Percent Increase
and Decrease
Master the formulas and strategies for solving percent change problems: from markups and discounts to population shifts and test scores.
What Is a Percent?
A percent means "per hundred." Writing \( 25\% \) is the same as writing the fraction \( \frac{25}{100} \) or the decimal \( 0.25 \). Every percent change problem starts from this idea: we are comparing a part to a whole, scaled to 100.
Defining Percent Change
When a quantity moves from an original value to a new value, the percent change tells us how large that shift is relative to where we started:
Why Percent Change Matters
Absolute change ("it went up by 10") tells only part of the story. Going from 50 to 60 is a \( 20\% \) increase, but going from 1{,}000 to 1{,}010 is only a \( 1\% \) increase: same absolute change, very different impact.
The Core Formulas
Visualizing Increase and Decrease
Increase vs. Decrease: Key Distinctions
| Percent Increase | Percent Decrease | |
|---|---|---|
| Direction | Value goes up | Value goes down |
| Formula numerator | New − Original | Original − New |
| Multiplier | \( 1 + \frac{p}{100} \) | \( 1 - \frac{p}{100} \) |
| Example: \( 25\% \) | Multiply by 1.25 | Multiply by 0.75 |
| Always relative to | The original value (denominator) | |
How to Solve Percent Change Problems
Example 1: Store Markup
A store buys a jacket for $80 and sells it for $100. What is the percent increase in price?
Example 2: Price Discount
A laptop originally costs $1{,}200. It goes on sale for $900. What is the percent decrease?
Example 3: Finding the New Value
A town has a population of 40{,}000. If the population increases by 15%, what is the new population?
Example 4: Finding the Original Value
After a 20% discount, a shirt costs $48. What was the original price?
Successive Percent Changes
When two percent changes happen one after another, you multiply the multipliers: do not add the percents.
Increase then Decrease by the Same Percent
A value that increases by \( p\% \) then decreases by \( p\% \) does not return to its original value. It ends up slightly lower.
Net: −1%
Net: −6.25%
Net: −25%
Common Mistakes to Avoid
Summary of Formulas
| Task | Formula / Method |
|---|---|
| Percent increase | \( \frac{\text{New} - \text{Original}}{\text{Original}} \times 100\% \) |
| Percent decrease | \( \frac{\text{Original} - \text{New}}{\text{Original}} \times 100\% \) |
| Find new value (increase) | \( \text{Original} \times (1 + \tfrac{p}{100}) \) |
| Find new value (decrease) | \( \text{Original} \times (1 - \tfrac{p}{100}) \) |
| Find original value | \( \frac{\text{New Value}}{\text{Multiplier}} \) |
| Successive changes | Multiply the individual multipliers |
Question 1 of 5
A bicycle originally costs $250. After a price increase, it now costs $300. What is the percent increase?
Question 2 of 5
A store reduces the price of a TV from $800 to $600. What is the percent decrease?
Question 3 of 5
After a 30% increase, a number becomes 260. What was the original number?
Question 4 of 5
A stock price increases by 40% and then decreases by 40%. What is the net percent change from the original price?
Question 5 of 5
A school's enrollment went from 1{,}500 to 1{,}800 students one year, then from 1{,}800 to 2{,}070 the next year. In which year was the percent increase greater?
Year 2: The increase is \( 2{,}070 - 1{,}800 = 270 \). The percent change is \( \frac{270}{1{,}800} = 15\% \).
Therefore, Year 1 had the greater percent increase.