Math
Percentage calculator
Three of the questions people actually ask about percentages, each with the arithmetic shown.
The three calculations
X as a % of Y = (X ÷ Y) × 100
Change from X to Y = ((Y − X) ÷ X) × 100
The third one is the one that trips people up, because the denominator is the starting value. Percentage change always measures against where you began, not where you ended.
Why an increase and a decrease don't cancel out
If a $100 item rises 20% it becomes $120. If that $120 then falls 20%, it does not return to $100 — it lands at $96, because the 20% is now being taken from a larger base. The same asymmetry explains why a stock that drops 50% needs a 100% gain to break even.
This matters in any situation where you're comparing a rise and a fall of the same stated size. They're only equal in percentage terms, never in absolute terms.
Percentage points are not percentages
If an interest rate moves from 4% to 5%, that's a rise of one percentage point — but a 25% increase in the rate itself. News reports and adverts frequently blur the two, and the difference is usually large enough to change what the number means. When you see a claim like "rates up 25%", check which of the two is being described.
Working out a discount
For a price cut, the shortcut is to calculate what you'll pay rather than what you'll save. An item reduced by 30% costs 70% of the original, so a $80 jacket at 30% off is 0.70 × 80 = $56. One multiplication instead of two steps, and less room for error.
Stacked discounts multiply rather than add. A 20% reduction followed by a further 10% off leaves you paying 0.8 × 0.9 = 0.72, so 28% off — not 30%.
Common questions
How do I add a percentage to a number?
Multiply by 1 plus the percentage as a decimal. Adding 15% to 200 is 200 × 1.15 = 230. To take 15% off instead, multiply by 0.85.
How do I calculate a tip?
For 15%, take 10% by moving the decimal point one place left, then add half of that again. On a $64 bill: $6.40 plus $3.20 is $9.60. For 20%, double the 10% figure.
What if the starting value is zero?
Percentage change is undefined when the starting value is zero, because the calculation would divide by zero. Any growth from nothing is infinite in percentage terms, which is why reports usually describe it in absolute numbers instead.