Percentage Calculator

Percentages cause more everyday numerical errors than any other common calculation, mostly because the word hides three genuinely different operations. "What is 20% of 50", "20 is what percent of 50" and "what is the percentage change from 20 to 50" all use the same vocabulary and none of the same arithmetic.

The six questions

1. What is X% of Y?

result = (X ÷ 100) × Y

A tip, a discount, a tax, a commission. You know the rate and the base and want the amount.

2. X is what percent of Y?

result = (X ÷ Y) × 100

A score, a share, a proportion. You know the part and the whole and want the rate.

3. Percentage change from X to Y

result = ((Y − X) ÷ X) × 100

Growth, decline, month-on-month movement. The sign tells you the direction; the base is always the starting value.

4. Increase X by Y%

result = X × (1 + Y ÷ 100)

Adding VAT, applying a markup, projecting growth.

5. Decrease X by Y%

result = X × (1 − Y ÷ 100)

Applying a discount, removing a fee, projecting a decline.

6. Reverse percentage — X after a Y% change, what was the original?

original = X ÷ (1 ± Y ÷ 100)

The one people get wrong most. If something costs 120 including 20% VAT, the pre-VAT price is 100, not 96. Subtracting 20% from the total applies the rate to the wrong base.

Percentage points versus percent

This distinction matters far more than it appears to. If a conversion rate moves from 2% to 3%:

• it rose by one percentage point;
• it rose by 50 percent.

Both statements are true and they describe completely different magnitudes. Interest rates, unemployment figures, poll movements and tax brackets should always be quoted in percentage points when discussing absolute movement, and in percent only when discussing relative change.

Compounding and why sequential changes do not cancel

A 25% decrease requires a 33.3% increase to recover. Starting from 100: down 25% gives 75; 75 up 33.3% gives 100. The asymmetry comes from the changing base — the decrease was measured against 100, the recovery against 75.

This is why investment losses hurt disproportionately, why a "10% off then 10% off" sale is 19% off in total, and why averaging percentage changes arithmetically produces wrong answers. For a series of changes, use the geometric mean.

Precision

Percentages are exact arithmetic, but the inputs often are not. A figure quoted as "about 15%" carried through four operations should not be reported to three decimal places. Match your output precision to your input precision — usually one or two significant figures is more honest than the full float the calculator can give you.