Percentage Calculator

How to Calculate Percentages — 5 Common Scenarios

Percentages show up in a handful of recurring situations: finding a percentage of a number (like a tip or discount), figuring out what percentage one number is of another, tracking how much something changed, comparing two values without a clear "before" and "after," and reversing a percentage to find an original value. This calculator covers all five with the exact formula for each.

Percentage Change vs. Percentage Difference

These two get confused constantly. Percentage change has a direction — it treats one value as the "original" and measures how far the other value moved from it:

$$\%\text{Change} = \frac{\text{New} - \text{Original}}{|\text{Original}|} \times 100$$

So ((100−80)/80)×100 = +25% going from 80 to 100, but ((80−100)/100)×100 = −20% going from 100 to 80. Percentage difference has no direction — it measures the gap relative to the average of the two values:

$$\%\text{Diff} = \frac{|A-B|}{|(A+B)/2|} \times 100$$

For the same 80 and 100: $\frac{|80-100|}{(80+100)/2} \times 100 = \frac{20}{90} \times 100 \approx 22.2\%$ — a single number regardless of which value is listed first, unlike the two different percentage-change results above.

How to Reverse a Percentage

If you know a final amount and the percentage that was added or subtracted to reach it, you can work backward to the original by dividing the final amount by (1 + percentage) if it was added, or by (1 − percentage) if it was subtracted. A 120-dollar price after a 20% markup means the original was 120 ÷ 1.20 = 100 dollars — not 120 × 0.80, which is a common mistake.

Common Percentage Mistakes

The biggest one: assuming a percentage increase followed by the same percentage decrease returns you to the original value. It doesn't, because each percentage applies to a different base. Another common error is confusing percentage points with percentage change — going from a 10% rate to a 15% rate is a 5 percentage-point increase, but a 50% relative increase (5 ÷ 10 × 100).

A Brief History of Percent

The word "percent" comes from the Latin "per centum," meaning "by the hundred." Calculations based on hundredths date back to ancient Rome, where taxes were often levied as a fraction of 100 units, and the practice continued through the Middle Ages in European trade and interest calculations. The modern "%" symbol evolved gradually from Italian merchants' handwritten abbreviation "per 100" (often written "p 100" or "p cento"), which was gradually compressed over the 15th and 16th centuries into a stylized symbol resembling two small circles separated by a slash — the form still used today. Percentages became especially important with the rise of banking, taxation, and statistics in the following centuries, since expressing any quantity as "out of 100" made comparing wildly different scales — interest rates, populations, test scores — far more intuitive than raw fractions.

Percentage Terms You Should Know

Percentage Point — a unit used when comparing two percentages directly; the difference between 10% and 15% is 5 percentage points, distinct from the 50% relative change between them.

Basis Point — one-hundredth of a percentage point, commonly used in finance for precision (100 basis points equal 1 percentage point).

Ratio vs. Percentage — a ratio compares two quantities directly (3 to 4); a percentage always expresses a quantity relative to 100, making it easier to compare across different totals.

Compounding — applying a percentage repeatedly over time, where each application is calculated on the new total rather than the original, causing growth (or decline) to accelerate.