# How to calculate gross item price minus sales tax

Say I have an item priced $\$120.00$(incl. sales tax) and the tax attributed is$20\%$. I can easily work out that the net cost (price minus tax) is$\text{\$}100.00$.

Equation for calculating $gross$ from $net + tax$ $$gross = \120.00$$ $$tax = 20\% \text{ or } 0.20$$

$$gross = net(1 + tax)$$

$$\therefore 120.00 = net(1 + 20\%) = net(1 + 0.20) = net \times 1.20$$

Equation for calculating $net$ from $gross$

Now with the tax rate the same ($20\%$) and the gross the same ($\$120.00$) you can add 1 to the denominator of the fraction for tax and say that $$net = gross - \left(\frac{gross}{\frac{100}{(100 \times tax)} + 1}\right)$$ $$= gross - \left(\frac{gross}{\frac{100}{(100 \times 0.20)} + 1}\right)$$ $$= gross - \left(\frac{gross}{\frac{100}{20} + 1}\right)$$ $$= gross - \left(\frac{gross}{5 + 1}\right)$$ $$= gross - \left(\frac{gross}{6}\right)$$ $$\therefore net = 120 - \left(\frac{\120}{6}\right) = \100$$ Pretty simple when you have this sort of situation. But what about when you have sales tax of$17.5\text{%}$You cannot apply the same rule. What would be the formula for working out the net value when: $$gross = 117.50$$ $$tax = 17.5\% \text{ or } 0.175$$ I would assume this equation would work without worrying if you have a whole number or not for your tax rate? ## 2 Answers As you have noted, $$P_{\text{net}} = (1 + r_{\text{tax}}) P_{\text{gross}}$$ where$r_{\text{tax}} = 0.20$in your case. What you want is to solve the above formula for$P_{\text{gross}}$, this is done simply by dividing both sides by$(1 + r_{\text{tax}})$, which gives: $$\frac{1}{1 + r_{\text{tax}}} P_{\text{net}} = P_{\text{gross}}.$$ In your case,$\frac{1}{1 + r_{\text{tax}}} = \frac{1}{1.2} \simeq 0.83$. To find how much tax is applied from net price, you want to subtract the above calculation for$P_{\text{gross}}$from$P_{\text{net}}$: $$P_{\text{net}}-P_{\text{gross}} =P_{\text{net}}-\frac{1}{1 + r_{\text{tax}}} P_{\text{net}} = \left(1 - \frac{1}{1 + r_{\text{tax}}}\right)P_{\text{net}}$$ In the case$r_{\text{tax}}=0.2$,$1 - \frac{1}{1 + r_{\text{tax}}} \simeq 16.67\%$For$r_{\text{tax}}=0.175$,$1 - \frac{1}{1 + r_{\text{tax}}} \simeq 14.89 \%$• Oh sorry, I thought you had approximated the 16.67 % by 17.5 %. You just need to apply formulae with$r_{tax} = 0.175$then. – Joce Commented Jun 21, 2018 at 13:05 The answer from @Joce, reminded me of basic algebra I haven't used since school and that was a long time ago. Correcting a few issues I have come to the following answer. As I have noted, $$gross=net(1+tax)$$ where$tax = 0.175$in the case for this answer. What you want is to solve the above equation for$net$, and this is done simply by dividing both sides by$(1 + tax)$, which gives: $$net = \frac{gross}{1 + tax}$$ With this equation, and the fact that: $$gross = \117.50$$ $$tax = 17.5\%\text{ or }0.175$$ we get: $$net = \frac{\117.50}{1 + 0.175} = \100.00$$ The formula $$net = \frac{gross}{1 + tax}$$ will work no matter what the tax rate is. At$12.5\%$tax which is$0.125\$, the formula would be

$$net = \frac{\112.50}{1 + 0.125} = \100.00$$