Question

Praksit makes a cuboid of sides $ 5cm,2cm,5cm $ . How many such cuboid will be needed to form a cube?

Answer 1

Hint: You need to arrange the cuboids in such a way that the total length of all the sides become equal at some point. For that, find multiples of the given dimensions of cuboid till you find a common multiple. You can use the concept of LCM for that.

Complete step-by-step answer:
It is given in the question that, Praksit makes a cuboid of sides $ 5cm,2cm,5cm $ .
We know that, for a cuboid of side lengths, $ l,b,h $ . Where, $ l,b,h $ are length, breadth and height respectively, the volume of cuboid is given by
 $ V = lbh $
Thus the volume of the cuboid made by Praksit will be,
 $ V = 5 \times 2 \times 5 $
 $ \Rightarrow V = 50\;c{m^3} $
Now, he has to arrange those cuboids in such a way that the combined could form a cube.
We know that a cube is a 3D figure whose all sides are equal.
Let us say that all the sides of the cube that Praksit needs to make are equal to $ a $
Then the volume of the cube will be
 $ V' = {a^3} $
Let us say that Praksit needs $ n $ cuboids that could form a cube.
Then we can say that the total volume of all the cuboids Praksit used is equal to the volume of the cube he formed using them.
 $ \Rightarrow V' = nV $
 $ \Rightarrow {a^3} = n50 $ . . . (1)
From the above equation, we can clearly observe that Praksit can form a cube only if $ 50n $ is cube of some positive integer.
So, we need to find the minimum value of $ n $ for which $ 50n $ is a perfect cube.
This can be done by using LCM.
We can add the cuboids to form a cube. Means, the size of the cube must be in the multiples of the sides of the cuboid.
So, we need a common number that will be a multiple of all the numbers that represent the sides of the given cuboid. And the best way to find such a multiple is by using LCM.
 $ lcm(5,2,5) = 5 \times 2 = 10 $
Therefore, 10 is a number common to all the sides of the cuboid. Therefore, the volume of the cube formed by using those cuboids will be
 $ {a^3} = {10^3} $
 $ \Rightarrow {a^3} = 1000 $
By substituting this value in equation (1), we get
 $ 1000 = 50n $
Dividing both the sides by 50 we get
 $ n = 20 $
Therefore, Praksit will need 20 cuboids of given dimension to form a cube.
So, the correct answer is “20 cuboids”.

Note: It was important in this question to understand that you need to use LCM to solve this question. You needed a cube. And the cube has all equal sides. Therefore, you needed all the sides of the cuboids to be equal at some point. You could not slice cuboids to make their sides equal. So you needed to find a common number which would be multiple of all the sides of a cuboid. LCM is the best way to do that. Praksit cannot use less than 20 cuboids. But he definitely can use more than 20 cuboids to form a larger cube. You just need to write the sides of a cube in the multiples of LCM to get a larger cube.