Percentage Calculator

Three different questions get called "a percentage problem", and they use three different formulas. This page asks you to pick which one you mean, because guessing from the numbers alone is how wrong answers happen. The first mode answers "what is X percent of Y". NIST Special Publication 811 defines the percent symbol as simply the number 0.01, so 15% is 0.15 and nothing more mysterious than that. Divide your percentage by 100 to get a decimal multiplier, then multiply the base value by it: 15% of 240 is 240 x 0.15 = 36. The page shows you the intermediate multiplier deliberately, because seeing 0.15 makes the operation obvious in a way that the symbol does not. The second mode reverses it: "the first value is what percentage of the second?" Divide the part by the whole and multiply by 100. So 36 out of 240 is 36 / 240 = 0.15, then x 100 = 15%. The whole cannot be zero - there is no meaningful share of nothing - and the schema rejects it rather than returning a nonsense figure. The third mode is percentage change, and it is the one that catches people out. The formula is the new value minus the original, divided by the magnitude of the original, times 100. NIST expresses relative difference as (R1 - R2) / R2, and the U.S. Census Bureau applies the same change-over-original relation, marks a zero starting estimate as undefined, and reverses the sign when the starting estimate is negative. Dividing by the absolute value of the original states that sign rule directly, so the sign of your result always follows the direction of the change rather than flipping because you started below zero. Now the non-obvious part. Percentage change is not symmetric, and treating it as though it were is probably the single most common numeric error in everyday reporting. Go from 100 to 150 and that is a 50% increase, because the denominator is 100. Come back down from 150 to 100 and that is not a 50% decrease - it is a 33.33% decrease, because the denominator is now 150. Reversing a move does not reverse its percentage. This is why a stock that falls 50% needs to rise 100% just to get back to even, and why "prices fell 20% then rose 20%" leaves you 4% below where you started rather than back at par. Whenever you see a percentage change quoted, the first question worth asking is what the denominator was. A related trap is confusing a percentage change with a percentage-point change. If an interest rate moves from 4% to 5%, that is a rise of one percentage point but a 25% increase in the rate itself. Both statements are true and they describe very different things. All the arithmetic here runs through exact decimal operations rather than binary floating point, so 0.1 + 0.2 behaves the way you expect it to and results are stable to six decimal places. If you are applying a percentage repeatedly rather than once - growth compounding year over year - the compound interest calculator handles the exponent properly, and for pay rises or deductions applied to earnings the salary calculator does the conversion between pay periods at the same time.