Question

The perimeter of a rectangle is $70$ cm. If its length is decreased by $5$ cm and its width is increased by $5$ cm, its area will increase by $50{\mathrm{cm}}^2$c. find the length and the width of the original rectangle?

Answer 1

Find the length and the width of the original rectangle:

Let $a$ and $b$ be the original length and width respectively of the rectangle.

The perimeter of a rectangle is $70$ cm.
$\begin{array}{rcl}P& =& 2a+2b\\ & \Rightarrow & 2a+2b=70\\ & \Rightarrow & 2b=70-2a\\ & \Rightarrow & b=35-a\end{array}$

Again, area of the rectangle is $A=a\cdot b$.

If its length is decreased by $5$ cm and its width is increased by $5$ cm, its area will increase by $50{\mathrm{cm}}^2$.

Use the above information to form the equation in length and width:
$$\begin{gathered} (a-5)(b+5)=a\cdot b+50 \\ \Rightarrow (a-5)(35-a+5)=a\cdot (35-a)+50 \\ \Rightarrow (a-5)(40-a)=35a-a^2+50 \\ \Rightarrow 40a-a^2-200+5a=35a-a^2+50 \\ \Rightarrow 45a-200=35a+50 \\ \Rightarrow 10a=250 \\ \Rightarrow a=25\mathrm{cm} \end{gathered}$$

Calculate the width as shown:

$$\begin{gathered} b=35-25\mathrm{cm} \\ =10\mathrm{cm} \end{gathered}$$

Hence, the length and the width of the original rectangle is $25$ cm and $10$ cm respectively.