CALCULATING PERCENT INCREASE AND DECREASE

When a quantity grows (gets bigger), then we can compute its PERCENT INCREASE:

[beautiful math coming... please be patient] $\text{PERCENT INCREASE} = \frac{\displaystyle{(\text{new amount} - \text{original amount})}} {\displaystyle\text{original amount}} $

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.

Recall that $\,100\% = 100\cdot\frac{1}{100} = 1\,$.
So, $\,100\%\,$ is just the number $\,1\,$,
and multiplying by $\,1\,$ doesn't change anything except the name of the number!

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

$\text{PERCENT INCREASE} = \frac{\displaystyle{(\text{new amount} - \text{original amount})}} {\displaystyle\text{original amount}}\cdot 100\% $

When a quantity shrinks (gets smaller), then we can compute its PERCENT DECREASE:

[beautiful math coming... please be patient] $\text{PERCENT DECREASE} = \frac{\displaystyle{(\text{original amount} - \text{new amount})}} {\displaystyle\text{original amount}} $
OR
$\text{PERCENT DECREASE} = \frac{\displaystyle{(\text{original amount} - \text{new amount})}} {\displaystyle\text{original amount}}\cdot 100\% $

Both formulas have the following pattern:

[beautiful math coming... please be patient] $\text{PERCENT INCREASE/DECREASE} = \frac{\displaystyle{\text{change in amount}}} {\displaystyle\text{original amount}} $
OR
$\text{PERCENT INCREASE/DECREASE} = \frac{\displaystyle{\text{change in amount}}} {\displaystyle\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).

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.

[beautiful math coming... please be patient] $\displaystyle\text{% increase} \ =\ \frac{(7-5)}{5} \ =\ \frac{2}{5} \ =\ 0.40 \ =\ 40\text{%}$
OR
$\displaystyle\text{% increase} \ =\ \frac{(7-5)}{5}\cdot 100\% \ =\ \frac{2}{5}\cdot 100\% \ =\ 2\cdot\frac{100}{5}\% \ =\ 2\cdot 20\% \ =\ 40\text{%}$

Note: No matter which version of the formula you choose to use,
be sure to give your answer as a PERCENT.
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.

[beautiful math coming... please be patient] $\displaystyle\text{% decrease} \ =\ \frac{(90-75)}{90} \ =\ \frac{15}{90} \ \approx\ 0.1667 \ =\ 16.67\text{%}$

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.

[beautiful math coming... please be patient] $\displaystyle\text{% decrease} \ =\ \frac{(16-13)}{16} \ =\ 0.1875 \ =\ 18.75\text{%}$
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Problems Involving Percent Increase and Decrease

 
 
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
(an even number, please)