Imagine melting a large block of ice to fill small identical ice cube trays; the total amount of water (volume) remains constant throughout the process.

The principle of conservation of volume dictates that the total quantity of matter does not change during melting and recasting. Since the volume of a cube is given by $V = a^3$ (where $a$ is the side length), the number of smaller cubes $N$ is calculated as the ratio of their volumes: $N = \frac{a_1^3}{a_2^3}$.

Always remember that scaling dimensions by a factor $k$ changes the volume by a factor of $k^3$, not $k$.