The problem it solves
VAT is charged as a percentage of the price before tax. That makes adding it easy: take 20% of 100, add it on, get 120.
Going the other way is where people slip. You have a receipt for 120 and you want to know how much of it was tax. The instinct is to take 20% of 120 — but 20% of 120 is 24, and the tax was 20. The percentage was charged on the smaller number, which is no longer in front of you.
The VAT fraction is the answer to “what proportion of the number I do have is tax?”
Where it comes from
If the net is 100% and the tax is 20% of it, then the gross is 120% of the net. So the tax, as a share of the gross, is:
vat fraction = rate / (100 + rate)
at 20% = 20 / 120 = 1/6
at 5% = 5 / 105 = 1/21
at 19% = 19 / 119 (does not reduce)
at 23% = 23 / 123 (does not reduce)
One sixth of 120 is 20. That is the whole idea, and it is exact rather than approximate — multiplying by 1/6 and dividing by 1.2 are the same operation written two ways.
Why authorities publish it rather than the division
Two reasons, and the second is the one that matters on an invoice.
It can be done without a calculator. HM Revenue & Customs prints the fraction in VAT Notice 700 precisely so a trader can work out the tax in a VAT-inclusive price on paper. A sixth of 59.99 is about 10; a division by 1.2 is not something anyone does in their head.
It rounds the right figure. Invoicing rules round the tax, then derive the net by subtraction. Taking the fraction gives you the tax directly, so the rounding lands where the rules put it. Dividing the gross by 1.2 gives you the net first, and rounding that instead produces invoices whose lines do not add up to their own totals — 12.03 including 20% VAT is exactly 10.025 net and 2.005 of tax, and rounding both independently gives 10.03 + 2.01 = 12.04.
Before you read on
A receipt shows 120 including 20% VAT. How much of it is tax?
20, which is one sixth of 120. The tax was charged on the 100 before it, not on the 120 in front of you, so taking 20% of the total overstates it by 4. The VAT fraction — the rate divided by 100 plus the rate — is the exact share of an inclusive price that is tax, and at 20% it happens to reduce to a sixth.
The fractions are usually untidy
The tidy ones are a coincidence of the British and Irish reduced rates:
| Rate | Fraction | Reduces to |
|---|---|---|
| 20% | 20/120 | 1/6 |
| 5% | 5/105 | 1/21 |
| 25% | 25/125 | 1/5 |
| 17.5% | 175/1175 | 7/47 |
| 19% | 19/119 | 19/119 |
| 23% | 23/123 | 23/123 |
| 13.5% | 135/1135 | 27/227 |
| 4.8% | 48/1048 | 6/131 |
The last two are Irish, and they are the reason a fraction has to be reduced carefully rather than with a naive greatest common divisor: 13.5 is not an integer, so both halves have to be scaled by ten before they can share a divisor. Get that wrong and 23% still looks perfectly correct while every fractional rate is silently wrong.
The mistake this all prevents
Subtracting the rate from the gross. It is wrong by the rate squared:
at 20% the naive answer is 4% below the true net
at 23% the naive answer is 5.3% below the true net
On a 179.99 receipt at 20%, taking 20% off gives 143.99. The real price before VAT is 149.99 — six pounds adrift, on one line.
Common questions
Is the VAT fraction an approximation?
No. It is exact. The fraction is the rate divided by 100 plus the rate, so at 20% it is 20/120, which reduces to one sixth exactly. Multiplying a VAT-inclusive price by it gives the same answer as dividing by 1.2, to every decimal place. The only rounding anywhere is the final step to a whole penny.
Why do I need it if I can just divide?
You do not, arithmetically — they are the same operation. The fraction exists because it can be done in your head on a receipt, and because it puts the rounding on the tax rather than on the net, which is where invoicing rules put it. Dividing the gross and subtracting rounds the wrong figure.
What is the fraction at 19% or 23%?
19/119 and 23/123, and neither reduces any further — 19 is prime and does not divide 119, and 23 is prime and does not divide 123. British guidance makes the VAT fraction sound like a tidy trick because 20% happens to give one sixth. That is the lucky case, not the normal one.