Find price without VAT from price with VAT and VAT rate

Question

I need to calculate what number my number was before it had added percentages.

I have a price including VAT, and I need to find out what to price was before VAT.

In Denmark our VAT Rate is $25\%$; So if I have a product which cost $100$DKK the total price would be $125$DKK; If I want to find the price without VAT I could say $125$DKK$\,\times\,0.80$.

The problem I have is that I have many differents VAT rates >>Germany have $19\%$<< .

So have can I find the price without VAT if I know the price with VAT and the VAT Rate?

Answer 1

Adding VAT can be done by multiply with $1+\text{vat rate}$, and removing it by dividing with $1+\text{vat rate}$.

Answer 2

Given an initial price P and VAT r, then the cost is C= P+ rP= P(1+ r). Dividing both sides by 1+ r, P= C/(1+ r). In your example, with P= 100 and r= 0.25, C= 100(1.25)= 125. Going the other way, P= 125/(1.25)= 100. Of course, 1.25= 5/4 so 1/1.25= 4/5= 0.80.

If the tax were 19% or 0.19, then with P= 100, C= 100(1.19)= 119. Going the other way, P= 119/1.19= 100. 1.19 is not as simple a fraction as 1.25= 5/4 so 1/1.19 cannot be written as easily as 4/5= 0.80. 1/1.19= 0.840336134453781512 which could be approximated by 0.84.