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Percentages and VAT explained: adding, removing and stacking

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Percentages look like simple arithmetic, and most of the time they are. The mistakes come from one detail that is easy to lose: every percentage is a percentage of something, and that something changes from one step to the next. This guide covers the three basic calculations, the difference between percent and percentage points, how to add and remove VAT without error, what a “we pay the VAT” offer is really worth, and why two discounts in a row are worth less than their sum.

The three basic calculations

Almost every percentage question is one of three, and each has one formula:

The third one is where most errors start. The change is always measured against the starting value, so going from 120 to 150 is a 25% increase, while going back from 150 to 120 is a 20% decrease.

Percentages & change

Percent and percentage points

When the values being compared are themselves percentages, there are two different ways to describe the change. If an interest rate goes from 4% to 5%, it has risen by one percentage point, the plain difference between the two. It has also risen by 25%, because 1 is a quarter of 4. Both statements are true; they answer different questions, and confusing them makes a change look four times bigger or smaller than it is.

The same applies to tax rates, unemployment figures and interest: “VAT rises by two points” means from 21% to 23%, not by 2% of 21%.

Up and down are not symmetrical

Adding 20% and then removing 20% does not bring you back. 100 plus 20% is 120; 20% of 120 is 24, so taking it off leaves 96. The second percentage is taken from a larger number, so it removes more than the first one added.

The same asymmetry means that a fall needs a bigger rise to recover: after a 50% drop, a value has to double, a 100% rise, to return to where it started. To undo a percentage increase you divide by the same factor you multiplied by; you never subtract the same percentage.

Adding and removing VAT

VAT is a percentage of the price before tax, the net price. Adding it is a multiplication: with 23% VAT, a net price of 100 becomes 100 × 1.23 = 123.

Removing it is the step that goes wrong. The VAT in a price of 123 is not 23% of 123, which would be 28.29; it is 23. The correct way is to divide the gross price by the same factor: 123 ÷ 1.23 = 100, and the VAT is the difference. The VAT is therefore a smaller share of the final price than the rate suggests:

VAT rateFactor: multiply to add, divide to removeShare of the final price that is VAT
5%1.054.76%
10%1.109.09%
19%1.1915.97%
20%1.2016.67%
21%1.2117.36%
22%1.2218.03%
23%1.2318.70%

The last column is the rate divided by 100 plus the rate: for 23%, 23 ÷ 123 = 18.70%. It is the fastest way to find the VAT inside a price: a shelf price of 49.99 with 23% VAT contains 49.99 × 0.1870 = 9.35 of VAT.

Tax & discounts

What “we pay the VAT” is worth

Shops sometimes advertise that they pay the VAT: you pay the price without tax. With 23% VAT, an item marked 123 costs 100. That is a discount of 23 on 123, or 18.70%, not 23%. The discount such an offer is really worth is always the share in the last column of the table above, never the VAT rate itself. It is still a real discount; it is just smaller than the number on the poster suggests.

Two discounts are not their sum

A discount of 20% followed by another 10% is not a 30% discount. The second one applies to the price already reduced: 100 minus 20% is 80, and 10% of 80 is 8, so you pay 72. The total discount is 28%.

In general, discounts combine by multiplying what remains: 0.80 × 0.90 = 0.72, so 72% of the price is left. The order does not matter, 10% then 20% also gives 72, but the sum always overstates the result, and the gap grows with the size of the discounts: 50% and then another 50% leaves a quarter of the price, not nothing.

Discount calculator

Standard VAT rates

These are the standard rates in September 2026 in the countries whose languages nTools is published in, and in the United States, Japan and Australia. Every country also has reduced rates for some goods and services, such as food, books or energy, so check the rate that applies to what you are buying or selling.

CountryStandard rate
Portugal23% (22% in Madeira, 16% in the Azores)
Spain21%
France20%
Germany19%
Italy22%
United Kingdom20%
United StatesNo VAT; state sales tax from 0% to 7.25%, plus local taxes
Japan10% (consumption tax)
Australia10% (GST)

In the United States, five states have no statewide sales tax, California has the highest state rate at 7.25%, and counties and cities add their own, so the average combined rate is above 10% in Louisiana. The tax is usually added at the till instead of being included in the shelf price; in Australia, as in Europe, it is included.

Brazil has no single VAT yet. Its consumption taxes are being replaced by two new ones, the federal CBS and the state and municipal IBS, under the reform enacted by Constitutional Amendment 132/2023 and Complementary Law 214/2025. 2026 is a test year with rates of 0.9% and 0.1%, and the old taxes are phased out until 2033. The difference matters for the arithmetic: ICMS, the main state tax today, is included in its own base, so a rate of 18% amounts to about 21.95% on the price without it, while CBS and IBS are calculated on the price without them, like VAT in Europe.

Frequently asked questions

How do I remove 23% VAT from a price?

Divide by 1.23. A price of 61.50 is 50.00 without VAT, and the VAT is 11.50. Taking 23% off the gross price gives the wrong answer.

Why doesn’t adding 20% and then removing 20% get me back?

Because the second 20% is taken from a bigger number. 100 plus 20% is 120, and 20% of 120 is 24, so you land on 96. To undo an increase, divide by the same factor.

Are two discounts of 10% the same as 20%?

No. They leave 0.90 × 0.90 = 81% of the price, a total discount of 19%.

Does nTools know my country’s VAT rate?

No. The calculators work with the rate you enter, so they suit any country and any reduced rate, but you have to choose the rate that applies.

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