Percentage Calculations Explained: Increase, Decrease and Difference

Percentages are the maths people most often get wrong in public, usually with money attached. The trouble is that one word covers at least four genuinely different calculations, and using the wrong one produces an answer that looks reasonable and is not.

1. What is X% of Y?

The simplest case. Multiply the number by the percentage and divide by 100.

15% of 80 = 80 × 15 ÷ 100 = 12

The shortcut worth memorising: 10% is dividing by ten, and everything else builds from there. 15% is 10% plus half of 10%. 20% is 10% doubled. 5% is half of 10%. Most mental percentage maths is just this decomposition.

2. X is what percent of Y?

Divide, then multiply by 100.

12 is what percent of 80? → 12 ÷ 80 × 100 = 15%

The mistake here is dividing the wrong way round. The number after "of" always goes on the bottom.

3. Percentage increase and decrease

This is where errors start costing money. The formula is:

((new − old) ÷ old) × 100

The denominator is always the original value. Going from 50 to 75 is a 50% increase, because the change of 25 is measured against the starting 50.

Now the part that catches everyone: going from 75 back to 50 is not a 50% decrease. It is 25 ÷ 75 = 33.3%. Increases and decreases are not symmetric, because the base changes.

Practical consequence: a stock that falls 50% needs to rise 100% to recover. A price that rises 25% needs to fall 20% to return to where it started.

4. Percentage difference (not the same thing)

When neither number is "the original" — comparing two measurements, two products, two regions — you want percentage difference, which uses the average as the base:

(|a − b| ÷ ((a + b) ÷ 2)) × 100

This is symmetric, which is the point. The difference between 50 and 75 is 40% either way round.

Using percentage change where you mean percentage difference is one of the most common errors in reporting, and it makes comparisons look larger or smaller depending on which value you happen to put first.

5. Percentage points versus percent

If an interest rate moves from 4% to 6%, that is an increase of two percentage points — and also an increase of 50 percent. Both statements are true and they describe the same event. Headlines that pick whichever sounds more dramatic are technically not lying, which is exactly why the distinction is worth knowing.

6. Stacked discounts do not add

"30% off, then an extra 20% at the till" is not 50% off.

100 → 30% off → 70 → 20% off → 56

You paid 56% of the original, so the real discount is 44%. Stacked percentages multiply: 0.70 × 0.80 = 0.56. The more discounts you stack, the further the intuitive sum drifts from the truth — and always in the shop's favour.

7. Removing a percentage that is already included

To find the pre-tax price from a total that already includes 18% tax, you do not subtract 18%. You divide:

Base = Total ÷ 1.18

Subtracting 18% from a gross figure of 118 gives 96.76, not 100. This one error appears constantly in invoices and spreadsheets.

Quick reference

Question	Formula
X% of Y	Y × X ÷ 100
X is what % of Y	X ÷ Y × 100
% increase / decrease	(new − old) ÷ old × 100
% difference	|a − b| ÷ ((a+b)/2) × 100
Add X% to a value	Y × (1 + X/100)
Remove an included X%	Y ÷ (1 + X/100)

All six are built into the percentage calculator, and the stacking case has its own discount calculator if you want to check a sale before you buy.