Question

The edges of a cube are 20 cm. How many small cubes of 5 cm can be formed from this cube?
(A) 4
(B) 32
(C) 64
(D) 100

Answer 1

Hint: Use the formula of volume of a cube, \[\text{Volume = }\left( \text{edge} \right)\times \left( \text{edge} \right)\times \left( \text{edge} \right)\] . Using this formula, get the volume of the bigger cube and the smaller cube. Assume the number of smaller cubes be x. The, calculate the volume of x number of smaller cubes. Since the smaller cubes are formed from the bigger cube, the volume of all smaller cubes is equal to the volume of the bigger cube. Now, solve it further.

Complete step by step solution:

According to the question, it is given that the edges of the cube are equal to 20 cm. We have to find the number of cubes whose edges are equal to 5 cm that can be formed from the bigger cube.
The edges of the bigger cube = 20 cm ………………(1)
The edges of the smaller cube = 5 cm ………………….(2)
We know the formula to get the volume of a cube, \[\text{Volume = }\left( \text{edge} \right)\times \left( \text{edge} \right)\times \left( \text{edge} \right)\]………………..(3)
From equation (1), we have the edges of a bigger cube.
Now, putting the value of edges of the bigger cube in equation (3), we get
\[\text{Volume of bigger cube= }\left( \text{edge} \right)\times \left( \text{edge} \right)\times \left( \text{edge} \right)\]
\[\Rightarrow \text{Volume of bigger cube=}\left( 20\text{ cm} \right)\times \left( 20\text{ cm} \right)\times \left( 20\text{ cm} \right)\] …………………………(4)
From equation (2), we have the edges of a smaller cube.
Now, putting the value of edges of the smaller cube in equation (3), we get
\[\text{Volume = }\left( \text{edge} \right)\times \left( \text{edge} \right)\times \left( \text{edge} \right)\]
\[\Rightarrow \text{Volume =}\left( 5\text{ cm} \right)\times \left( 5\text{ cm} \right)\times \left( \text{5 cm} \right)\] …………………………(5)
Let the number of smaller cubes be x.
From equation (5), we have
The volume of 1 smaller cube = \[\left( 5\text{ cm} \right)\times \left( 5\text{ cm} \right)\times \left( \text{5 cm} \right)\] .
The volume of x number of smaller cubes = \[x\left( 5\text{ cm} \right)\times \left( 5\text{ cm} \right)\times \left( \text{5 cm} \right)\] ……………………………(6)
Since the smaller cubes are formed from the bigger cube, the volume of all smaller cubes is equal to the volume of the bigger cube. So,
Volume of the bigger cube = Volume of all smaller cubes ………………….(7)
From equation (4), equation (6), and equation (7), we have
Volume of the bigger cube = Total volume of all smaller cubes
\[\begin{align}
  & \Rightarrow \left( 20\text{ cm} \right)\times \left( 20\text{ cm} \right)\times \left( 20\text{ cm} \right)=x\left( 5\text{ cm} \right)\times \left( 5\text{ cm} \right)\times \left( \text{5 cm} \right) \\
 & \Rightarrow \dfrac{\left( 20\text{ cm} \right)\times \left( 20\text{ cm} \right)\times \left( 20\text{ cm} \right)}{\left( 5\text{ cm} \right)\times \left( 5\text{ cm} \right)\times \left( \text{5 cm} \right)}=x \\
 & \Rightarrow 4\times 4\times 4=x \\
 & \Rightarrow 64=x \\
\end{align}\]
Therefore, the number of smaller cubes is 64.
Hence, the correct option is (C).

Note: We can also solve this question directly by using the formula.
\[\text{Number of smaller cubes=}\dfrac{\text{Volume of bigger cube}}{\text{Total volume of smaller cube}}\]
The volume of smaller cube = \[\left( 5\text{ cm} \right)\times \left( 5\text{ cm} \right)\times \left( \text{5 cm} \right)\] .
The volume of bigger cube = \[\left( 20\text{ cm} \right)\times \left( 20\text{ cm} \right)\times \left( 20\text{ cm} \right)\] .
Now, the number of smaller cubes = \[\dfrac{\left( 20\text{ cm} \right)\times \left( 20\text{ cm} \right)\times \left( 20\text{ cm} \right)}{\left( 5\text{ cm} \right)\times \left( 5\text{ cm} \right)\times \left( \text{5 cm} \right)}=4\times 4\times 4=64\] .
Therefore, the number of smaller cubes is 64.
Hence, the correct option is (C).