There are a whole range of VAT-inclusive prices that cannot be mathematically arrived at if the tax is calculated from the VAT-free price. One example is 3.99. These prices are also possible, but then the VAT-free price must be interpreted as rounded.
VAT is calculated from the VAT-free price. The VAT amount can be calculated using the formula:
VAT Amount = Pre-VAT Price * VAT %
If the VAT rate is 20% and the VAT-free price is £100, the VAT amount will be £20 and the VAT-inclusive price will be £120. The VAT-inclusive price is the VAT-free price + VAT amount.
| Pre-VAT Price | £100 |
| VAT % | 20% |
| VAT Amount | £20 |
| Sale price | £120 |
However, when selling to consumers, the starting point is often that you want the price including VAT to be something. The VAT amount can also be easily calculated from the VAT-taxed price, and then the formula is:
VAT Amount = Sale price - (Sale price / (1 + VAT %))
This formula is derived using the information that Sale price = Pre-VAT Price + VAT Amount.
So, for example, let's take a product whose final price is £100, when the VAT rate is 20%.
VAT Amount = £100 - (£100 / (1 + 20%)) = £16.666666666667
We get the VAT-free price when we subtract the VAT amount from the VAT-inclusive price.
We get the following as the final result:
| Pre-VAT Price | £83.333333333333 |
| VAT % | 20% |
| VAT Amount | £16.666666666667 |
| Sale price | £100 |
As we can see, it is easy to end up with a lot of decimals, when we calculate the VAT amount from the VAT-inclusive price. In invoices and receipts, the numbers are rounded to two decimals and the above becomes:
| Pre-VAT Price | £83.33 |
| VAT % | 20% |
| VAT Amount | £16.67 |
| Sale price | £100 |
However, due to rounding, the situation gets interesting for some VAT-inclusive prices.
Let's take as an example that we want the VAT-inclusive price of a product to be £3.99. As a result of the calculation described above, we get:
| Pre-VAT Price | £3.325 |
| VAT % | 20% |
| VAT Amount | £0.665 |
| Sale price | £3.99 |
If we now round both VAT-free price and VAT amount, we end up with VAT-inclusive price that is more than their sum. So we need to round first the VAT amount and then make sure the VAT-inclusive price is the sum of VAT-free price and VAT amount.
And when these are rounded, we get:
| Pre-VAT Price | £3.32 |
| VAT % | 20% |
| VAT Amount | £0.67 |
| Sale price | £3.99 |
However, if we take the resulting VAT-free price of £3.32 and calculate the VAT using it, we get:
VAT Amount = £3.32 * 20% = £0.664, which is rounded to £0.66
When calculating from the VAT-free price of £3.32, the final rounded figures are:
| Pre-VAT Price | £3.32 |
| VAT % | 20% |
| VAT Amount | £0.66 |
| Sale price | £3.98 |
Looking at the last two tables above, it seems that the same price excluding VAT gives different results.
In fact, there are a whole range of VAT-inclusive prices that simply cannot be arrived at if VAT is calculated from the VAT-free price (this is how one should do it) and if the VAT-free price has two decimal places precision and if (this is important): the VAT-free price is considered exact and not rounded.
Of course, a VAT-inclusive price of £3.99 is entirely possible. In the calculator on this site, you can get these by using the advanced mode, which will give you two alternative answers, both of which are correct.
If you want to end up with a VAT-inclusive price of £3.99, this will happen when the VAT-free price is £3.325. In such a case, the value rounded to two decimal places is £3.32. The important thing is that this is seen as a rounded value. And as we have seen, it is also in this case adjusted so that the sum of VAT-free price and VAT amount equals VAT-inclusive price.
All VAT-inclusive prices are therefore mathematically completely possible, there are no impossible prices.
Author:
Published: 3.3.2025
In Europe there is VAT - in the US sales tax
In Europe, companies belonging to the production chain account for the VAT paid by the consumer to the state in proportion to their value added. The US sales tax, on the other hand, is billed entirely by the company that made the consumer sale. Sales tax is paid to local operators, not to the state.
History of taxation
Many of today's taxes were already in use in Antiquity. Deciding on taxation has moved from autocrats to parliaments and the level of taxation has risen. In earlier times taxes were used to wage wars, today taxes are used to maintain welfare states.