Question

Manoj’s income increased by $20\%$ and then decreased by $20\%$. What is the total percentage change in Manoj’s income?

Answer 1

Hint: Let us assume that the Manoj’s income is x. Then we are going to write the income of Manoj when it is increased by $20\%$ and the income would be $x\left( 1+\dfrac{20}{100} \right)$. After that, the income is decreased by $20\%$ so we are going to find the decrease in income in the current increment of income of Manoj which will look as follows: $x\left( 1+\dfrac{20}{100} \right)\left( 1-\dfrac{20}{100} \right)$. Now, we are asked to find the total percentage change which we are going to find by subtracting the current one with the x and then divide this subtraction by x followed by multiplication with 100.

Complete step by step answer:
In the above problem, Manoj’s income is incremented and decremented. And we have asked to calculate the percentage change in Manoj’s income.
Let us assume that the Manoj’s income is x and then the value of increment in Manoj’s income by $20\%$ is calculated as follows:
$x\left( 1+\dfrac{20}{100} \right)$
Now, the Manoj’s income is decreased by $20\%$ so we are going to find the value of the decrease in Manoj’s income by applying the percentage decrease in the above expression and we get,
$x\left( 1+\dfrac{20}{100} \right)\left( 1-\dfrac{20}{100} \right)$
There is an algebraic identity which is as follows:
$\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$
Using this algebraic identity in the above expression we get,
$\begin{align}
  & x\left( {{1}^{2}}-{{\left( \dfrac{20}{100} \right)}^{2}} \right) \\
 & =x\left( 1-{{\left( \dfrac{1}{5} \right)}^{2}} \right) \\
 & =x\left( 1-\dfrac{1}{25} \right) \\
\end{align}$
Taking 25 as LCM in the denominator in the above expression and we get,
$\begin{align}
  & =x\left( \dfrac{25-1}{25} \right) \\
 & =x\left( \dfrac{24}{25} \right) \\
\end{align}$
This is the final value of the Manoj’s income so we are going to find the final percentage change by using this expression and subtracting this expression from x and we get,
$\begin{align}
  & \left( \dfrac{24}{25} \right)x-x \\
 & =x\left(\dfrac{24}{25}-1 \right) \\
 & =x\left( \dfrac{24-25}{25} \right) \\
 & =x\left( \dfrac{-1}{25} \right) \\
\end{align}$
Dividing the above expression to x we get,
$\begin{align}
  & \dfrac{-x}{25\left( x \right)} \\
 & =\dfrac{-1}{25} \\
\end{align}$
Multiplying 100 to the above expression and we get,
$\begin{align}
  & \dfrac{-1}{25}\times 100 \\
 & =-4\% \\
\end{align}$
Hence, we have found the percentage change in income as $-4\%$.

Note: The mistake that could be possible in the above problem is that you might be thinking that first $20\%$ increase has occurred then $20\%$ decrease means net percentage will be 0 but this is the wrong approach to do the problem. Go for the right approach which we have shown above.