which is made of unit cubes that are NOT painted:


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 Post subject: Re: math: 3D geometry, probability.
PostPosted: Mon Dec 06, 2010 1:26 pm 
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Why will the number of cubes painted at least on one side ie the cubes on the outside not be 600, 10 × 10 on each side × 6 sides?


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 Post subject: Re: math: 3D geometry, probability.
PostPosted: Mon Dec 06, 2010 1:41 pm 
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 Post subject: Re: GMAT Probability
PostPosted: Sun Dec 09, 2012 5:24 am 
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I spent a lot of time on this problem and I question your total number of cubes that comprise the 10 inch cube. There may be a 1000 faces on the outside of the cube, but the cubes at the corners account for 3 of those faces each and the cubes at the edges account for 2 of those faces each. Using similar logic to that used to find the number of cubes at the interior of the block. I calculated that the total number of cubes that make up the block is 760 not 1000. I would appreciate it if you would get back to me with a response. I have been struggling with the quantitative section and would appreciate knowing if my calculations were faulty.


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 Post subject: Re: GMAT Probability
PostPosted: Sun Dec 09, 2012 5:42 am 
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