Percent change describes how much a quantity grows or shrinks compared with its original value. It is useful because many real situations are measured relative to a starting amount, such as prices, populations, grades, and measurements. A 10 dollar increase matters more for a 20 dollar item than for a 200 dollar item, so the comparison to the original value is essential.
Percent change gives a common language for increases, decreases, discounts, tax, tips, markup, and growth rates.
The percent change formula compares the difference between the new value and the original value to the original value. A positive result means the quantity increased, and a negative result means it decreased. You can also use percent multipliers, such as multiplying by 1.20 for a 20% increase or by 0.75 for a 25% decrease.
These multipliers make it faster to find sale prices, final bills, and values after repeated changes.
Key Facts
- Percent change = (new value - original value) / original value x 100%
- Amount of change = new value - original value
- Percent increase happens when new value > original value.
- Percent decrease happens when new value < original value.
- New value after an increase = original value x (1 + percent rate)
- New value after a decrease = original value x (1 - percent rate)
Vocabulary
- Original value
- The starting amount used as the comparison base in a percent change problem.
- New value
- The final amount after an increase, decrease, discount, tax, tip, or other change.
- Percent change
- The change in a quantity expressed as a percent of the original value.
- Markup
- An increase added to a cost or original price, often used to set a selling price.
- Discount
- A decrease from the original price, usually written as a percent off.
Common Mistakes to Avoid
- Using the new value as the denominator, which is wrong because percent change must be compared to the original value.
- Forgetting to multiply by 100, which leaves the answer as a decimal instead of a percent.
- Treating a decrease as a positive increase, which hides the direction of the change and can give the wrong interpretation.
- Adding percent changes directly after repeated changes, which is wrong because each change may be applied to a new base value.
Practice Questions
- 1 A backpack costs 50. What is the percent increase?
- 2 A jacket originally costs $80 and is discounted by 25%. What is the sale price?
- 3 A phone price increases by 10% one month and then decreases by 10% the next month. Is the final price equal to the original price, greater than it, or less than it? Explain.