Price Calculation Paradox: How to cover tax and fees when these values depend upon one another

Question

I have a real-world math problem pertaining to a pricing formula, a paradox.

In this formula, two adjustments are needed, but both depend on knowing the result of each other first.

I need to apply an adjustment to cover tax:

$$ \begin{align} P_{tax-adjusted} = P_{fee-adjusted} \times 1.1 \end{align}$$

I also need to apply another adjustment to cover fees.

$$ \begin{align} P_{fee-adjusted} = P_{tax-adjusted} \times \frac{1}{0.88} \end{align}$$

But both the tax and fee adjustment depend on knowing each other first, so you end up in an infinite cycle of having to adjust one for the other. How do I resolve this paradox?

Edit:

For more context

Fee is 12% of final sale price

Tax is 10% of final sale price

You can see how this creates a dilemma. Fee adjustment depends on knowing the tax-adjusted price, and tax adjustment depends on knowing the fee-adjusted price.

Answer 1

I think you have the wrong equations, if I understand the problem correctly. Let $P$ be the net sales price (before adjustment) and $G$ be the gross sales price. Let $T$ be the tax, and $F$ be the fee. Then we have $$\begin{align}G&= P+T+F\\ T&=.1G\\F&=.12G\end{align}$$

We get $$G={P\over.78}$$

Answer 2

Your two equations are inconsistent. The first implies that $$ \frac{t}{f} = 1.1 $$ (with the obvious abbreviation for the unknowns). The second implies that $$ \frac{t}{f} = 0.88 $$ So there is no exact solution. You can get close with any value of that ratio between $1.1$ and $0.88\ldots$.