A solid cube of side 12 cm is cut into 8 identical cubes. What will be the side of the new cube? Also, find the ratio between the surface area of the original cube and the total surface area of all the small cubes formed.

Side of the original cube = 12 cm

Let s be the side of each smaller cube.

As we know, the volume of a cube = side³

Volume of the original cube = Volume of 8 identical cubes

⇒ (12)³ = 8 x s³

⇒ 1,728 = 8 x s³

⇒ s³ = $\dfrac{1,728}{8}$

⇒ s³ = 216

⇒ s = $\sqrt[3]{216}$

⇒ s = 6 cm

The surface area of a cube = 6 x side²

Ratio of the total surface area of the original cube to the total surface area of all the smaller cubes =

$= \dfrac{6 \times (6)^2}{6 \times (12)^2}\\[1em] = \dfrac{\cancel{6} \times 36}{\cancel{6} \times 144}\\[1em] = \dfrac{36}{144}\\[1em] = \dfrac{1}{2}$

Hence, the side of each smaller cube is 6 cm and the ratio of the total surface area of the original cube to that of all the smaller cubers is 1 : 2.