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.
These concepts are thoroughly explored on this page.
When a quantity grows (gets bigger), then we can compute its PERCENT INCREASE: $\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, 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 $\,100\% = 100\cdot\frac{1}{100} = 1\,$. By the way, there's a very optimistic percent T-shirt here. | Visualizing Percent Increase
NOTE: $$\text{percent increase} = \frac{\displaystyle{(\text{new} - \text{original})}} {\displaystyle\text{original}} $$ becomes $$75\% = \frac{\displaystyle{(\text{new} - \text{original})}} {\displaystyle\text{original}} $$ and solving for ‘new’ gives: $$ \text{new} = \text{original} + 75\%(\text{original}) $$ |
Percent DecreaseWhen a quantity shrinks (gets smaller), then we can compute its PERCENT DECREASE: $\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: $\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, Note also that the numerator in these formulas is always a POSITIVE number | Visualizing Percent Decrease
NOTE: $$\text{percent decrease} = \frac{\displaystyle{(\text{original} - \text{new})}} {\displaystyle\text{original}} $$ 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.
$\displaystyle\text{% increase} \ =\ \frac{(7-5)}{5} \ =\ \frac{2}{5} \ =\ 0.40 \ =\ 40\text{%}$
OR
$\displaystyle\text{% increase} \ =\ \frac{(7-5)}{5}\cdot 100\% \ =\ \frac{2}{5}\cdot 100\% \ =\ 2\cdot\frac{100}{5}\% \ =\ 2\cdot 20\% \ =\ 40\text{%}$
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:
$$ \text{% increase} \quad=\quad \frac{\$7 - \$5}{\$5} \quad=\quad \frac{\$2}{\$5} \quad= \overbrace{\frac 25}^{\text{units have cancelled}} =\quad 0.40 \quad=\quad 40\% $$
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.
$\displaystyle\text{% decrease} \ =\ \frac{(90-75)}{90} \ =\ \frac{15}{90} \ \approx\ 0.1667 \ =\ 16.67\text{%}$
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.
$\displaystyle\text{% decrease} \ =\ \frac{(16-13)}{16} \ =\ 0.1875 \ =\ 18.75\text{%}$