Question

The opposite pairs of sides of a square are increased by 40% and 30% respectively. The area of the resulting rectangle exceeds the area of the square by:
A) 42%
B) 62%
C) 82%
D) 72%

Answer 1

Hint: We will find the area of the original square first. Then we will find the length and breath of the rectangle formed by increasing the side by 40% and 30% respectively. For that you need to use the percentage formula. Then we will find the area of the new rectangle formed. Then we will find the difference between the two rectangles and then find the percentage difference and area of square.

Complete step-by-step answer:
Area of a square is given by:
A = $a^2$, where a = length of side
Let the side of square be x
Area of square =$x^2$
Now, increase in sides of squares result in a rectangle with:
Now the length becomes =\[x + \dfrac{{40}}{{100}}x = \dfrac{7}{5}x\] ​
and the breadth becomes =\[x + \dfrac{{30}}{{100}}x = \dfrac{{13}}{{10}}x\]
A rectangle is a quadrilateral and also a parallelogram.
Let us consider a rectangle with length L units and width W units.
Area of the above rectangle = L × W
Therefore, area of rectangle =\[\dfrac{7}{5}x\; \times \dfrac{{13}}{{10}}x\; = \dfrac{{91}}{{50}}{x^2}\]
The increase in the area will be the area of the square subtracted from the area of the rectangle​.
Which implies, the increase in area =\[\dfrac{{91}}{{50}}{x^2} - {x^2} = \dfrac{{41}}{{50}}{x^2}\]​
Now we will find the percentage increase by applying the formula of percentage.
The Percentage Formula is given as,
Percentage = \[\left( {\dfrac{{Value}}{{Total{\text{ }}Value}}} \right){\text{ }} \times {\text{ }}100\]
Percentage increase =\[\dfrac{{41}}{{50}}{x^2} \times \dfrac{{1}}{{{x^2}}} \times 100\; = 82\% \]
The area of the resulting rectangle exceeds the area of the square by 82%.

So, option (C) is the correct answer.

Note: The key in such questions is
To be aware of the formula of the area for various figures viz. square, rectangle etc.
Be careful of the squaring of the change term. Most of the students commit a mistake in this step in a hurry to solve the question.
Also, remember, a square is also a rectangle and the answer will be the same whether or not you assume the same values for length and width. Computation becomes easier when you assume the same values for length and width. I assumed it x you can also assume a value as 1 or 10 etc.