Question

Salary of a person is first increased by \[20\% \], then it is decreased by \[20\% \]. Change in his salary is
A.\[4\% \] Decrease
B.\[4\% \] Increase
C.\[8\% \] Decrease
D.\[20\% \] Increase

Answer 1

Hint: This is a simple percentage question. To solve the above question, first we will obtain increased salary by assuming the initial salary as a variable. Then we will derive the new decreased salary of the person by applying the percentage method on the salary received after the increased percentage. Find the difference in initial salary and salary obtained at the end to find the change and convert into percentage.
* We write \[m\% \] of $x$ as \[\dfrac{m}{{100}} \times x\]

Complete step by step answer:
Let the initial salary of a person be \[x\] … (1)
According to the question, the salary of the person was increased by \[20\% \]
We calculate the salary of person after the increase in percentage of salary as initial salary plus \[20\% \] of initial salary
\[ \Rightarrow \] Salary after increased percentage \[ = x + 20\% \, of \, x\]
Using the method of percentage we can write
 \[ \Rightarrow \] Salary after increased percentage \[ = x + \dfrac{{20}}{{100}}x\]
Take LCM in RHS of the equation
\[ \Rightarrow \] Salary after increased percentage \[ = \dfrac{{100x + 20x}}{{100}}\]
\[ \Rightarrow \] Salary after increased percentage \[ = \dfrac{{120x}}{{100}}\]
\[\therefore \] Salary after \[20\% \] increase in initial salary \[ = \dfrac{{120x}}{{100}}\] … (2)
Now salary is decreased by \[20\% \]
We will decrease \[20\% \] of the salary obtained in equation (2)
\[ \Rightarrow \] Salary after \[20\% \] decrease \[ = \dfrac{{120x}}{{100}} - 20\% \left( {\dfrac{{120x}}{{100}}} \right)\]
\[ \Rightarrow \] Salary after \[20\% \] decrease \[ = \dfrac{{120x}}{{100}} - \dfrac{{20}}{{100}} \times \left( {\dfrac{{120x}}{{100}}} \right)\]
Take\[\dfrac{{120x}}{{100}}\]common
\[ \Rightarrow \] Salary after \[20\% \] decrease \[ = \dfrac{{120x}}{{100}}\left( {1 - \dfrac{{20}}{{100}}} \right)\]
Take LCM inside the bracket
\[ \Rightarrow \] Salary after \[20\% \] decrease \[ = \dfrac{{120x}}{{100}}\left( {\dfrac{{100 - 20}}{{100}}} \right)\]
\[ \Rightarrow \] Salary after \[20\% \] decrease \[ = \dfrac{{120x}}{{100}} \times \dfrac{{80}}{{100}}\]
Cancel the same factors from numerator and denominator
\[ \Rightarrow \] Salary after \[20\% \] decrease \[ = \dfrac{{12x \times 8}}{{100}}\]
\[ \Rightarrow \] Salary after \[20\% \] decrease \[ = \dfrac{{96x}}{{100}}\] … (3)
Change in salary is given by subtracting initial salary from final salary and dividing the answer by initial salary
Subtract equation (1) from equation (3) in numerator and write salary from equation (1) in denominator.
\[ \Rightarrow \] Change in salary \[ = \dfrac{{\dfrac{{96x}}{{100}} - x}}{x}\]
Take LCM in numerator of RHS
\[ \Rightarrow \] Change in salary \[ = \dfrac{{96x - 100x}}{{100x}}\]
\[ \Rightarrow \] Change in salary \[ = \dfrac{{ - 4x}}{{100x}}\]
Cancel the same terms from numerator and denominator.
\[ \Rightarrow \] Change in salary \[ = \dfrac{{ - 4}}{{100}}\]
To convert the fraction into percentage we multiply the fraction by 100
\[ \Rightarrow \] Percentage change in salary \[ = \left( {\dfrac{{ - 4}}{{100}} \times 100} \right)\% \]
Percentage change in salary \[ = - 4\% \]
Since the sign is negative, the percentage indicates a decrease in salary.
\[\therefore \] Change in his salary is \[4\% \] decrease.

Hence, Option A is correct.

Note:
Many times students make the mistake of not writing the percentage with respect to the salary, keep in mind in these types of questions we take percentage with reference to salary, so we add or subtract accordingly. Also, add the value of percentage whenever it is stated that there was an increased percentage and subtract the percentage whenever it is stated that there was decreased percentage.
Also, we don’t write the fraction having denominator 100 in simplest form so we can cancel that value of 100 directly when we use the formula of percentage.