A cuboid with dimensions 24 cm × 9 cm × 8 cm is melted and re-cast into smaller solid cubes each of side 3 cm. How many such cubes can be formed?

Introduction / Context:
This question deals with conversion of one solid shape into another by melting and re casting, a common topic in mensuration and volume based aptitude problems. The total volume is conserved in the process. Here, a cuboid is melted and turned into smaller cubes, and the number of cubes is determined by dividing the volume of the original cuboid by the volume of one cube.

Given Data / Assumptions:
• Original solid is a cuboid of dimensions 24 cm × 9 cm × 8 cm.• New solids are cubes each with side 3 cm.• There is no loss of material during melting and re casting.• We need the number of cubes formed.

Concept / Approach:
The key idea is that volume is conserved. Let V1 be the volume of the original cuboid and V2 be the volume of one cube. The number of cubes N is N = V1 / V2. For the cuboid, V1 = length * breadth * height. For each cube, V2 = side^3. By substituting the given dimensions, we can calculate the value of N directly.

Step-by-Step Solution:
Step 1: Compute volume of the cuboid.V1 = 24 * 9 * 8 cubic centimetres.24 * 9 = 216.V1 = 216 * 8 = 1728 cm^3.Step 2: Compute volume of one small cube.Side of cube = 3 cm.V2 = 3^3 = 3 * 3 * 3 = 27 cm^3.Step 3: Compute the number of cubes.N = V1 / V2 = 1728 / 27.Step 4: Perform the division.1728 / 27 = 64.

Verification / Alternative check:
We can factor 1728 as 64 * 27, so the division 1728 / 27 equals 64 exactly. This confirms that the number of cubes is an integer and matches our calculation. Since there is no leftover volume, the assumption of volume conservation is satisfied and the result is logically consistent.

Why Other Options Are Wrong:
Option B: 56 implies an incorrect division or a mistaken volume for either the cuboid or the cube.Option C: 48 undercounts the cubes and would require a larger cube side or a smaller cuboid volume.Option D: 40 is much smaller than the correct answer and does not arise from any correct arithmetic based on the given dimensions.

Common Pitfalls:
Learners sometimes use the sum of dimensions instead of their product when calculating volume, or they forget to cube the side for the cube volume. Another mistake is to reverse the division, computing 27 / 1728 instead of 1728 / 27. Writing the formulas clearly and performing the arithmetic carefully helps avoid such errors.

Final Answer:
The number of cubes that can be formed is 64.