Why does removing VAT not give you back the original price?
Because the percentage is charged on the net price, not the gross. Adding 20% to 100 gives 120, but taking 20% off 120 gives 96, not 100. To remove VAT you divide by 1.20 rather than subtracting.
Updated 2026-08-24
The asymmetry that catches everyone
Add twenty percent to 100 and you get 120. Take twenty percent off 120 and you get 96. The two operations do not undo each other, and this is the single most common arithmetic mistake in invoicing.
The reason is that the percentage is always calculated on the net price. Twenty percent of 100 is 20, so the gross is 120. But twenty percent of 120 is 24 — a bigger number, because 120 is a bigger base. Subtracting it overshoots by exactly the tax on the tax.
To go the other way you have to reverse the multiplication rather than repeat it. The gross is the net times 1.20, so the net is the gross divided by 1.20. And 120 divided by 1.20 is exactly 100.
The two formulas are therefore not mirror images: **gross = net × (1 + rate)** going up, **net = gross ÷ (1 + rate)** coming down. Anyone subtracting a percentage to strip tax is producing a number that is too low, and the error grows with the rate — at 25% it is over-deducting by more than five percent of the total.
What share of a gross price is tax
A useful and slightly counterintuitive fact: at a twenty percent rate, tax is not twenty percent of what the customer pays. It is 16.67 percent.
The arithmetic is straightforward once the direction is clear. On a gross of 120, the tax is 20, and 20 divided by 120 is 0.1667. The general form is rate ÷ (1 + rate) — so a twenty percent rate takes one sixth of the gross, and a twenty-five percent rate takes exactly one fifth.
This matters whenever you are working backwards from money received. A trader with 6,000 of gross takings at twenty percent owes 1,000 in VAT, not 1,200. Applying the headline rate to gross receipts overstates the liability by a fifth, which is a large error to make in the direction of overpaying.
It is also the source of the familiar shortcut. At twenty percent, the VAT in a gross figure is that figure divided by six — 120 ÷ 6 = 20. The divisor is 1 ÷ (rate ÷ (1 + rate)): six at twenty percent, five at twenty-five percent, twenty-one at five percent.
Which price you are looking at
Most of the confusion in practice is not arithmetic but not knowing which of the two numbers you have.
Consumer prices in the UK and the EU are almost always shown gross — the shelf price includes VAT and is what you hand over. Business-to-business quoting, wholesale price lists and most invoicing software default to net, with tax added as a separate line. So the same product legitimately appears as 100 in one context and 120 in another, and neither is wrong.
The US works differently in a way that trips people up in both directions. Sales tax is generally added at the register rather than shown on the shelf, and it varies by state, county and sometimes city, so a single national rate does not exist. It is also a genuinely different tax: sales tax is charged once at the final sale, while VAT is collected at every stage with businesses reclaiming what they paid on inputs.
When a figure is ambiguous, the cheap check is whether it divides cleanly. A price like 119.99 is almost certainly gross and consumer-facing; a round 100.00 on a quote is almost certainly net. On any invoice, the reliable move is to find the line that names the tax and see whether the total above it or below it matches the number you were given.
Rounding, and where the pennies go
Tax calculations rarely land on whole units, and how you round is not entirely a matter of preference.
The main decision is whether to calculate tax per line or on the invoice total. Rounding each line and then summing gives a different answer from summing first and rounding once, sometimes by a penny or two per line. Tax authorities generally specify which method they expect, and the important part is applying one consistently — an invoice whose lines do not sum to its total is the kind of thing that gets queried.
For a multi-line invoice, calculating on the total is usually both simpler and closer to the true figure, because per-line rounding errors accumulate in whichever direction the rounding rule leans rather than cancelling out.
Watch the direction of a discount, too. A discount applied before tax reduces both the net and the tax; applied after, it reduces only what the customer pays while leaving your declared tax unchanged, which is generally not what you meant and not what the authority expects. Discount first, then tax.
Working backwards from what you were paid
The practical case most people meet is reconstructing a net figure from a payment that already includes tax — reconciling a bank statement, splitting a receipt, or working out what a job actually earned.
The procedure is one division. Take the amount received, divide by one plus the rate, and that is the net. The difference between the two is the tax. On 480 received at twenty percent: 480 ÷ 1.2 is 400, so 80 was tax and 400 was income.
The mistake to avoid is the one from the first section — 480 minus twenty percent is 384, which would understate the net by 16 and the tax by the same amount in the opposite direction. Over a year of invoices that error compounds into a materially wrong set of accounts.
A quick sanity check catches most slips: the net must always be smaller than the gross, the tax must be smaller than the net at any rate under a hundred percent, and net plus tax must return exactly the gross you started with. If those three do not hold, the direction of the calculation was wrong.