Finding GST Amount from GST Inclusive Price In New Zealand

Question

In New Zealand, GST (Goods and Services Tax) is equal to 15%.

The following formula can be used to work out the GST Amount from the GST Inclusive Price:

    GST-Inclusive-Price X 3 ÷ 23 = GST-Amount

So if a product has a GST-Inclusive price of $115, the GST Amount could be worked out by:

    115 X 3 / 23 = 15

So my question is: Why does this formula work? How would one come up with a formula like this?

Answer 1

The price without GST corresponds to $100\%$. The price inclusive the GST ($P^G$) corresponds to $100\%+15\%=115\%$. And the GST ($T^G$) in dollars corresponds to $15\%$.

$P^G \ \widehat = \ 115\% $

$ T^G \ \widehat = \ 15\%$

Therefore $T^G=P^G\cdot \frac{15}{115}$

Cancelling out 5

$T^G=P^G\cdot \frac{3}{23}$

If $P^G$ is equal to $\$115$, then $T^G$ is equal to $\$115\cdot \frac{3}{23}=\$ 15$