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Manager  Joined: 10 Feb 2011
Posts: 102
A large cube consists of 125 identical small cubes, how  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 63% (01:23) correct 37% (01:57) wrong based on 481 sessions

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A large cube consists of 125 identical small cubes, how many of the small cubes are exposed in air?

(A) 64
(B) 72
(C) 98
(D) 100
(E) 116
Math Expert V
Joined: 02 Sep 2009
Posts: 61287
Re: 226. A large cube consists of 125 identical small cubes, how  [#permalink]

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banksy wrote:
226. A large cube consists of 125 identical small cubes, how many of the small cubes are exposed in air?
(A) 64
(B) 72
(C) 98
(D) 100
(E) 116

As the cube consists of 125=5^3 identical small cubes then its sides are built with 5 small cubes. The sides of the inner cube are built with 3 small cube (5 - 2 small edge cubes), thus there are 125-3^3=98 cubes exposed.

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Manager  Joined: 14 Feb 2011
Posts: 142
Re: 226. A large cube consists of 125 identical small cubes, how  [#permalink]

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OK - let me try and explain this.

The question says that a large cube is actually made up of 125 smaller cubes.

Since 125 = $$5^3$$, it implies that each side of the large cube is actually made up of 5 cubes. For simplicity, if large cube has side 5 cm, then small cube has side one cm and hence large cube with volume 125 cu. cm. is actually 125 cubes of 1 cu. cm. each.

Now, the question is how many cubes are exposed, which simply means that how many of such small cubes constitute outer surface (which is exposed to air)?

So, we can work this out by finding those cubes that do not have any of the surfaces exposed to air.

Lets visualise the cube as 5 squares of 25 cubes each placed over one another, Now top and bottom square are clearly the ones with all cubes exposed to air.

For squares 2, 3 and 4, 16 cubes that make the four sides are exposed to air, but 9 cubes that are enclosed within these sides are not exposed. So total of 9*3 = 27 cubes are not exposed to air.

Hence, 125 - 27 = 98 cubes are exposed to air.

You can quickly visualise that except a 3x3 cube within the larger cube, everything else is exposed, so you can quickly do $$5^3$$ - $$3^3$$ and derive the answer, as Bunuel has done.
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Re: A large cube consists of 125 identical small cubes, how  [#permalink]

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banksy wrote:
A large cube consists of 125 identical small cubes, how many of the small cubes are exposed in air?

(A) 64
(B) 72
(C) 98
(D) 100
(E) 116

The detailed solution with more learning is right here...

Exposed in air = 125 - 27 = 98

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Intern  Joined: 22 Feb 2010
Posts: 32
Re: 226. A large cube consists of 125 identical small cubes, how  [#permalink]

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2
beyondgmatscore wrote:
OK - let me try and explain this.

The question says that a large cube is actually made up of 125 smaller cubes.

Since 125 = $$5^3$$, it implies that each side of the large cube is actually made up of 5 cubes. For simplicity, if large cube has side 5 cm, then small cube has side one cm and hence large cube with volume 125 cu. cm. is actually 125 cubes of 1 cu. cm. each.

Now, the question is how many cubes are exposed, which simply means that how many of such small cubes constitute outer surface (which is exposed to air)?

So, we can work this out by finding those cubes that do not have any of the surfaces exposed to air.

Lets visualise the cube as 5 squares of 25 cubes each placed over one another, Now top and bottom square are clearly the ones with all cubes exposed to air.

For squares 2, 3 and 4, 16 cubes that make the four sides are exposed to air, but 9 cubes that are enclosed within these sides are not exposed. So total of 9*3 = 27 cubes are not exposed to air.

Hence, 125 - 27 = 98 cubes are exposed to air.

You can quickly visualise that except a 3x3 cube within the larger cube, everything else is exposed, so you can quickly do $$5^3$$ - $$3^3$$ and derive the answer, as Bunuel has done.

