The 'plus VAT' line that starts every quote dispute.
Why doesn't 'taking 20% off' £1,200 give you back £1,000?
If you multiplied to add it, you must divide to undo it.
Experiment
Hands-on experiment
Predict first — add 20% VAT to a £1,000 net price and you get £1,200. Now subtract 20% from £1,200 — do you get back to £1,000?
Try both undos yourself (total: £1,200)
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Add 20% VAT to a £1,000 net price and you get £1,200. Now subtract 20% from £1,200? You get £960 — not the £1,000 you started with.
The reason: the percent's base changed. The 20% you added was based on £1,000, but the 20% you subtracted is based on £1,200.
Businesses writing quotes, freelancers setting fees — disputes over 'VAT included or excluded?' are everywhere. This lab organizes that confusion into structure. (We use the UK's 20% VAT as the running example.)
Misconception
Common misconceptions
Subtract 20% from the VAT-inclusive price and you get the net price.
20% of £1,200 is £240, so subtracting gives £960 — off by £40. The net price is £1,200 ÷ 1.2 = £1,000. A value made by multiplying must be undone by dividing.
The VAT is 20% of the VAT-inclusive price.
VAT is '20% of the net price' = '1/6 of the total'. That's why the VAT line on a receipt looks smaller than 20% of the total.
Formula
Writing it as math
The experiment's two modes (add / reverse) are one equation seen from both sides. The whole story: multiplication and division are inverse operations.
Adding VAT
'Adding 20%' means multiplying by 1.2.
Recovering the net price
Undoing a multiplication is a division. Subtracting 20% from the total (×0.8) is not the inverse of ×1.2.
The VAT inside the total
Total = net × 6/5, so the VAT (1/5 of net) is 1/6 of the total. In £1,200, the VAT is exactly £200.
In Real Life
Where you meet it in real life
Freelancer quotes
'£1,000 (VAT excluded)' and '£1,000 (VAT included)' differ by about £167 in what you actually keep. That's why it must be confirmed before signing.
The VAT line on receipts
The VAT on a cafe receipt is 1/6 of the total, not 20%. On a £6 coffee, the VAT is £1 — not £1.20.
A business's VAT return
You remit the VAT on sales minus the VAT on purchases. The 1/6 inside your revenue that looks like 'your money' belonged to the state from the start.
Comparing duty-free prices
When comparing duty-free and high-street prices, 1/6 of the high-street price is VAT. Now you can compute exactly how much you're saving.
Math Behind
The math behind this
Related lab
Functions
×1.2 and ÷1.2 are functions that undo each other — an inverse-function pair. The feeling of 'undoing' is what math formalizes as the inverse function.
Go to the Functions lab →Connection
Labs connect
Previous lab
Discounts
Curious about the structure of subtracting multiplication (discounts)? Head to the previous lab.
← Discounts labRecommended next
Compound Interest
Unlike VAT's single multiplication, compound interest repeats the same multiplication year after year. What happens when multiplication stacks up over time?
Go to the Compound Interest lab →