# Calculating Percent Increase and Decrease

Want more practice with percents and related concepts?

When a quantity grows (gets bigger), then we can compute its percent increase.

When a quantity shrinks (gets smaller), then we can compute its percent decrease.

## Percent Increase

When a quantity grows (gets bigger), then we can compute its percent increase:

\begin{align} &\text{PERCENT INCREASE}\cr &\qquad = \frac{(\text{new amount} - \text{original amount})} {\text{original amount}} \end{align}

Some people write this formula with $\,100\%\,$ at the end, to emphasize that since it is percent increase, it should be reported as a percent.

So, here's an alternative way to give the formula:

\begin{align} &\text{PERCENT INCREASE}\cr &\qquad = \frac{(\text{new amount} - \text{original amount})} {\text{original amount}}\cdot 100\% \end{align}

Recall that $\,100\% = 100\cdot\frac{1}{100} = 1\,.$ So, $\,100\%\,$ is just the number $\,1\,.$ Multiplying by $\,1\,$ doesn't change anything except the name of the number! (See examples below.)

By the way, there's a very optimistic percent T-shirt here. Wear it and watch people smile!

## Visualizing Percent Increase

Percent to increase by:

Type a nonnegative number in the box above, and then:

Note:
If $\,\text{percent increase} = 75\%\,$,
then the formula
$$\text{percent increase} = \frac{(\text{new} - \text{original})} {\text{original}}$$
becomes
$$75\% = \frac{(\text{new} - \text{original})} {\text{original}}$$
and solving for ‘new’ gives:
$$\text{new} = \text{original} + 75\%(\text{original})$$

## Percent Decrease

When a quantity shrinks (gets smaller), then we can compute its percent decrease:

\begin{align} &\text{PERCENT DECREASE}\cr &\qquad = \frac{(\text{original amount} - \text{new amount})} {\text{original amount}} \end{align}

or

\begin{align} &\text{PERCENT DECREASE}\cr &\qquad = \frac{(\text{original amount} - \text{new amount})} {\text{original amount}}\cdot 100\% \end{align}

Both formulas have the following pattern:

$$\text{Percent Inc/Dec} = \frac{\text{change in amount}} {\text{original amount}}$$

or

$$\text{Percent Inc/Dec} = \frac{\text{change in amount}} {\text{original amount}}\cdot 100\%$$

Note that when you compute percent increase or decrease, you always compare how much a quantity has changed to the original amount.

Note also that the numerator in these formulas is always a positive number (or zero, if the quantity doesn't change at all).

## Visualizing Percent Decrease

Percent to decrease by:

Type a number between $\,0\,$ and $\,100\,$ in the box above, and then:

Note:
If $\,\text{percent decrease} = 25\%\,$,
then the formula
$$\text{percent decrease} = \frac{(\text{original} - \text{new})} {\text{original}}$$
becomes
$$25\% = \frac{(\text{original} - \text{new})} {\text{original}}$$
and solving for ‘new’ gives:
$$\text{new} = \text{original} - 25\%(\text{original})$$

## Examples

Question: A price rose from $\,\$5\,$to$\,\$7\,.$ What percent increase is this?
Solution: Which is the original price?
Answer: $\,\$5\,This will be the denominator. \begin{align} &\text{% increase} \ =\ \frac{(7-5)}{5}\cr &\qquad =\ \frac{2}{5} \ =\ 0.40 \ =\ 40\text{%} \end{align} or \begin{align} &\text{% increase} \ =\ \frac{(7-5)}{5}\cdot 100\%\cr\cr &\qquad =\ \frac{2}{5}\cdot 100\% \ =\ 2\cdot\frac{100}{5}\%\cr\cr &\qquad =\ 2\cdot 20\% \ =\ 40\text{%} \end{align} Notes: No matter which version of the formula you choose to use, be sure to give your answer as a percent. The units have been suppressed (left out) in the calculations above. This is common practice when it is known that units will cancel, since it makes things look simpler. Here is the same result, with the units in place: \begin{align} &\text{% increase} \ =\ \frac{\7 - \5}{\5}\cr\cr &\qquad=\ \frac{\2}{\5} \ = \overbrace{\frac 25}^{\text{units have cancelled}}\cr\cr &\qquad =\ 0.40 \ =\ 40\% \end{align} In a correct use of the formulas for percent increase and decrease, the units of the numerator and denominator will always be the same, so the units will always cancel. Question: A quantity decreased from\,90\,$to$\,75\,.$What percent decrease is this? Solution: Which is the original quantity? Answer:$\,90\,This will be the denominator. \begin{align} &\text{% decrease} \ =\ \frac{(90-75)}{90}\cr\cr &\qquad =\ \frac{15}{90} \ \approx\ 0.1667 \ =\ 16.67\text{%} \end{align} Note: In the exercises below, if an answer does not come out exact, then it is rounded to two decimal places. Question: An item went on sale for\,\$13\,$ from $\,\$16\,.$What percent decrease is this? Solution: Which is the original price? Answer:$\,\$16\,$
This will be the denominator.
\begin{align} &\text{% decrease} \ =\ \frac{(16-13)}{16}\cr\cr &\qquad =\ 0.1875 \ =\ 18.75\text{%} \end{align}