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

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.

• A little more context would help us to understand your problem. How much is the fee, tax rate, etc, for instance? The two equations are contradictory. Jan 5, 2019 at 16:10
• From the additional information you gave we can derive the following two equations: $\color{blue}{\textrm{ sales price}=1.1\cdot \textrm{ (sales price-taxes)}}$ and $\color{blue}{\textrm{ sales price=} 1.12 \cdot \textrm{ (sales price - fee)}}$. If you know the sales price you can derive the taxes from the first equation and the fee from the second equation. Jan 5, 2019 at 17:04
• But how do we formulate one single equation in which taxes and fees are both factored into the final price? Jan 5, 2019 at 17:20
• I have to correct my equations. From the additional information you gave we can derive the following two equations: $\color{blue}{0.9\cdot \textrm{ sales price}= \textrm{ sales price-taxes}}$ and $\color{blue}{0.88\cdot \textrm{ sales price=} \textrm{ sales price - fee}}$. If you know the sales price you can derive the taxes from the first equation and the fee from the second equation. Jan 5, 2019 at 17:21

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

• Simple and elegant and offers clarifies the problem. Jan 5, 2019 at 17:24
• @ptrcao But your equations still clarify nothing since they are still contradictory. I can only hope that you have understood the topic. Jan 5, 2019 at 17:35
• @callculus Yes, I believe I didn't formulate the question properly originally - this is my fault - but this answer does provide me with a workable solution for the real-life situation. I thank you and the answerer. Jan 5, 2019 at 17:57

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 $$0.88\ldots$$.

• Good point. Just a question though - shouldn't $\frac{t}{f} = 0.88$ in the second instance? (You might have read the fraction the wrong way around.) Jan 5, 2019 at 15:44
• @ptrcao: you are correct. That makes the disagreement worse. Jan 5, 2019 at 16:38
• I have fixed my error, just to keep the record straight. @saulsplatz 's answer fills in the missing data. Jan 5, 2019 at 22:32