If each side of a rectangle is increased by 50%, then find the percentage increase in its area.

Let's consider a rectangle with length L and width W, and its initial area is given by:

${\mathrm{A}}_1=\mathrm{L}\times \mathrm{W}$

If each side of the rectangle is increased by 50%, the new length ($\mathrm{L}\prime$) and new width ($\mathrm{W}\prime$) will be:

$\mathrm{L}\text{'}=1.5\times \mathrm{L}$

$\mathrm{W}\prime =1.5\times \mathrm{W}$

The new area (${\mathrm{A}}_2$​) is given by the product of the new length and new width:

${\mathrm{A}}_2=\mathrm{L}\prime \times \mathrm{W}\prime$ 

${\mathrm{A}}_2=(1.5\times \mathrm{L})\times (1.5\times \mathrm{W})$

${\mathrm{A}}_2=2.25\times (\mathrm{L}\times \mathrm{W})$

${\mathbf{A}}_{\mathbf{2}}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{25}\mathbf{\times }{\mathbf{A}}_{\mathbf{1}}\mathbf{}$

Now, let's find the percentage increase in the area:

$\text{Percentage Increase}=\left(\frac{\text{New Area}-\text{Initial Area}}{\text{Initial Area}}\right)\times 100$

Substitute the values:

$\text{Percentage Increase}=\left(\frac{2.25\times {\mathrm{A}}_1-{\mathrm{A}}_1}{{\mathrm{A}}_1}\right)\times 100$

$\text{Percentage Increase}=\left(\frac{1.25\times {\mathrm{A}}_1}{{\mathrm{A}}_1}\right)\times 100$

 $\mathbf{\text{Percentage Increase}}\mathbf{=}\mathbf{125}\mathbf{\%}$

Therefore, if each side of a rectangle is increased by 50%, its area will be increased by 125%.