Question

A square of side x is taken. A rectangle is cut out from this square such that one side of the rectangle is half that of the square and the other is \[\dfrac{1}{3}\] of the first side of the rectangle. What is the area of the remaining portion?
A) \[\left( {\dfrac{3}{4}} \right){x^2}\]
B) \[\left( {\dfrac{7}{8}} \right){x^2}\]
C) \[\left( {\dfrac{{11}}{{12}}} \right){x^2}\]
D) \[\left( {\dfrac{{15}}{{16}}} \right){x^2}\]

Answer 1

Hint: Find the area of the square by taking a side square and the area of a rectangle by multiplying the length by width then subtract the area of the rectangle from the area of the square.

Complete step by step answer:
First, we will draw the figure to understand the question,

A rectangle is cut out from the square means the area of the square is larger than the rectangle so out the task is to subtract the area of the rectangle from the area of the square but for this first, we have to recognize the sides of the rectangle.

As this is given that one side of the rectangle is half of the side of the square, so one side of the rectangle is \[\dfrac{x}{2}\].

Now, another side of the rectangle is \[\dfrac{1}{3}\] of the first side of the rectangle so, we will multiply \[\dfrac{1}{3}\] by \[\dfrac{x}{2}\] to find the another side of the rectangle, we get \[\dfrac{1}{3} \cdot \dfrac{x}{2}\]

Now we have \[\dfrac{x}{2}\] as one side of rectangle and \[\dfrac{1}{3} \cdot \dfrac{x}{2}\] as another side.

We are multiplying both the sides of the rectangle to find the area of rectangle \[{A_r}\],

\[
  {A_r} = \dfrac{x}{2} \cdot \left( {\dfrac{1}{3} \cdot \dfrac{x}{2}} \right) \\
   = \dfrac{{{x^2}}}{{12}} \\
\]
One side of the square is x.

Now, we will find the area of square \[{A_s}\] by multiplying two sides of the square,

\[{A_s} = {x^2}\]

Now we will subtract the area of rectangle from the area of square to find the area of the remaining portion,

\[
  A = {x^2} - \dfrac{{{x^2}}}{{12}} \\
   = \dfrac{{12}}{{12}} \cdot {x^2} - \dfrac{{{x^2}}}{{12}} \\
   = \dfrac{{12{x^2} - {x^2}}}{{12}} \\
   = \dfrac{{11{x^2}}}{{12}} \\
\]

Thus, the area of the remaining portion is \[\dfrac{{11{x^2}}}{{12}}\]square units.

Hence, the correct option is (C).

Note:
Here the rectangle is cut out from the square so the area of the rectangle will be subtracted from the area of the square. The area of the square is larger than the area of the rectangle.