A very good explanation indeed. I tried a different approach and ended up with incorrect answer. Please help me correct with my mistake here.
Let us consider that there are five individual small cubes that make up the edge of the cube. I have added one picture for representation.
Assume that the cube is made up of 5 small individual cubes as shown in the picture. This is only one face of the 6 faces of the cube. We consider the outer cubes as below:
Blue Cubes - The 3X3=9 blue faces have no corners common and repeat at each of the faces of the cube. Thus, there are in total 3X3X6 = 54 blue cubes in total
Green Cubes - We have 12 (3X4) Green cubes on one face that share their edges with other cubes. We will have similar 12 Green cubes on the other side. Thus, we have 24 Green cubes.
Orange Cubes - We have 4 such cubes that are at corners and share their edges with cubes at other sides. We will have 4X2=8 such orange cubes.
Total cubes facing the air = Blue + Green + Orange = 54+24+8=86
Please tell me my mistake. Thanks.
Attachments cube.JPG [ 5.07 KiB | Viewed 62115 times ]

Intern  Joined: 17 Nov 2010
Posts: 2
Re: 226. A large cube consists of 125 identical small cubes, how  [#permalink]

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Bunuel wrote:
banksy wrote:
226. A large cube consists of 125 identical small cubes, how many of the small cubes are exposed in air?
(A) 64
(B) 72
(C) 98
(D) 100
(E) 116

As the cube consists of 125=5^3 identical small cubes then its sides are built with 5 small cubes. The sides of the inner cube are built with 3 small cube (5 - 2 small edge cubes), thus there are 125-3^3=98 cubes exposed.

Haven't understood the solution. Can you please explain further?

Thanks
Manager  Joined: 18 Oct 2010
Posts: 60
Re: 226. A large cube consists of 125 identical small cubes, how  [#permalink]

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i understand neither the mathematic question and its solution. can you please explain?
Manager  Joined: 25 Sep 2012
Posts: 226
Location: India
Concentration: Strategy, Marketing
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GMAT 2: 680 Q48 V34
Re: 226. A large cube consists of 125 identical small cubes, how  [#permalink]

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holidevil wrote:
beyondgmatscore wrote:
OK - let me try and explain this.

The question says that a large cube is actually made up of 125 smaller cubes.

Since 125 = $$5^3$$, it implies that each side of the large cube is actually made up of 5 cubes. For simplicity, if large cube has side 5 cm, then small cube has side one cm......

A very good explanation indeed. I tried a different approach and ended up with incorrect answer. Please help me correct with my mistake here.
Let us consider that there are five individual small cubes that make up the edge of the cube. I have added one picture for representation.
Assume that the cube is made up of 5 small individual cubes as shown in the picture. This is only one face of the 6 faces of the cube. We consider the outer cubes as below:
Blue Cubes - The 3X3=9 blue faces have no corners common and repeat at each of the faces of the cube. Thus, there are in total 3X3X6 = 54 blue cubes in total
Green Cubes - We have 12 (3X4) Green cubes on one face that share their edges with other cubes. We will have similar 12 Green cubes on the other side. Thus, we have 24 Green cubes.
Orange Cubes - We have 4 such cubes that are at corners and share their edges with cubes at other sides. We will have 4X2=8 such orange cubes.
Total cubes facing the air = Blue + Green + Orange = 54+24+8=86
Please tell me my mistake. Thanks.

if anyone is facing the same problem...

In this case we are not counting 3 small cubes which are between the bottom and top orange cubes (for each orange cube). We can clearly understand that counting by above method can lead to an error and Bunuel's method is way much better. If you understand the logic behind Bunuel's method, this question will be a cakewalk.
Math Expert V
Joined: 02 Sep 2009
Posts: 61287
Re: 226. A large cube consists of 125 identical small cubes, how  [#permalink]

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Bunuel wrote:
banksy wrote:
226. A large cube consists of 125 identical small cubes, how many of the small cubes are exposed in air?
(A) 64
(B) 72
(C) 98
(D) 100
(E) 116

As the cube consists of 125=5^3 identical small cubes then its sides are built with 5 small cubes. The sides of the inner cube are built with 3 small cube (5 - 2 small edge cubes), thus there are 125-3^3=98 cubes exposed.

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Re: A large cube consists of 125 identical small cubes, how  [#permalink]

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