Percent Change
Percent Change
Percent change measures how much a quantity has grown or shrunk, expressed as a percentage of its original value. It's one of the most common ways real-world (and contest) problems disguise plain arithmetic.
The Percent Change Formula
The general formula is:
\text{Percent change} = \frac{\text{new value} - \text{original value}}{\text{original value}} \times 100\%A positive result is a percent increase; a negative result is a percent decrease. The key detail students often get wrong: the denominator is always the original (starting) value, never the new value.
Percent Increase and Decrease
To increase a value v by p\%, multiply by \left(1 + \frac{p}{100}\right). To decrease v by p\%, multiply by \left(1 - \frac{p}{100}\right). For example, increasing 80 by 25\% gives 80 \times 1.25 = 100, and decreasing 80 by 25\% gives 80 \times 0.75 = 60. Notice these are not symmetric — increasing then decreasing by the same percent does not return you to the original value.
Discounts and Markups
A discount is a percent decrease applied to a price; a markup is a percent increase. A \$40 item discounted 20\% costs 40 \times 0.80 = \$32. A \$40 item marked up 20\% costs 40 \times 1.20 = \$48.
Percent-of-a-Percent Traps
A classic contest trap: successive percent changes do not add together directly. If a price increases 10\% and then decreases 10\%, the net effect is not 0\%. Starting from 100: after a 10\% increase, the price is 110; after a 10\% decrease of that new price, it's 110 \times 0.90 = 99 — a net 1\% decrease, not zero! This happens because the second percent is taken of a different base value than the first. Whenever you see multiple percent changes in a row, apply them one at a time, each time using the most recently updated value as the new base — never add the percents together.
Example
A shirt's price is increased by 30\% to \$52. What was the original price?
Step 1: Let the original price be p. A 30\% increase means the new price is 1.3p.
Step 2: Set up the equation: 1.3p = 52
Step 3: Solve: p = \frac{52}{1.3} = 40.
The original price was \$40. (Check: 40 \times 1.3 = 52. \checkmark)
Key Takeaways
- Percent change = (new - original) \div original, times 100\% — the denominator is always the original value.
- Increase by p\%: multiply by (1+\frac{p}{100}). Decrease by p\%: multiply by (1-\frac{p}{100}).
- Successive percent changes do not simply add — apply them one at a time to the running value.
- To find an original value from a changed one, divide by the multiplier rather than reversing the percent by subtraction.