A Large Cube Consists Of 125 Identical Small Cubes GMAT Problem Solving

Question: 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

“A large cube consists of 125 identical small cubes''- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. GMAT quant section is designed to check the candidates’ talents in logical thinking and mathematical calculations. The students must determine the answer choice that is correct by doing suitable calculations. The students must hold an adequate knowledge of mathematics to solve GMAT Problem Solving questions. The calculative mathematical problems regarding the GMAT Quant topic in the problem-solving part can be cracked with valid mathematical skills. The candidates can practice more questions by answering from the book “501 GMAT Questions”.

Solution and Explanation:

Approach Solution 1:

The problem statement informs that:
Given:

• A large cube includes 125 identical small cubes.

Find Out:

• The number of small cubes exposed to the air.

Since the cube contains 125 identical small cubes, therefore, we can say,
125= 5^3.
We know that the formula of the volume of a cube is, V=(side)^3
Therefore, it can be concluded that each side of the large cube is made of 5 small cubes.

If the side of the large cube is 5 cm, then the side of the small cube is 1cm.
The large cube with a volume of 125 cu. cm means the volume of 125 cubes is 1cu.cm.

The question asks to find the number of small cubes exposed, which denotes finding the number of such small cubes that constitute the exterior surface.
Therefore, to solve the answer, we can find those cubes whose surfaces are not exposed to air.
Let’s analyse the cube as 5 squares of 25 cubes and each cube is positioned over the other.
Therefore, the top and lowermost squares are indeed the ones with all cubes exposed to air.

In the case of squares 2, 3 and 4, 16 cubes that drive the four sides are exposed to air.
However, 9 cubes that are wrapped within these sides are not exposed.
So a total of cubes that are not exposed to air = 9*3 = 27 cubes

Hence, No. of cubes exposed to air= 125 - 27 = 98

Correct Answer: (C)

Approach Solution 2:

The problem statement suggests that:

Given:

• A large cube contains 125 identical small cubes.

Find Out:

• The number of small cubes exposed to the air.

The cube contains 125 identical small cubes
Where 125= 5^3.
Therefore, the sides of the large cube are constructed with 5 small cubes.
Thus the sides of the internal cube are made with (5-2) = 3 small cubes (since there are 2 small edge cubes)

Hence, number of cubes exposed to air= 125 - 3^3 =125 - 27 = 98 cubes

Correct Answer: (C)

Approach Solution 3:

The problem statement declares that:

Given:

• A large cube contains 125 identical small cubes.

Find Out:

• The number of small cubes exposed to the air.

There are 125 identical cubes
Therefore, it means N=5. This indicates that each face of the large cube will hold 5 cubes horizontally and 5 cubes vertically.
Let’s imagine a large cube trimmed on each face with 4 horizontal lines and 4 vertical lines making 5x5 small cubes per face.

The cube holds 8 vertices that can be exposed to air.
Therefore, (N-2) will hold 2 faces revealed to the air.
The total no. of sides of a cube =12
hence= 12*(5-2)= 36
Each face will hold (n-2)2 cubes with one face revealed to air.
Total number of faces we have= 6 faces that is 6* (n-2)^2= 54

Total faces exposed to air= 8+36+54= 98

No. of in-centre cubes = (N-2)^3=27

Hence, the number of cubes exposed to air is= 125 - 27 = 98 cubes

Correct Answer: (C)

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