The ratio between the lengths of the edges of two cubes are in the ratio 3 : 2. Find the ratio between their :

(i) total surface area

(ii) volume.

It is given that the ratio between the lengths of the edges of two cubes are in the ratio 3 : 2.

Let the edges of two cubes be 3a and 2a.

As we know that total surface area of cube = 6 x side²

Ratio of total surface area of two cubes = $\dfrac{\text{total surface area of 1st cube}}{\text{total surface area of 2nd cube}}$

$= \dfrac{6 \times (3a)^2}{6 \times (2a)^2}\\[1em] = \dfrac{6 \times 9a^2}{6 \times 4a^2}\\[1em] = \dfrac{\cancel{6} \times 9a^2}{\cancel{6} \times 4a^2}\\[1em] = \dfrac{9\cancel{a^2}}{4\cancel{a^2}}\\[1em] = \dfrac{9}{4}$

Hence, the ratio of total surface area of 2 cubs is 9 : 4.

(ii) As we know that volume of cube = side³

Ratio of volume of two cubes = $\dfrac{\text{Volume of 1st cube}}{\text{Volume of 2nd cube}}$

$= \dfrac{(3a)^3}{(2a)^3}\\[1em] = \dfrac{27a^3}{8a^3}\\[1em] = \dfrac{27\cancel{a^3}}{8\cancel{a^3}}\\[1em] = \dfrac{27}{8}$

Hence, the ratio of volume of 2 cubes is 27 : 8.