Percentage of a value, a proportion, or a change
A percentage means a quantity per hundred, but the correct calculation depends on which value serves as the base. This calculator handles three distinct questions: X percent of Y, X as a percentage of Y, and percentage change from an old value to a new one. Choose the question first, then enter the numbers in the matching order. The arithmetic is unit-independent: it works for scores, prices or quantities as long as the values being compared refer to the same kind of measurement.
Three formulas, three meanings
X% of Y = X / 100 × Y. X is what percent of Y = X / Y × 100. Percentage change = (new − old) / old × 100. A positive change means an increase and a negative change a decrease when the old value is positive. The old value is always the denominator for change; reversing the inputs gives a different question.
A zero denominator makes a ratio undefined. There is no ordinary percentage growth rate from zero to a positive value, and the page does not calculate a proportion of a zero whole. A change from 50 to 100 is +100%, while 100 back to 50 is −50%. The different bases explain why a percentage increase and its reversal do not cancel numerically.
Avoid common comparison mistakes
Percentage points and percentage changes are different. A rate rising from 5% to 6% increases by one percentage point but by 20% relative to its original rate. When explaining results, name the original value so the comparison is clear. The page's change mode is directional; it is not a separate symmetric percent-difference tool.
Apply chained discounts sequentially rather than adding their advertised rates. A 20% discount followed by 10% off the reduced price leaves 72% of the original price, a total discount of 28%, not 30%. Keep intermediate calculations unrounded and round only the final amount as appropriate. For specialised tax, compound-interest or loan calculations, use the matching calculator rather than assuming a single percentage operation covers the entire calculation.
Worked examples
A USD 80 item discounted by 15%
In X% of Y mode, enter 15 and 80. The discount is 0.15 × 80 = USD 12. Subtract that from USD 80 to get USD 68; the calculator's result is the discount amount, not the final selling price.
Scores and price changes
45 out of 60 gives 45 / 60 × 100 = 75%. A price rising from 80 to 100 gives (100 − 80) / 80 × 100 = 25%. The first is a proportion; the second is a change using 80 as its base.
Frequently asked questions
Why is 20% up followed by 20% down not zero change?
The second percentage applies to a new base. Starting at 100 gives 120, then 96: an overall 4% decrease.
Can a percentage exceed 100%?
Yes. A change from 10 to 30 is a 200% increase. Whether values over 100% make sense depends on what the proportion describes.
Is a one-percentage-point rise a 1% increase?
Not necessarily. Moving from 5% to 6% is one percentage point but a 20% relative increase.
Why does change from zero have no result?
The percentage-change formula divides by the old value. Dividing by zero is undefined, so there is no ordinary percentage change from that base.