What is the percent increase from 25 to 40?

Hint: Assume the first number as ${{x}_{1}}$ and the second number as ${{x}_{2}}$. Find the increase in the numerical value by subtracting ${{x}_{2}}$ from ${{x}_{1}}$. Now, find the fractional increase in the data by dividing the obtained difference with the initial number ${{x}_{1}}$. Finally, to find the percentage increase, multiply the obtained fraction with 100 to get the answer.

Complete step by step answer:
Here we have been given that a number increases from 25 to 40 and we are asked to calculate the percentage increase in the number.
Let us assume the initial number 25 as ${{x}_{1}}$ and the final number 40 as ${{x}_{2}}$. So the increase in the quantity will be the difference of the final and initial number. Therefore subtracting ${{x}_{1}}$ from ${{x}_{2}}$ we get,
$\begin{align}
  & \Rightarrow {{x}_{2}}-{{x}_{1}}=40-25 \\
 & \Rightarrow {{x}_{2}}-{{x}_{1}}=15 \\
\end{align}$
Now, the fractional increase will be the ratio of the above difference and the initial number, so we get,
$\begin{align}
  & \Rightarrow \dfrac{\left( {{x}_{2}}-{{x}_{1}} \right)}{{{x}_{1}}}=\dfrac{15}{25} \\
 & \Rightarrow \dfrac{\left( {{x}_{2}}-{{x}_{1}} \right)}{{{x}_{1}}}=\dfrac{3}{5} \\
\end{align}$
Finally, to change a fraction into the percentage we multiply it with 100 which will give the percentage increase from the fractional increase. So multiplying the above relation with 100 we get,
$\begin{align}
  & \Rightarrow \dfrac{\left( {{x}_{2}}-{{x}_{1}} \right)}{{{x}_{1}}}\times 100\%=\dfrac{3}{5}\times 100\% \\
 & \Rightarrow \dfrac{\left( {{x}_{2}}-{{x}_{1}} \right)}{{{x}_{1}}}\times 100\%=60\% \\
\end{align}$

Hence, the percentage increase from 25 to 40 is 60%.

Note: Always remember that the percentage increase or decrease is calculated upon the initial number and not the final number. This is the reason that we always take the initial number in the denominator of the fractional increase. Note that for the decrease in the number we subtract the final number from the initial number and find the percentage decrease as $\dfrac{\left( {{x}_{1}}-{{x}_{2}} \right)}{{{x}_{1}}}\times 100\%$. These concepts are used in the chapter ‘profit and loss’.