Question

What percent decrease in salary would exactly cancel out the \[20\] percent increase?
A.\[16\dfrac{2}{3}\% \]
B.\[18\% \]
C.\[20\% \]
D.\[33\dfrac{1}{3}\% \]

Answer 1

Hint: In order to determine the percent of a given number, we can use the following formula:
\[p(\% ) = \dfrac{x}{y} \times 100\] . Then, we have to find the offsetting percentage decrease in salary which will cancel the \[20\] percent increase. We can assume the unknown percentage decreases as \[x\] and initial salary as \[y\] and form an equation to solve it.

Complete step-by-step answer:
Let us understand the concept of percent first.
Calculating percentages is a simple mathematical procedure. If you need to find a ratio or a component of a quantity as a proportion of another quantity, you can express it as a percentage.
To find out the percent of a given number, we can use the following formula:
\[p(\% ) = \dfrac{x}{y} \times 100\]
Where:
\[x\]= Number for which percentage is to be found out;
\[y\]= Total or whole number of given data
Now we can solve the sum as follows:
Let the original salary be \[y\] and percentage decrease be \[x\].
Salary after the \[20\] percent increase will be:
\[y + 20\% \]
\[ = y + \dfrac{{20}}{{100}}y\]
\[ = 1.2y\]
We have to form the equation such that \[x\% \] decrease will offset \[20\] percent increase. It can be written as follows:
\[1.2y - x\% = y\]
\[1.2y - \dfrac{x}{{100}}(1.2y) = y\]
\[1.2y - y = 0.012xy\]
\[0.2y = 0.012xy\]
Dividing the equation, we get,
\[x = \dfrac{{0.2y}}{{0.012y}}\]
Converting it into fraction and removing the decimals, we get,
\[x = \dfrac{{20}}{{10}} \times \dfrac{{1000}}{{12}}\]
\[x = 16.6667 = 16\dfrac{2}{3}\% \]
Thus, Option (A) \[16\dfrac{2}{3}\% \] is the correct answer.
So, the correct answer is “Option A”.

Note: We can solve the sum without converting into decimal. For example- Increased salary can be written as \[\dfrac{6}{5}y\]. Now the equation will be-
\[\dfrac{6}{5}y(1 - \dfrac{x}{{100}}) = y\]
\[1 - \dfrac{x}{{100}} = \dfrac{5}{6}\]
\[\dfrac{x}{{100}} = 1 - \dfrac{5}{6}\]
Rationalizing the denominators on the right-hand side, we get,
\[\dfrac{x}{{100}} = \dfrac{{6 - 5}}{6} = \dfrac{1}{6}\]
Cross-multiplying with \[100\] on the other side, we get,
\[x = \dfrac{{100}}{6} = \dfrac{{50}}{3}\]
Now we just have to convert the fraction into a mixed number by division and we will get \[16\dfrac{2}{3}\% \].