Question

The length of a 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: Find the value of the new length and breadth of the rectangle by using the initial length and breadth and the change in them. Then use the new values to find the new area and find the percentage change in area using the relation between the initial and the final area and multiplies by 100 .

Complete step-by-step answer:
Given, the length of a given rectangle is increased by 20% and the breadth is decreased by 20%
Let us consider the initial length of the rectangle= \[l\]
Then the increase in length= \[\dfrac{{20}}{{100}}l\]
The new length is given by \[{l_1} = l + \dfrac{{20}}{{100}}l\]
\[ \Rightarrow {l_1} = l + \dfrac{1}{5}l\]
\[ \Rightarrow {l_1} = \dfrac{{6l}}{5}\]
Let us consider the initial length of the breadth= \[b\]
Then the decrease in breadth = \[ - \dfrac{{20}}{{100}}l\]
The new breadth is given by \[{b_1} = b - \dfrac{{20}}{{100}}b\]
\[ \Rightarrow {b_1} = b - \dfrac{1}{5}b\]
\[{b_1} = \dfrac{{4b}}{5}\]
The initial area is given by \[A = lb\]
The area of the rectangle after the change is given by \[{A_1} = \dfrac{{6l}}{5} \times \dfrac{{4b}}{5}\]
\[ \Rightarrow {A_1} = \dfrac{{24}}{{25}}lb\]
Percentage decrease in area is given by %decrease = [(original area -new area) * 100]/lb
\[\dfrac{{[(lb - 24lb/25) \times 100]}}{{lb}}\]
\[
   = \dfrac{{\dfrac{{lb}}{{25}} \times 100}}{{lb}} \\
   = 4\% \\
\]
Therefore, D. decreases by 4% is the correct solution.

Note: In these types of questions first form the equation according to given values and then apply the percentage change formula to get an answer. Percentage formula is used to find the amount or share of something in terms of 100.