Percentages show up in prices, grades, tips, interest rates and statistics. A handful of formulas covers nearly every situation, and knowing the traps keeps you from making costly mistakes.
Finding a percentage of a number
Multiply the number by the percentage divided by 100. A 15% discount on $80 is 80 × 0.15 = $12, so the sale price is $68.
Percentage change
Use (new − old) ÷ old × 100. If a price rises from $80 to $100, the change is 20 ÷ 80 = 25% higher. Notice the asymmetry: a fall from $100 back to $80 is only 20%, because the base is now larger. Gains and losses of the same percentage do not cancel out.
Percentage points versus percent
If an interest rate rises from 4% to 5%, it increased by one percentage point, but by 25% in relative terms. News reports and ads sometimes blur the two, so ask which one is meant.
Stacked discounts
A 20% discount followed by another 10% discount is not 30% off. After the first discount you pay 80% of the price, and the second takes 10% off that amount: 0.80 × 0.90 = 0.72. The total discount is 28%.
Working backwards from a final price
If an item costs $60 after a 20% discount, the original price is 60 ÷ 0.80 = $75. To remove a percentage, divide by one minus that percentage rather than adding it back. The same idea applies to sales tax: a total of $107 with 7% tax means a pre-tax price of 107 ÷ 1.07 = $100.
Rounding and zero denominators
Percent change is undefined when the starting value is zero, since you cannot divide by zero. A good calculator should say so rather than show a misleading number. Round only at the end, not in the middle of a multi-step calculation.
Shares of a total
To find what percent one number is of another, divide the part by the whole and multiply by 100. If you scored 42 out of 50, that is 42 ÷ 50 = 0.84, or 84%. If 18 of 24 students passed, the pass rate is 75%.
Raises, growth and repeated change
A 5% raise on $50,000 is $52,500. A second 5% raise is applied to the new amount, giving $55,125, not $55,000. Repeated percentage changes multiply rather than add, which is the same idea behind compound interest.
Mental math shortcuts
Ten percent is the number with the decimal moved one place left. One percent is two places left. Five percent is half of ten percent, and 25% is a quarter. To find 15% of $64, take 10% ($6.40) and add half of it ($3.20) for $9.60. These shortcuts are handy for checking a calculator result or a bill.
Reading claims carefully
"50% more" and "50% less" are not opposites, because they start from different bases. "Up to 70% off" may apply to a single item. A risk that rises from 2 in 1,000 to 3 in 1,000 is a 50% relative increase, but only 0.1 percentage points in absolute terms. When a statistic sounds dramatic, ask what the starting number was and how many people or items were measured.
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Percentage Calculator · Discount Calculator · Sales Tax Calculator
This guide is for general information only. It is not financial, medical, or legal advice.