Question

A 5m cube is cut into as many 1 cm cubes as possible. What is the ratio of the surface area of the larger cube to that of the sum of the surface areas of the smaller cubes?
$
  (a){\text{ 1:6}} \\
  (b){\text{ 1:5}} \\
  (c){\text{ 1:25}} \\
  (d){\text{ 1:125}} \\
$

Answer 1

Hint – In this question let the larger cube be broken down into n smaller number of cubes of 1cm. The volume of this single cube will be equal to n time’s volume of the smaller cube. Then use the direct formula of the surface area of the cube to find the ratio of the surface area.

Complete step-by-step answer:

As we know that the volume of the cube is equal to the side cube.
So it is given that the side of the bigger cube = 5 cm.
So the volume ($V_1$) of the bigger cube is $ = {\left( 5 \right)^3} = 125$ $cm^3$.
Now we cut 1 cm cube from the 5 cm cube so the volume ($V_2$) of the smaller cube $ = {\left( 1 \right)^3} = 1$ $cm^3$.
Now let there are (n) such possible cubes.
So n times the volume of a smaller cube is equal to the volume of a bigger cube.
$ \Rightarrow n{V_2} = {V_1}$
$ \Rightarrow n \times 1 = 125$
$ \Rightarrow n = 125$
So there are 125 such cubes possible.
Now as we know that the cube has six faces as shown in figure and each surface represents the square so the area of the square is side square.
So the surface area of the cube is = 6 $ \times $the area of the square and the area of the square is ${(side)^2}$.
So the surface area of the bigger cube is ${\left( {S.A} \right)_1} = 6{\left( 5 \right)^2}$ $cm^2$.
And the surface area of the smaller cube is ${\left( {S.A} \right)_2} = 6{\left( 1 \right)^2} = 6$ $cm^2$.
So the surface area of 125 smaller cubes = ${\left( {S.A} \right)_3} = 125 \times 6$ $cm^2$.
So the ratio of surface area of the larger cube to that of the sum of the surface areas of the smaller cubes is
$\dfrac{{{{\left( {S.A} \right)}_1}}}{{{{\left( {S.A} \right)}_3}}} = \dfrac{{6{{\left( 5 \right)}^2}}}{{125 \times 6}} = \dfrac{{25}}{{125}} = \dfrac{1}{5}$
So this is the required ratio.
Hence option (B) is correct.

Note – The curved surface area is the area of all the curved regions of the solid. If the object has both curved surfaces and flat surfaces then the curved surface area will be the area of only the curved surface but the lateral surface area will include the curved area and the flat area excluding the base. The key point here was the multiplication of the number of cubes in which the larger cube is broken while finding the ratio of surface area, as the larger cube is used to make this 125 number of smaller cubes.