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# The outside surface of a white cube with a volume 64 is painted purple

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Manager
Joined: 24 Dec 2016
Posts: 95
Location: India
Concentration: Finance, General Management
WE: Information Technology (Computer Software)
The outside surface of a white cube with a volume 64 is painted purple  [#permalink]

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03 Apr 2017, 22:57
3
00:00

Difficulty:

65% (hard)

Question Stats:

58% (02:17) correct 42% (02:43) wrong based on 85 sessions

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The outside surface of a white cube with a volume 64 is painted purple. The cube is then cut evenly into smaller 2x2x2 cubes. such that the sides that originally faced outside will remain white. What is the ratio of the total purple surface area and the total white surface area of the smaller cubes ?

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

Hi, This is one of the simpler questions on the Kaplan math workbook but for some reason I'm getting confused.
As per me, the answer should be
A as there would be 4 purple and 2 white sides
but the OA is
Can somebody help me understand this ?
Math Expert
Joined: 02 Aug 2009
Posts: 7961
Re: The outside surface of a white cube with a volume 64 is painted purple  [#permalink]

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04 Apr 2017, 18:39
1
1
Shruti0805 wrote:
The outside surface of a white cube with a volume 64 is painted purple. The cube is then cut evenly into smaller 2x2x2 cubes. such that the sides that originally faced outside will remain white. What is the ratio of the total purple surface area and the total white surface area of the smaller cubes ?

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

Hi, This is one of the simpler questions on the Kaplan math workbook but for some reason I'm getting confused.
As per me, the answer should be
A as there would be 4 purple and 2 white sides
but the OA is
Can somebody help me understand this ?

Hi shruti,

I do not think it is very simple, but the key is to visualise what is happening..
Volume of 64 means sides of 4...
Outside sides are painted purple, so area of purple colour = 4*4*6

Now the cube is cut into 2*2*2, so number of cubes cut =$$\frac{64}{2*2*2}=8$$

To find the white sides, we can take tedious route of finding white sides per cube..
Cube in corners will have 3 sides facing outwards so 3 purple and 3 white.
Edges cube will have 2 purple and 4 white and SO ON

Easier way out..
Total surface area of all 8 cubes= 8*(2*2*6)
Out of these purple coloued are 4*4*6, so remaining are white coloured..
White coloured = 8*2*2*6-4*4*6=8*4*6-4*4*6=4*6*(8-4)=4*4*6..

So both white and purple coloued are 4*4*6
Ratio =$$\frac{4*4*6}{4*4*6}=1/1$$
And 1:1
C

Hope it helps
_________________
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Joined: 03 Sep 2018
Posts: 7
Re: The outside surface of a white cube with a volume 64 is painted purple  [#permalink]

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16 Oct 2018, 11:34
more simply:

large cube is 4x4x4. So as the smaller cubes are 2x2x2 there are 2 small cubes (4/2=2) for each of the sides of the big cube. If you visualize this you will realize that the small cubes are all corner pieces and so they each have 3 purple and 3 white sides.

So the ratio is 1:1 -> C
Re: The outside surface of a white cube with a volume 64 is painted purple   [#permalink] 16 Oct 2018, 11:34
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