If a price rose 25%, what discount returns it to the original?

20%. The increase was calculated on the smaller original figure and the decrease is calculated on the larger new one, so the reversing percentage is always smaller than the original rise.

Do 20% off and an extra 10% off come to 30% off?

No — 28%. You pay 0.80 × 0.90 = 0.72 of the list price. Stacked percentage discounts multiply, so the combined discount is always less than the sum.

How do I find the pre-tax price from a tax-inclusive total?

Divide by (1 + the tax rate). At 20% tax, divide the gross by 1.20. Subtracting 20% from the gross is the common mistake and understates the net price.

Is a percentage point the same as a percent?

No. Moving from 4% to 6% is a two percentage point rise and a 50% relative increase. Both are correct, so check which one a statistic means before repeating it.

Percentages, discounts and tax: the errors that cost money

Money7 min readPublished 17 September 2026

Percentages are the most-used piece of mathematics in ordinary life and the most reliably mishandled. The errors aren't arithmetic slips — they come from a small number of specific misconceptions that almost everyone shares, and each one has a predictable direction.

This guide walks through the four that cost real money: reversing a percentage change, stacking discounts, extracting tax from an inclusive figure, and confusing percentages with percentage points.

A percentage always refers to something

Every percentage is a fraction of a specific base, and almost every percentage error comes from losing track of which base. '20% off' is 20% of the list price. '20% more' is 20% of the original amount. When the base changes between two operations, the percentages stop behaving the way intuition expects.

This is why a 20% increase followed by a 20% decrease doesn't return you to where you started. $100 rises to $120, then falls by 20% of $120 — which is $24, not $20 — leaving $96. The two percentages were calculated against different bases, so they don't cancel.

Change	Reversal needed	Why
+10%	−9.09%	The base is now larger
+25%	−20%	25% of the original is 20% of the new figure
+100%	−50%	Doubling is undone by halving
−50%	+100%	A halved value must double to recover
−90%	+900%	This is why large losses are so hard to recover

Discounts multiply, they don't add

'Extra 10% off already reduced prices' does not mean 10% more off the original. Each discount applies to whatever the price is when it's applied, so they compound downward rather than summing.

A 30% discount followed by a 20% coupon leaves you paying 0.70 × 0.80 = 0.56 of the list price. That's 44% off, not 50%. The gap widens with larger discounts: two 50% discounts leave you paying 25%, a 75% total reduction rather than the 100% that addition would suggest.

The order of two percentage discounts doesn't matter — multiplication is commutative. But order matters a great deal when a flat-amount coupon is involved. On a $100 item with 20% off and a $15 coupon: percentage first gives $65, coupon first gives $68. Which one applies is decided by the retailer's checkout, not by you, so it's worth watching the running total.

Pulling tax out of an inclusive total

Adding tax is easy and everyone gets it right: multiply by (1 + rate). Extracting it is where invoices go wrong, because the instinct is to take the rate off the gross figure — and that is always wrong, in a consistent direction.

At 18% tax, a gross invoice of 1,180 has a net value of 1,180 ÷ 1.18 = 1,000, and tax of 180. Taking 18% off the gross instead gives 967.60, understating the net by 32.40. The error happens because 18% of the larger gross figure is a bigger number than 18% of the net — you're subtracting the wrong quantity.

Add tax: net × (1 + rate). 1,000 at 18% → 1,180.
Extract tax: gross ÷ (1 + rate). 1,180 at 18% → 1,000 net, 180 tax.
Never: gross × (1 − rate). This understates the net every time.
The same arithmetic applies whether it's called GST, VAT or sales tax — only the rate and the name differ.

Percentages versus percentage points

When an interest rate moves from 4% to 6%, that is a rise of two percentage points and a relative increase of 50%. Both descriptions are accurate, and they sound wildly different, which is exactly why the distinction gets exploited in headlines and marketing.

The rule: use percentage points when comparing two percentages, and percentages when comparing two quantities. 'Unemployment rose 50%' and 'unemployment rose one percentage point' can describe the same move from 2% to 3%. Whenever a percentage change of something that is itself a percentage is quoted without specifying which, assume the more dramatic framing was chosen deliberately.

Quick checks worth doing in your head

10% of anything is the decimal point moved one place. 5% is half of that, 20% is double it — most mental percentage work reduces to those three.
x% of y always equals y% of x. 16% of 25 is awkward; 25% of 16 is obviously 4.
To reverse-solve a total: divide the part by the percentage as a decimal. If 45 is 15% of something, it's 45 ÷ 0.15 = 300.
'Up to 70% off' means at least one item is 70% off. It says nothing about the item you're holding.
'Buy one get one free' is 50% off across two items, and no discount at all if you only wanted one.

Frequently asked questions

If a price rose 25%, what discount returns it to the original?: 20%. The increase was calculated on the smaller original figure and the decrease is calculated on the larger new one, so the reversing percentage is always smaller than the original rise.
Do 20% off and an extra 10% off come to 30% off?: No — 28%. You pay 0.80 × 0.90 = 0.72 of the list price. Stacked percentage discounts multiply, so the combined discount is always less than the sum.
How do I find the pre-tax price from a tax-inclusive total?: Divide by (1 + the tax rate). At 20% tax, divide the gross by 1.20. Subtracting 20% from the gross is the common mistake and understates the net price.
Is a percentage point the same as a percent?: No. Moving from 4% to 6% is a two percentage point rise and a 50% relative increase. Both are correct, so check which one a statistic means before repeating it.