You know how to find a percent of a number and how to find the whole from a part. Percent change puts those together. It answers: compared with where we started, how big was the change?
Key idea
$$\text{percent change} = \frac{\text{amount of change}}{\text{original amount}} \times 100\%$$ It is an increase when the new amount is bigger and a decrease when it is smaller.
Why it works
A percent always compares a part with a whole. "By what percent did the price go up?" means "the increase is what percent of the price we started with?" So the original amount is the whole, the 100%. After an increase of 25%, the new amount is $100\% + 25\% = 125\%$ of the original.
Worked examples
Worked example: An increase
A video game's price went from \$40 to \$50. Find the percent increase.
- Change: $50 - 40 = 10$ dollars.
- Divide by the original price, \$40: $\frac{10}{40} = 0.25$.
- $0.25 = 25\%$.
Answer: A 25% increase
Worked example: A decrease
A club had 80 members last year and has 60 now. Find the percent decrease.
- Change: $80 - 60 = 20$ members fewer.
- Divide by the original, 80: $\frac{20}{80} = 0.25 = 25\%$.
Answer: A 25% decrease
Worked example: Finding the new amount
A \$60 coat goes up in price by 15%. What is the new price?
- The new price is $100\% + 15\% = 115\%$ of the old price.
- $1.15 \times 60 = 69$.
- Check: 15% of 60 is 9, and $60 + 9 = 69$ ✓.
Answer: \$69
Common mistake
Dividing by the new amount. For \$40 to \$50, $\frac{10}{50} = 20\%$ is wrong. Check: 20% of the original \$40 is only \$8, and $40 + 8 = 48$, not 50. Always divide by the amount you started with.
The same change can be different percents. Going from 40 to 50 is a 25% increase, but going from 50 back to 40 is a 20% decrease, because the starting amount changed.