Add or remove VAT, at any rate

What this works out

VAT is charged as a percentage of the price before tax. That makes adding it easy and taking it back off harder than it looks, because the percentage on a VAT-inclusive price is a fraction of a number that is no longer in front of you.

The tool does both directions and shows all three figures — the price before VAT, the tax, and the price including it — so whichever one you started with, the other two are there. It also states the VAT fraction for the rate, which is the number tax authorities publish so the sum can be done on a receipt without a calculator at all.

It holds no rates, and that is the point. This page used to carry the standard rate of 32 European countries. A table like that is wrong somewhere for most of the year, because rates move at short notice and mid-year, and a stale rate on a page somebody trusts is worse than no rate at all. Those rates now live in a tool for each country — UK VAT, German VAT and Irish VAT so far — each with its own bands, its own rounding rule and its own dated sources. What is left here is the arithmetic, which is the same in every country and cannot go out of date. It is the right page for a rate nothing here covers, for a historic rate, and for a band read straight off an invoice.

The method

Adding VAT is a multiplication: the tax is the rate applied to the price before tax, and the total is the two added together.

Removing it is a division. A price of 120 that includes 20% VAT is 120% of the price before tax, so the price before tax is 120 divided by 1.2. The equivalent shortcut is the VAT fraction: the rate divided by 100 plus the rate. At 20% that is 20/120, or one sixth, so the tax in any 20% VAT-inclusive price is a sixth of it. HM Revenue & Customs publishes it in VAT Notice 700 and every other authority publishes its own; the tool shows it in lowest terms for whatever rate you type, which is often less tidy than one sixth — 19% gives 19/119, and 13.5% gives 27/227.

Only the VAT is rounded, and the third figure is derived from the other two. That is a deliberate choice rather than an oversight. A price of 12.03 including 20% VAT is exactly 10.025 before tax and 2.005 of tax; round both to the nearest penny on their own and you get 10.03 and 2.01, which add to 12.04 — a penny more than the price on the receipt. Rounding the tax and subtracting cannot produce that, so the three numbers always reconcile.

The arithmetic is done in decimal, not in binary floating point. 19% of 13.50 is exactly 2.565 and rounds to 2.57, but the nearest double to 2.565 sits fractionally below it, so a naive implementation reports 2.56. On a page whose entire output is a tax figure, one penny is the whole product.

There is also a rounding option, and it is offered rather than chosen because this page belongs to no country. Some authorities let a trader issuing invoices round the VAT down to a whole penny rather than to the nearest, on the grounds that it is tax-neutral: the supplier’s output tax and the customer’s input tax move together. Others do not. Which applies to you is a rule you file under, so the tool exposes the choice instead of making it.

A worked example

The figures below carry no currency symbol on purpose. VAT arithmetic does not depend on the currency, so the tool follows whichever currency your locale uses, and a worked example that hard-coded one would contradict the result panel on every other locale.

Adding. 100 before VAT at 20%:

VAT	20.00
Price before VAT	100.00
Price including VAT	120.00
VAT fraction	1/6

Removing. A receipt for 179.99 that includes 20% VAT:

VAT	30.00
Price before VAT	149.99
Price including VAT	179.99

Checked against the VAT fraction:

vat = 179.99 x 20 / 120
    = 179.99 / 6
    = 29.998333...
    = 30.00 to the nearest penny

net = 179.99 - 30.00
    = 149.99

Taking 20% off 179.99 instead gives 143.99, which is 6.00 short. That gap is the reason this tool exists.

The penny cases. 13.50 plus VAT at 19% is 2.57 of tax and 16.07 in total — not the 2.56 a floating-point calculation reports. And 12.03 including 20% VAT is 2.01 of tax on 10.02, the two adding back to 12.03 exactly.

A column of prices, and why the total is not what a spreadsheet says

Paste a column into the price list and the table applies the rate to every line. The example the tool starts with is four VAT-inclusive prices at 20%:

Line	Before VAT	VAT	Including VAT
Line 1	10.82	2.17	12.99
Line 2	3.74	0.75	4.49
Line 3	26.25	5.25	31.50
Line 4	7.29	1.46	8.75
Total	48.10	9.63	57.73

The total tax is 9.63, and one sixth of 57.73 is 9.62. Both are correct arithmetic and only the first is the right answer for an invoice, because rounding does not distribute over addition: a VAT invoice states a tax amount per line, so the total it has to reconcile against is the one built from those rounded figures. A spreadsheet that keeps full precision down the column and rounds SUM once reports the second, and the gap grows with the number of lines rather than with their size.

These are the same numbers asserted in this tool’s test file, so if the formula ever changes without this page changing with it, the build fails.

Before you read on

Four VAT-inclusive lines total 57.73 at 20%. Adding the tax line by line gives 9.63. One sixth of 57.73 is 9.62. Which figure belongs on the invoice?

• A9.62 — the total is what the customer pays, so work from it.

  It is correct arithmetic on the wrong input. A VAT invoice does not state one tax figure for the whole document.

• B9.63 — the sum of the rounded lines.

  Yes. The invoice states tax per line, so the total has to be built from those.

• CEither — a penny is a rounding difference.

  It is a penny on four lines. On four hundred it is not, and it grows with the count rather than the amounts.

9.63. Both are correct arithmetic and only one is the right answer for an invoice, because rounding does not distribute over addition: a VAT invoice states a tax amount per line, so the total it has to reconcile against is the one built from those rounded figures. A spreadsheet that keeps full precision down the column and rounds SUM once reports 9.62, and the gap grows with the number of lines rather than with their size.

1. gross = net + vat

   The only relationship there is. Everything below is this rearranged.

2. vat = net x rate / 100

   The tax was charged on the net — the figure you do not have when you are holding a receipt.

3. gross = net x (100 + rate) / 100

   Substitute. A 20%-inclusive price is 120% of the net, which is why subtracting 20% cannot get back to it.

4. vat = gross x rate / (100 + rate)

   Rearrange for the figure you can compute. At 20% that is 20/120 — one sixth — and the rate over 100 plus the rate at every other rate there is.

What it does not do

It does not know any country’s rates, and that is deliberate rather than unfinished — see above. It does not decide which band a supply falls in, handle the reverse charge, margin schemes, the flat rate scheme, partial exemption or import VAT, and it does not convert currencies. Every line in the price list is taxed at the same rate, so a mixed-rate invoice has to be run once per band.

For a country whose bands, rounding rule and sources are worth stating properly, there is a tool for it rather than an option on this one. For an ordinary percentage of a number, without the inclusive-price arithmetic, a percentage calculator is the plainer tool and the right one.

Adding VAT to a price that excludes it:
  vat   = round(net x rate / 100)
  gross = net + vat

Removing VAT from a price that includes it, by the VAT fraction:
  vat = round(gross x rate / (100 + rate))
  net = gross - vat

where
  rate  = the rate as a percentage, so 20 rather than 0.2
  round = to the nearest 0.01, or towards zero when
          "round the VAT down" is chosen

At 20% the VAT fraction is 20/120, which is one sixth.

Only the VAT is rounded. The third figure is derived from
the other two, so net + vat = gross to the penny, always.

Down a pasted column, each line is rounded and the totals
are the sums of the rounded lines — never the rounding of
the summed lines, which is a different number.