One Mathematical Cat
the Truth and Language Series in Mathematics

Dr. Carol JVF Burns
Calculating Percent Increase and Decrease

When a quantity grows (gets bigger), then we can compute its PERCENT INCREASE. When it grows from an original amount to a bigger new amount, then: $$\text{PERCENT INCREASE} = \frac{\displaystyle{(\text{new amount} - \text{original amount})}} {\displaystyle\text{original amount}}$$

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

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When a quantity shrinks (gets smaller), then we can compute its PERCENT DECREASE. When it shrinks from an original amount to a smaller new amount, then: $$\text{PERCENT DECREASE} = \frac{\displaystyle{(\text{original amount} - \text{new amount})}} {\displaystyle\text{original amount}}$$ OR $$\text{PERCENT DECREASE} = \frac{\displaystyle{(\text{original amount} - \text{new amount})}} {\displaystyle\text{original amount}}\cdot 100\%$$

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Both formulas have the following pattern: $$\text{PERCENT INCREASE/DECREASE} = \frac{\displaystyle{\text{change in amount}}} {\displaystyle\text{original amount}}$$ OR $$\text{PERCENT INCREASE/DECREASE} = \frac{\displaystyle{\text{change in amount}}} {\displaystyle\text{original amount}}\cdot 100\%$$

When you compute percent increase or decrease,
you always compare how much a quantity has changed
to the original amount.
The numerator in all these formulas is always a positive number
(or zero, if the quantity doesn't change at all).
EXAMPLES
A price rose from \$5 to \$7. What percent increase is this?

Which is the original price?   Answer: $\,\$5$This will be the denominator. $$\text{% increase} = \frac{(7-5)}{5} = \frac{2}{5} = 0.40 = 40\text{%}$$ OR $$\text{% increase} = \frac{(7-5)}{5}\cdot 100\% = \frac{2}{5}\cdot 100\% = 2\cdot\frac{100}{5}\% = 2\cdot 20\% = 40\text{%}$$ No matter which version of the formula you choose to use, be sure to give your answer as a PERCENT. A quantity decreased from$\,90\,$to$\,75\,$. What percent decrease is this? Which is the original quantity? Answer:$90$This will be the denominator. $$\text{% decrease} = \frac{(90-75)}{90} = \frac{15}{90} \approx 0.1667 = 16.67\text{%}$$ In the exercises below, if an answer does not come out exact, then it is rounded to two decimal places. An item went on sale for$\,\$13\,$ from $\,\$16\,$. What percent decrease is this? Which is the original price? Answer:$\,\$16$
This will be the denominator.
It's not always the first number to appear—be careful!

$$\text{% decrease} = \frac{(16-13)}{16} = 0.1875 = 18.75\text{%}$$
MIXED EXERCISES
WORKSHEETS
Coming soon!