Question

The length of a rectangle is increased by 60%. By what percent should the width be decreased to maintain the same area?
(a) $33\dfrac{1}{2}\%$
(b) $35\dfrac{1}{2}\%$
(c) $37\dfrac{1}{2}\%$
(d) $39\dfrac{1}{2}\%$

Answer 1

Hint: Area of a rectangle is a definite quantity which is dependent on the respective length and breadth of the rectangle. So, to maintain a constant area while increasing the length will demand decrease in breadth of the rectangle and for increasing the breadth, we have to decrease the length of the rectangle. In this case we have increased the length, so to nullify any change in area we will decrease the breadth.

Complete step-by-step answer:
Area of a closed body can be defined as the space occupied by a flat surface of the body. It defines the extent up to which the body exists in the two-dimensional space. The expression of area of a body is in square units whether it may be meter, centimeter, inch etc.
For example, area of a square lamina of side a unit is nothing but ${{a}^{2}}$ square units.
Now the expression of area of a rectangle having dimensions of length as l unit and breadth as b unit $=l\times b$ square units.
According to our problem, let initial length of rectangle = L units
Initial width of the rectangle = B units
Hence, initial area of rectangle would be $L\times B$ square units.
New length of rectangle $=L+0.6L=1.6L$
Since area is constant, assuming new breadth be ${B}'$
$\begin{align}
  & A=1.6L\times {B}' \\
 & \therefore L\times B=1.6L\times {B}' \\
 & \therefore {B}'=\dfrac{B}{1.6} \\
\end{align}$
So, decrease in width would be:
$\begin{align}
  & B-{B}'=B-\dfrac{B}{1.6} \\
 & \Rightarrow B-{B}'=\dfrac{0.6B}{1.6} \\
\end{align}$
So, the percentage decrease in width would be:
\[\begin{align}
  & \dfrac{B-{B}'}{B}=\dfrac{0.6}{1.6} \\
 & \text{Percent decrease }=\dfrac{0.6}{1.6}\times 100 \\
 & \text{Percent decrease }=37\dfrac{1}{2} \\
\end{align}\]
Therefore, on increasing the length to 60 percent we must decrease the breadth by 37.5 percent.
Hence, option (c) is correct.

Note:The key step for solving this problem is the knowledge of the area of the rectangle and effects associated with the increase or decrease of dimension of the rectangle. Since the length of the rectangle is increased by 60%, we can directly write that the new length of the rectangle would be 100+60=160% of L or we can write as 1.6L. If it was reduced by 60%, we could have written as 100-60=40% of L or 0.4L. Such small tricks can help save time in competitive exams.