Question

If the length of a rectangle is decreased by $40\%$ and the breadth is increased by $30\%$, then find the $\%$ change in the area of the rectangle.

• A. $22\%$ Increase
• B. $22\%$ Decrease
• C. $10\%$ Decrease
• D. No change

Answer 1

The correct option is B $22\%$ Decrease

Let $x$ be the length and $y$ be the breadth of the rectangle.

The original area of the rectangle will be $xy$.

After $40\%$ decrease in the length, length of the new rectangle will be,

$\begin{array}{cc}{x}^{\prime }& =x\left(1-\frac{40}{100}\right)\\ & =\frac{3}{5}x\end{array}$

After $30\%$ increase in the length, length of the new rectangle will be,

$\begin{array}{cc}{y}^{\prime }& =y\left(1+\frac{30}{100}\right)\\ & =\frac{13}{10}y\end{array}$

$\begin{array}{cc}\text{Area of new rectangle}& =x^{\prime }{y}^{\prime }\\ & =\frac{3x}{5}\times \frac{13y}{10}\\ & =\frac{39xy}{50}\end{array}$

$\begin{array}{cc}\text{Percentage change}& =\frac{\left(\frac{39xy}{50}-xy\right)}{xy}\times 100\\ & =-22\%\end{array}$

Here negative sign indicates that the area of the final rectangle is less than the original rectangle which means a $22\%$ decrease in the area of the rectangle takes place.