Question

How many squares with the side of 2 cm cover the surface of a rectangle with a length of 12 cm and a width of 2 cm?
(A) 2
(B) 3
(C) 5
(D) 6

Answer 1

Hint: Assume that the number of squares required to cover the surface of the rectangle is x. The side of the square is 2 cm. Use the formula, \[Area=Side\times Side\] and calculate the area of the square. Now, calculate the area of x number of squares. The length and width of the rectangle is 12 cm and 2 cm respectively. Use the formula, \[Area=Length\times Width\] and get the area of the rectangle. Since the surface of the rectangle is covered with the x number of the square, the area of the x number of the square is equal to the area of the rectangle. Now, solve it further and get the value of x.

Complete step-by-step answer:
According to the question, we have a square of side 2 cm and a rectangle of length equal to 12 cm and width equal to 2 cm.
First of all, assume that the total number of the square of side 2 cm required to cover the surface of the rectangle of length equal to 12 cm and width equal to 2 cm is x.

The total number of square = x …………………………….(1)
The side of the square = 2 cm …………………….(2)
The length of the rectangle = 12 cm ………………………….(3)
The width of the rectangle = 2 cm …………………………….(4)
We know the formula of the area of the rectangle, \[Area=Length\times Width\] …………………………(5)
From equation (3) and equation (4), we have the length and width of the rectangle.
Putting the values of the length and width of the rectangle from equation (3) and equation (4) respectively, in the formula shown in equation (5), we get
\[Area=12cm\times 2cm=24c{{m}^{2}}\] ………………………………(6)
We know the formula of the area of the square, \[Area=Side\times Side\] …………………………(7)
From equation (2), we have the side of the rectangle.
Putting the value of the side of the square in the formula shown in equation (7), we get
\[Area=2cm\times 2cm=4c{{m}^{2}}\] ………………………………(8)
From equation (8), we have the area of a square.
So, the area of x number of square = \[4x\,c{{m}^{2}}\] ………………………..…(9)
Since the x number of the square is covering the whole rectangle so, the area of x number of the square must be equal to the area of the rectangle.
So, The area of x number of square = The are of the rectangle ………………………..(10)
Now, from equation (6), equation (9), and equation (10), we get
\[\begin{align}
  & \Rightarrow 4x=24 \\
 & \Rightarrow x=\dfrac{24}{4} \\
 & \Rightarrow x=6 \\
\end{align}\]
Therefore, the number of squares of side 2 cm required to cover the surface of the rectangle having length and width equal to 12 cm and 2 cm respectively are 6.
Hence, option (D) is the correct one.

Note: In this question, one might think that the perimeter of the x number of the square must be equal to the perimeter of the rectangle. This is wrong. Since the squares are covering the surface of the rectangle, so the area of x number of the square must be equal to the area of the rectangle.