How to calculate gross item price minus sales tax

Question

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?

Answer 1

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

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

Answer 2

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 $r_{\text{tax}}=0.175$, $1 - \frac{1}{1 + r_{\text{tax}}} \simeq 14.89 \%$