CALCULATING PERCENT INCREASE AND DECREASE

Percent Increase

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.

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\% $

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.)

Visualizing Percent Increase

percent to increase by:

NOTE:
If $\,\text{percent increase} = 75\%\,$,
then the formula $$\text{percent increase} = \frac{\displaystyle{(\text{new} - \text{original})}} {\displaystyle\text{original}} $$ becomes $$75\% = \frac{\displaystyle{(\text{new} - \text{original})}} {\displaystyle\text{original}} $$ 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:

[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).

Visualizing Percent Decrease

percent to decrease by:

NOTE:
If $\,\text{percent decrease} = 25\%\,$,
then the formula $$\text{percent decrease} = \frac{\displaystyle{(\text{original} - \text{new})}} {\displaystyle\text{original}} $$ becomes $$25\% = \frac{\displaystyle{(\text{original} - \text{new})}} {\displaystyle\text{original}} $$ 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.

[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

 
 
CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10 11 12
AVAILABLE MASTERED IN PROGRESS

 
(an even number, please)