Question

By what percent will the area of a square change if its side is increased by 10%?

Answer 1

Hint: In this type of question we have to use formulas for the area of a square. We know that the area of the square is given by, \[Area={{\left( Side \right)}^{2}}\]. Here, we consider the length of the side of a square equal to say \[x\]. Then by using formulas we can obtain an area for a new square. As we have to find the percent increase in the area we will use \[\%\text{ }increase=\dfrac{New\text{ }Area-Old\text{ }Area}{Old\text{ }Area}\times 100\].

Complete step by step solution:
Now, we have to find a $\%$ increase in the area of a square if its side is increased by $10\%$.
Let us suppose that the side of the square is given by \[x\]. Hence, area of the square is given by,
\[\begin{align}
  & \Rightarrow Area={{\left( Side \right)}^{2}} \\
 & \Rightarrow Area={{x}^{2}} \\
\end{align}\]
Now, as we have given that the side of the square is increased by 10\%, hence the new side will be
\[\begin{align}
  & \Rightarrow \text{New Side}=\text{Old Side}+10\% \text{ of Old Side} \\
 & \Rightarrow \text{New Side}=x+10\%\text{of}\text{ }x \\
 & \Rightarrow \text{New Side}=x+\dfrac{10}{100}\times x \\
 & \Rightarrow \text{New Side}=x+0.1x \\
 & \Rightarrow \text{New Side}=1.1x \\
\end{align}\]
Hence, the corresponding area of the square also gets changed. Thus the area of new square is given by,
\[\begin{align}
  & \Rightarrow \text{New Area}={{\left( New\text{ }Side \right)}^{2}} \\
 & \Rightarrow \text{New Area}={{\left( 1.1x \right)}^{2}} \\
 & \Rightarrow\text{New Area}=1.21{{x}^{2}} \\
\end{align}\]
As we need to find the increased area of the square,
\[\begin{align}
&\Rightarrow \%\text{ Increase}=\dfrac{New\text{ }Area-Old\text{ }Area}{Old\text{ }Area}\times 100 \\
 & \Rightarrow \%\text{ increase}=\dfrac{1.21{{x}^{2}}-{{x}^{2}}}{{{x}^{2}}}\times 100 \\
 & \Rightarrow \%\text{ increase}=\dfrac{0.21{{x}^{2}}}{{{x}^{2}}}\times 100 \\
 & \Rightarrow \%\text{ increase}=0.21\times 100 \\
 & \Rightarrow \%\text{ increase}=21\% \\
\end{align}\]
Hence, the area of the square gets increased by $21\%$ if its side is increased by $10\%.$

Note: In this type of question students may make mistakes in calculation of the new side, they have to remember to add the old side along with a percent increase to obtain the new side. Also, as the required result is in the form of percentage so that students have to calculate $\%$ increase in the area.