A cube has a volume of 72. If it is divided into 8 equal cub

ardz24 wrote:A cube has a volume of 72. If it is divided into 8 equal cubes, the ratio of an edge of a smaller cube to an edge of the original cube is

(A) 1 : 2
(B) 1: 3
(C) 1 : 3√2
(D) 2 : 9
(E) 1 : 9

I'm confused how to set up the formulas here. Can any experts help?

If the size of the edge of original cube is = a, its volume would be = a^3
And if the size of the edge of new cube is =b, its volume would be = b^3
Then ratio of volume of the new cube to the volume of original cube would be b^3 : a^3
In the present case a^3=72 and b^3= 72/8 = 9
Therefore b^3 : a^3 = 9 : 72
Or b^3 : a^3 = 1 : 8
Taking cube root of both the sides we get
b : a = 1 : 2
hence A is the correct answer

A cube has a volume of 72. If it is divided into 8 equal cubes, the ratio of an edge of a smaller cube to an edge of the original cube is

(A) 1 : 2
(B) 1: 3
(C) 1 : 3√2
(D) 2 : 9
(E) 1 : 9

I'm confused how to set up the formulas here. Can any experts help?

Hi ardz24,
Let's take a look at your question.

Volume of original cube = 72
Original cube is divided into 8 equal cubes, therefore, volume of each of the new cube will be:
$$=\frac{72}{8}=9$$

Let, edge length of original cube be 'x' and edge length of each of the new cubes is 'y', then,
$$x^3=72$$
$$y^3=9$$

Volume of new cube : Volume of Original cube
$$y^3\ :\ \ x^3 = 9\ :\ 72$$
$$y^3\ :\ \ x^3 =1\ :\ 8$$

But we are asked to find the ratio of an edge of a smaller cube to an edge of the original cube, i.e., x : y
$$y^3\ :\ \ x^3 =1\ :\ 8$$
$$y^3\ :\ \ x^3 =1^3\ :\ 2^3$$
$$y\ :\ \ x =1\ :\ 2$$

Therefore, Option A is correct.

Hope it helps.
I am available if you'd like any follow up.