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Three calculations, three different questions

Most percentage confusion comes from reaching for the wrong operation, so it helps to be clear which question you are asking.

"What is X% of Y" finds a part of a known total — a 20% tip on a 45 bill, or 15% VAT on a price. "X is what percent of Y" goes the other way, expressing one number as a proportion of another: 37 correct answers out of 50 is 74%. "Percent change from X to Y" measures movement between two values, which is the one people get wrong most often.

Percentage change is not symmetrical

This is the single most useful thing to understand about percentages. A rise and the fall that undoes it are never the same percentage, because each is measured against a different starting point.

Go from 100 to 150 and that is a 50% increase. Go back from 150 to 100 and that is a 33.3% decrease — the same 50 units, a different base. It follows that a 50% loss needs a 100% gain to recover, and a 20% discount followed by a 20% increase does not return you to the original price. It leaves you 4% below it.

This is why an investment down 50% needs to double to break even, and why chaining discounts is not additive: 20% off then a further 10% off is not 30% off, it is 28%.

Percentage points versus percent

These are different units and conflating them produces genuinely misleading statements. If an interest rate moves from 2% to 3%, that is a rise of one percentage point — but a 50% increase in the rate.

Both descriptions are true and they sound wildly different, which is exactly why the distinction gets exploited in advertising and reporting. When a figure is itself a percentage, always say whether a change is expressed in points or as a relative percentage.

Working backwards from a total that includes tax

A common and genuinely error-prone task: you have a price that already includes 20% tax and need the pre-tax figure. Subtracting 20% is wrong, because the 20% was calculated on the smaller pre-tax amount, not on the total you are holding.

Divide instead. A total of 120 including 20% tax is 120 ÷ 1.20 = 100 pre-tax, with 20 of tax. Subtracting 20% from 120 would have given 96, which is 4 short. The same applies to reversing any percentage increase: divide by one plus the rate rather than subtracting the rate.

A couple of shortcuts worth knowing

Percentages are commutative, which is surprisingly useful for mental arithmetic: X% of Y always equals Y% of X. Working out 4% of 75 is awkward; 75% of 4 is obviously 3. Same answer, far less effort.

For tips, 10% is a decimal point shift, so 20% is that doubled and 15% is 10% plus half of it. And to find any percentage quickly, 1% is the number with the decimal moved two places left — multiply up from there.

Frequently asked questions

Why is a 50% increase not cancelled by a 50% decrease? #

Because each is measured against a different base. 100 up 50% is 150; 150 down 50% is 75, not 100. Reversing a 50% rise needs a 33.3% fall, and recovering from a 50% loss needs a 100% gain.

How do I remove tax from a price that already includes it? #

Divide rather than subtract. For 20% tax, divide the total by 1.20. Subtracting 20% is wrong because the tax was calculated on the smaller pre-tax figure — on a total of 120 you would get 96 instead of the correct 100.

What is the difference between percent and percentage points? #

If a rate moves from 2% to 3%, that is one percentage point, but a 50% relative increase. Both are true and sound very different, which is why the distinction matters whenever the underlying figure is itself a percentage.

Is 20% off then 10% off the same as 30% off? #

No. The second discount applies to the already-reduced price, so you pay 0.8 × 0.9 = 72% of the original — a 28% discount, not 30%. Stacked discounts always come out smaller than their sum.

How do I calculate a percentage increase? #

Subtract the old value from the new, divide by the old value, and multiply by 100. From 80 to 100: (100 − 80) ÷ 80 × 100 = 25%. Dividing by the new value instead is the usual mistake.

Is there a quick way to do percentages in my head? #

Two tricks. X% of Y equals Y% of X, so 4% of 75 becomes the much easier 75% of 4 = 3. And 1% is just the decimal point moved two places left, which you can then multiply up.