Question : A square of side $p$ is taken. A rectangle is cut out from this square such that the length of one side of the rectangle is equal to half of the length of one side of the square and the length of another side of the rectangle is equal to $\frac{1}{3}$rd of the length of the first side of the rectangle. What is the area of the portion of the square that remained after the rectangle was cut out?
Option 1: $\frac{7}{8} p^2$
Option 2: $\frac{3}{4} p^2$
Option 3: $\frac{11}{12} p^2$
Option 4: $\frac{15}{16} p^2$
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Correct Answer: $\frac{11}{12} p^2$
Solution : Use the formula, The area of the rectangle = Length × Breadth The area of the square = Side 2 According to the question, The length of the rectangle $=\frac{1}{2}\times p=\frac{p}{2}$ The breadth of the rectangle $=\frac{1}{3}\times \frac{p}{2}=\frac{p}{6}$ The area of the rectangle $=\frac{p}{2} \times \frac{p}{6}=\frac{p^2}{12}$ The area of the square $=p^2$ The area of the portion of the square that remained after the rectangle was cut out = The area of the square – The area of the rectangle ⇒ $p^2-\frac{p^2}{12}=\frac{11}{12} p^2$ Hence, the correct answer is $\frac{11}{12} p^2$.
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