In this guide
Percentages appear everywhere: discounts, pay raises, interest, test scores and price changes. The formulas are short, but a few traps catch almost everyone. This guide covers the essentials with numbers you can verify.
The three basic formulas
- X% of Y: X ÷ 100 × Y. So 15% of 80 is 12.
- X is what percent of Y: X ÷ Y × 100. So 12 is 15% of 80.
- Percentage change: (new − old) ÷ old × 100.
Percentage increase
A price rising from $80 to $100 is a change of $20. Divide by the old price: 20 ÷ 80 = 0.25, or a 25% increase. A salary going from $50,000 to $52,500 is 2,500 ÷ 50,000 = 5%.
Percentage decrease
The same formula gives a negative number for a fall. From $100 to $80 is (80 − 100) ÷ 100 = −20%. Notice that going up 25% and going down 20% connect the same two prices. The percentage is always measured against the starting value, so increases and decreases are not mirror images.
Why +10% then −10% does not return to the start
Multiply the factors: 1.10 × 0.90 = 0.99. A 10% rise followed by a 10% fall leaves you 1% below where you began. Likewise, 100 up 50% is 150, and 150 down 50% is 75.
Working backwards: the original price
If an item costs $60 after a 25% discount, it was not $60 plus 25%. Divide by the remaining fraction: 60 ÷ 0.75 = $80.
Percent versus percentage points
If an interest rate rises from 4% to 5%, that is a rise of 1 percentage point, but a 25% relative increase (1 ÷ 4). News headlines often mix the two up.
Markup versus margin
Suppose an item costs a shop $60 and sells for $100. The profit is $40. The markup is 40 ÷ 60 = 66.7% of cost, while the margin is 40 ÷ 100 = 40% of the selling price. They are different numbers for the same profit.
Mental math shortcuts
- 10% of a number: move the decimal point one place left.
- 5% is half of 10%, and 20% is double 10%.
- 15% is 10% plus 5%. A 15% tip on $48 is 4.80 + 2.40 = $7.20.
Common mistakes
- Dividing by the new value instead of the old value when finding a change.
- Adding percentages that apply to different bases.
- Reading a rise of 50 percentage points as a 50% rise.
Try the calculators
- Percentage Calculator: Percent of, what percent, percent change
- Discount Calculator: Sale price and how much you save
- Salary Raise Calculator: New pay and percentage raise
General information only, not professional advice. See our editorial policy.