Question

The length of the given rectangle is increased by $20\% $ and the breadth is decreased by $20\% $, then the area
A. Remains the same
B. Increases by $5\% $
C. Decreases by $5\% $
D. Decreases by $4\% $

Answer 1

Hint: We can assume the length and breadth of the rectangle as l and b. Then we can calculate the new length and breadth using the given percentages. Then we can calculate the areas of the original rectangle and new rectangle using the equation $A = l \times b$. Now we can find the change in the area using the equation$\Delta A = A' - A$ and percentage of change using $\dfrac{{\Delta A}}{A} \times 100\% $

Complete step by step answer:

Let the length and breadth of the rectangle be l and b respectively.
Then the area of the rectangle is given by,$A = l \times b$
The length has increased by $20\% $. So, the new length is given by,
$l' = l + 20\%$ of l
$ \Rightarrow l' = l + \dfrac{{20}}{{100}}l$
On simplification we get,
$ \Rightarrow l' = \dfrac{{120}}{{100}}l = \dfrac{6}{5}l$
The breadth is decreased by $20\% $. So, the new breadth is given by,
$b' = b - 20\% $ of b
 $ \Rightarrow b' = b - \dfrac{{20}}{{100}}b$
On simplification we get,
$ \Rightarrow b' = \dfrac{{80}}{{100}}b = \dfrac{4}{5}b$
Hence the new area is given by, $A' = l' \times b'$
On substituting the values, we get,
$A' = \dfrac{6}{5}l \times \dfrac{4}{5}b$
After multiplication we get,
$A' = \dfrac{{24}}{{25}}l \times b$
The change in area is given by,
$\Delta A = A' - A$
$ \Rightarrow \Delta A = \dfrac{{24}}{{25}}l \times b - l \times b$
$ \Rightarrow \Delta A = - \dfrac{1}{{25}}l \times b$
As the change in area is negative, the area decreases
The change in percentage is given by, $\dfrac{{\Delta A}}{A} \times 100\% $
On substituting, we get,
$\% = \dfrac{{ - \dfrac{1}{{25}}l \times b}}{{l \times b}} \times 100\% $
$ = - \dfrac{1}{{25}} \times 100\% = - 4\% $
So, the area decreases by 4%.
Therefore, the correct answer is option D.

Note: While calculating the increment and decrement, the sign of the percentage must be taken care of. For finding the rate of change or percentage of change of some parameters, we write the new value in terms of the original value. While calculating the change in the area, the order is important. It is always the initial value subtracted from the final value. If the change is negative, we can say that its value is decreased. For finding the percentage of change, the change must be divided with the initial value, not with the changed value. If the change is negative, the percentage of change also will be negative.