Increase, or Decrease, by a Percentage

In this section we learn how to increase, or decrease, a quantity by a percentage.
For example we'll learn how to decrease the cost a of a $\$42$ pair of jeans by $15\%$, or how to increase the price of a smartphone by $12\%$.

We start by learning the formula for increasing, or decreasing, by a percentage. We'll then work through some examples.

Formula to Increase, or Decrease, by a Percentage

Formula to Increase by a Percentage

To increase a quantity $A$ by $X\%$ we need to add $X\%$ of $A$ to $A$, that's: \[A+\frac{X}{100}\times A = A\begin{pmatrix}1+\frac{X}{100}\end{pmatrix}\] Formula to Decrease by a Percentage

To decrease a quantity $A$ by $X\%$ we need to subtract $X\%$ of $A$ from $A$, that's: \[A-\frac{X}{100}\times A = A\begin{pmatrix}1-\frac{X}{100}\end{pmatrix}\] Both of these formula will become clearer with the following examples.

Example

Increase $80$ by $20\%$.

Solution

Using the formula to increase by a percentage we calculate this as follows: \[\begin{aligned} 80\begin{pmatrix}1+\frac{20}{100} \end{pmatrix} &= 80\begin{pmatrix} 1 + 0.2 \end{pmatrix} \\ & = 80 \times \begin{pmatrix}1.2\end{pmatrix} \\ &= 80 \times 1.2 \\ & = 96 \end{aligned}\] So $80$ increased by $20\%$ equals $96$.

Example

Decrease $420$ by $5\%$.

Solution

Using the formula to decrease by a percentage we calculate this as follows: \[\begin{aligned} 420\begin{pmatrix}1-\frac{5}{100}\end{pmatrix} &= 420\begin{pmatrix}1 - 0.05 \end{pmatrix} \\ &= 420 \begin{pmatrix}0.95 \end{pmatrix} \\ &= 420 \times 0.95 \\ &= 399 \end{aligned}\] So $420$ decreased by $5\%$ is equal to $399$.

Example

After negotiating with her boss, Cathy obtained an $8\%$ pay rise.
Given that her hourly income used to be $\$30$, how much will it be after her raise?

Solution

Using the formula to increase by a percentage we calculate Cathy's raise as follows: \[\begin{aligned} 30\begin{pmatrix}1+\frac{8}{100} \end{pmatrix} &= 30\begin{pmatrix}1+0.08 \end{pmatrix} \\ &= 30 \begin{pmatrix}1.08 \end{pmatrix} \\ &= 30 \times 1.08 \\ &= 32.4 \end{aligned}\] So after her pay-rise, Cathy's hourly income is $\$32.40$.

Example

During a sale, a $\$98$ pair of shoes' price is decreased by $40\%$.
Calculate the new price.

Solution

Using the formula to decrease by a percentage we calculate this as follows: \[\begin{aligned} 98\begin{pmatrix}1-\frac{40}{100}\end{pmatrix} &= 98 \begin{pmatrix} 1 - 0.4 \end{pmatrix}\\ &= 98\begin{pmatrix} 0.6 \end{pmatrix} \\ &= 98 \times 0.6\\ &= 58.8\end{aligned}\] $98$ decreased by $40\%$ equals $58.8$, so the price of the pair of shoes, after the discount, is $\$58.80$.