Find the original value without VAT

Question

How to find out the amount without VAT?

$$£\,3.325 + 20\,\% = £\,3.99$$

but, if I do $£3.99 - 20\% = £3.192$ instead of $£3.325$, how can I find out $£3.325$ out of $£3.99$?

Answer 1

Assume you want to add a percentage $p\%$ to a given number $x$. What you have denoted by the expression "$x+p\%$" for the final price $y_1$ means the following: $$y_1=x + \frac{p}{100}\cdot x = x\cdot\left(1 + \frac{p}{100}\right)$$

On the same way, "$x-p\%$" gives a different price $y_1$: $$y_2=x - \frac{p}{100}\cdot x = x\cdot\left(1 - \frac{p}{100}\right)$$

Thus, in your particular case, you have to recover $x$ from $y_1$ and, hence, you have to divide that quantity by $1.20$. Note that this is not the same as what you did. You multipled by $0.8$:

£3.99 / 1.20 = £3.325

£3.99 · 0.8 = £3.192

Answer 2

If we call the price without the tax 100%, we know that the price with tax is 100%+20%=120%. So the tax is actually 20% of 120%. So if you want to know the price without VAT, you do -16.667%, or you multiply the price with 5/6.

In this case £3.99*5/6 = £3.325.

This is a trick commonly used by salesmen: They have an action called 'VAT free'. Most people will think that they get 20% discount, but in reality, they are not even getting 17% discount.

Answer 3

$$3.325 + \frac{20}{100}3.325 =3.99$$ $$3.325\cdot1.2=3.99$$ $$\frac{3.99}{1.2}=3.325$$ Let $x$ be price without VAT then $$x+\frac{20}{100}x=\frac{6}{5}x$$ is the final price. Let $y$ be a price with VAT then the price without VAT $20\%$ is $$y-\frac{20}{100}y=\frac{4}{5}y$$