Question

A cube of side $5cm$ is cut into as many $1cm$ cubes as possible. What is the ratio of the surface areas of the original cube to that of the sum of the surface areas of the smaller cubes?

Answer 1

As we know that the surface area of a cube $=6a^2$, where $a$ is side of a cube.
Since, side of cube $=5cm$
$\therefore$ Surface area of the cube $=6\times (5{)}^2=6\times 25$
$=150c{m}^2$
Volume of the cube $=5^{3}c{m}^3$
$=125c{m}^3$
Now, volume of the cube with side $1cm=1^3=1c{m}^3$
Let there are $n$ such possible of cubes of side of length $1cm$
Then, $n\times 1=125$
$\therefore n=125$
So, there are $125$ such possible cubes.
Surface area of the cube with side $1cm$ $=6\times (1{)}^2=6c{m}^2$
$\therefore$ Surface area of $125$ cubes with side $1cm=125\times 6=750c{m}^2$
Ratio of the surface area of the original cube to that of the sum of the surface area of the smaller cubes
$=\frac{150}{750}=\frac{3}{15}=1:5$