Let's say I want to go shopping and all I have is $15.00.
The rate that I will be taxed at, on my purchases is 8.25%. I would like to know the formula to figure out what I can spend, less the taxes, so it will equal or come the closest to the $15 I have.
I tried $15.00 - 8.25% and that didn't work. So I know there has to be a formula I can use so I can apply this to all my shopping trips regardless of the amount I want to spend, the tax rate will stay constant.
Let $x$ be the amount you can spend. You need $$x + (8.25\%\cdot x) = \$15.00.$$ The left-hand side is equal to $x\cdot (1 + 0.0825)$, so $$x = \frac{\$15.00}{1.0825} = \$13.85.$$
Now let's check if this is correct. The tax on $\$13.85$ is $\$13.85\times 0.0825 = \$1.14$, and $\$13.85 + \$1.14 = \$14.99$, so it checks out. (Maybe you could spend as much as $\$13.86$ without trouble, but that depends on the way the cashier rounds off the fraction in the tax.)