A large cube has edge length 18 cm. It is to be cut up completely into smaller cubes, each having an edge length of 3 cm. What is the total number of such smaller cubes that can be obtained from the original cube?

Introduction / Context:
This question checks understanding of volume and subdivision of a solid. A big cube with a given edge length is cut into many identical smaller cubes. The goal is to count how many small cubes fit exactly inside the big one without wasting any material. Aptitude tests frequently use such problems to test spatial reasoning as well as ability to handle ratios of volumes and simple arithmetic. The easiest method is to use the fact that along each edge we can place an integer number of small cubes, and the total count is the cube of that number.

Given Data / Assumptions:

• Edge length of large cube = 18 cm.
• Edge length of each small cube = 3 cm.
• The large cube is fully cut into identical smaller cubes without gaps or wastage.
• We need the total number of small cubes.

Concept / Approach:
There are two equivalent approaches:
1) Linear count method: Number of small cubes along one edge = (edge of big cube) / (edge of small cube). 2) Volume method: Total number = (volume of big cube) / (volume of small cube). Both will lead to the same answer. The first method is usually faster and less prone to computational errors.

Step-by-Step Solution:
Step 1: Compute how many small edges fit along one large edge. Number along one edge = 18 / 3 = 6. Step 2: Use the fact that the cube is three dimensional. Total number of small cubes = 6 * 6 * 6. Total number = 6^3 = 216. Thus, the original cube can be exactly cut into 216 smaller cubes of side 3 cm.

Verification / Alternative check:
Using the volume method: Volume of big cube = 18^3 = 5832 cubic centimetres. Volume of each small cube = 3^3 = 27 cubic centimetres. Number of small cubes = 5832 / 27. This gives 216 again, confirming the answer.

Why Other Options Are Wrong:
Option A (36) is simply the square of 6 and ignores the third dimension. Option B (232) and Option D (484) are arbitrary numbers that do not correspond to any simple cube of an integer. Option E (128) would be 4^3, which would occur only if the edge ratio were 4 to 1 instead of 6 to 1.

Common Pitfalls:
A common mistake is to multiply by 2 dimensions instead of 3, leading to 36 not 216. Some students also confuse surface area with volume and try to divide surface areas, which does not give the correct count of small cubes. Always remember that you need to consider all three dimensions when counting how many smaller cubes fit inside a larger cube.

Final Answer:
The large cube can be cut into a total of 216 smaller cubes, each of edge 3 cm.