Question

What change in percent is made in the area of a rectangle by decreasing its length and increasing its breadth by $5\%$?
A). $25\%$ increase
B). $0.25\%$ increase
C). $0.25\%$ decrease
D). $2.5\%$ increase

Answer 1

Hint: We will use the area of the rectangle to solve this question. The area of the rectangle is given by \[A=l\times b\] where l is the length of the rectangle and b is the breadth of the rectangle. We will form two equations based on the given conditions that are the areas of the rectangle before making any changes and after making changes in it’s length and breadth.

Complete step-by-step solution:
Given that the length of a rectangle is reduced by 50% and the width is increased by 50%, we must calculate the change in percent of the rectangle's area.
To do this, we have to consider the variables of the length and breadth given rectangle
Consider the length of the rectangle be x and the breadth of the rectangle be y.
We know that,
Area of rectangle \[A=l\times b\]
Substituting the value of length and breadth of rectangle as variable x and y.
So, area of rectangle is given by \[A=x\times y\]
According to the given conditions in the question length is decreased by 50% and breadth is increased by 50%.
Consider new length be \[{{l}^{'}}\]
New length of rectangle \[{{l}^{'}}\]will be
\[{{l}^{'}}=x-\dfrac{5x}{100}\]
By simplifying further we get:
\[\Rightarrow {{l}^{'}}=\dfrac{95x}{100}\]
Consider new breadth be \[{{b}^{'}}\]
New breadth of rectangle \[{{l}^{'}}\]will be
\[{{b}^{'}}=x+\dfrac{5x}{100}\]
By simplifying further we get:
\[\Rightarrow {{b}^{'}}=\dfrac{105x}{100}\]
Then we calculate the new area of rectangle be \[{{A}^{'}}\]
New area of the rectangle using length \[{{l}^{'}}\] and breadth \[{{b}^{'}}\]
\[{{A}^{'}}=\left( \dfrac{95x}{100} \right)\left( \dfrac{105y}{100} \right)\]
By simplifying further we get:
\[{{A}^{'}}=\dfrac{9975}{10000}xy\]
After simplifying further we get:
\[{{A}^{'}}=0.9975xy\]
Now, we calculate the change in percentage in area would be
Difference in area \[=xy-0.9975xy\]
Difference in area \[=xy(1-0.9975)\]
Difference in area \[=xy(0.0025)\]
Percentage change in Area= $\dfrac{\text{Change in Area}}{\text{Initial value of Area}}\times 100$
\[\therefore \] The area of the rectangle decreased by \[0.25\%\]. So, the correct option is “option C”.

Note: The likelihood of inaccuracy in this question is determining the new area and the new % change in area by multiplying the old length and breadth by x and y. Always go for using the new length \[{{l}^{'}}\] and new breadth \[{{b}^{'}}\] to determine the new area \[{{A}^{'}}\].