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Percentage formulas at a glance

Every percentage problem links three numbers: a part, a whole and a rate, with part = rate ÷ 100 × whole. Knowing any two gives the third. The % sign stands for the number 0.01 (NIST Guide to the SI, §7.10.2), so dividing by 100 is all it takes to turn a percentage into a decimal: 7.5% is 0.075.

The last two rows use the same pair of numbers and disagree because they divide by different bases: 250 for the change, and the average, 265, for the difference.

The symbol Δ% in the formula means percent change. For signed inputs, percent difference uses the absolute average; it is undefined when that average is zero.

How to work out a percentage by hand

A student scores 43 out of 52 on a test. To express that as a percentage:

1. Divide the part by the whole: 43 ÷ 52 = 0.826923…
2. Multiply by 100: 82.6923…%.
3. Round once, at the end: 82.7%.

The calculator's "X is what % of Y" mode, with X = 43 and Y = 52, returns 82.6923%. Rounding in step 1 instead, to 0.83, reports 83%, which is 0.3 points too high and can be enough to cross a grade boundary.

For mental arithmetic, build the rate out of easy pieces. Moving the decimal point one place left gives 10%, two places gives 1%, half of 10% is 5%, and dividing by 4 or 8 gives 25% or 12.5%. A 15% tip on a bill of 84 is 8.40 (10%) plus 4.20 (5%), or 12.60. The tip and bill split calculator does the same sum and divides it between people.

Converting between fractions, decimals and percentages is the same operation in different notation. Multiply a decimal by 100 to get the percentage; divide the numerator by the denominator to get the decimal.

Percent change and percentage points

When the two values being compared are themselves percentages, there are two ways to state the change. A mortgage rate moving from 2.5% to 3.0% rose by 0.5 percentage points, found by plain subtraction, and by 20% in relative terms, found with the percent change formula (0.5 ÷ 2.5 × 100). In basis points, where one basis point is 0.01 percentage point, the move is 50 basis points.

A fall and a rise of the same percentage do not cancel, because the rise is measured from the lower value. The rise needed to recover from a fall of p% is p ÷ (100 − p) × 100:

The gap widens quickly: an investment that loses half its value has to double to break even.

The calculator divides a change by the absolute value of the starting number. Moving from −50 to −25 therefore reads as a 50% increase, because the value rose towards zero. A spreadsheet formula written as (new − old) ÷ old returns −50% for the same move, so check which convention a report uses before comparing figures. When the starting value is 0 no percent change exists; report the absolute change instead.

Reverse percentages: finding the original value

To undo a percentage change, divide by the multiplier that produced it. After a 15% discount a jacket costs 68. The sale price is 85% of the original, so the original was 68 ÷ 0.85 = 80. In the calculator, choose "Y is X% of what" with Y = 68 and X = 85.

The tempting shortcut, adding 15% of 68 back on, gives 68 × 1.15 = 78.20, which is 1.80 short, because the 15% was taken from 80, not from 68. The same error appears when removing sales tax or VAT from a gross price; the GST, VAT and sales tax calculator handles that case directly.

Markup and margin are a reverse-percentage pair. An item that costs 80 and sells for 100 has a markup of 25% (20 ÷ 80) and a margin of 20% (20 ÷ 100). The profit is the same 20 in both; only the base changes. The discount and markup calculator converts between the two.

Common percentage mistakes

Successive discounts multiply rather than add. A 20% discount followed by a further 10% off leaves 0.80 × 0.90 = 0.72 of the price, a total discount of 28%, not 30%. The order of the two discounts makes no difference.

"50% off, then an extra 50% off" takes three quarters off the price, not all of it. Other errors that recur:

• Adding successive increases. Two 10% rises multiply to 1.1 × 1.1 = 1.21, a 21% increase. Interest that is added to the balance each period grows the same way; the compound interest calculator runs that over many periods.
• Dividing a total change by the number of years. A price that rises from 100 to 150 over five years has gone up 50%, but not 10% a year: five 10% rises compound to 1.1⁵ = 1.61, a 61% increase. The steady yearly rate that produces 50% is 1.5^(1/5) − 1 = 8.45%. The scientific calculator evaluates 1.5^(1/5) - 1 in one line.
• Confusing "percent of" with "percent more than". 150% of 40 is 60, while 150% more than 40 is 40 + 60 = 100. In the calculator the first is "X% of Y" and the second is "Increase Y by X%".
• Averaging percentages with different bases. If 30% of 200 people in one town and 60% of 50 people in another answer yes, the combined share is (60 + 30) ÷ 250 = 36%. The simple average of 30% and 60%, 45%, gives the town of 50 the same weight as the town of 200.
• Decreasing by more than 100%. A positive quantity cannot fall by more than 100% and stay positive. "Decrease Y by X%" with X above 100 returns a negative number: 50 decreased by 120% is −10. If that is not what the situation allows, the rate or the base is wrong.
• Rounding the rate before using it. A rate rounded to 8% instead of 8.45% understates five years of growth by more than 3 percentage points, since 1.08⁵ = 1.469. Keep full precision through the calculation and round only the figure you report.