+10% · VAT
If you multiplied to add it, you must divide to undo it.
Why doesn't 'taking 20% off' £1,200 give you back £1,000?
What are we trying to understand?
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.)
Change one assumption at a time
The result follows the inputs and assumptions. Check the units and model before treating it as an actual outcome.
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 the calculation another way
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.
Why the calculation works, and when
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.
Try the idea in another setting
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.
Now explain it in your own words
Follow the mathematics behind the calculation
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 →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 →