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












2














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.






arithmetic economics







share|cite|improve this question
























  • 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.
    – callculus
    2 days ago






  • 1




    From the additional information you gave we can derive the following two equations: $color{blue}{textrm{ sales price}=1.1cdot textrm{ (sales price-taxes)}}$ 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.
    – callculus
    2 days ago












  • But how do we formulate one single equation in which taxes and fees are both factored into the final price?
    – ptrcao
    2 days ago












  • I have to correct my equations. From the additional information you gave we can derive the following two equations: $color{blue}{0.9cdot textrm{ sales price}= textrm{ sales price-taxes}}$ and $color{blue}{0.88cdot textrm{ sales price=} textrm{ sales price - fee}}$. If you know the sales price you can derive the taxes from the first equation and the fee from the second equation.
    – callculus
    2 days ago


















2














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.






arithmetic economics







share|cite|improve this question
























  • 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.
    – callculus
    2 days ago






  • 1




    From the additional information you gave we can derive the following two equations: $color{blue}{textrm{ sales price}=1.1cdot textrm{ (sales price-taxes)}}$ 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.
    – callculus
    2 days ago












  • But how do we formulate one single equation in which taxes and fees are both factored into the final price?
    – ptrcao
    2 days ago












  • I have to correct my equations. From the additional information you gave we can derive the following two equations: $color{blue}{0.9cdot textrm{ sales price}= textrm{ sales price-taxes}}$ and $color{blue}{0.88cdot textrm{ sales price=} textrm{ sales price - fee}}$. If you know the sales price you can derive the taxes from the first equation and the fee from the second equation.
    – callculus
    2 days ago
















2












2








2







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.






arithmetic economics







share|cite|improve this 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.






arithmetic economics




arithmetic economics






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 days ago









Blue

47.7k870151




47.7k870151










asked 2 days ago









ptrcaoptrcao

155213




155213












  • 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.
    – callculus
    2 days ago






  • 1




    From the additional information you gave we can derive the following two equations: $color{blue}{textrm{ sales price}=1.1cdot textrm{ (sales price-taxes)}}$ 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.
    – callculus
    2 days ago












  • But how do we formulate one single equation in which taxes and fees are both factored into the final price?
    – ptrcao
    2 days ago












  • I have to correct my equations. From the additional information you gave we can derive the following two equations: $color{blue}{0.9cdot textrm{ sales price}= textrm{ sales price-taxes}}$ and $color{blue}{0.88cdot textrm{ sales price=} textrm{ sales price - fee}}$. If you know the sales price you can derive the taxes from the first equation and the fee from the second equation.
    – callculus
    2 days ago




















  • 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.
    – callculus
    2 days ago






  • 1




    From the additional information you gave we can derive the following two equations: $color{blue}{textrm{ sales price}=1.1cdot textrm{ (sales price-taxes)}}$ 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.
    – callculus
    2 days ago












  • But how do we formulate one single equation in which taxes and fees are both factored into the final price?
    – ptrcao
    2 days ago












  • I have to correct my equations. From the additional information you gave we can derive the following two equations: $color{blue}{0.9cdot textrm{ sales price}= textrm{ sales price-taxes}}$ and $color{blue}{0.88cdot textrm{ sales price=} textrm{ sales price - fee}}$. If you know the sales price you can derive the taxes from the first equation and the fee from the second equation.
    – callculus
    2 days ago


















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.
– callculus
2 days ago




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.
– callculus
2 days ago




1




1




From the additional information you gave we can derive the following two equations: $color{blue}{textrm{ sales price}=1.1cdot textrm{ (sales price-taxes)}}$ 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.
– callculus
2 days ago






From the additional information you gave we can derive the following two equations: $color{blue}{textrm{ sales price}=1.1cdot textrm{ (sales price-taxes)}}$ 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.
– callculus
2 days ago














But how do we formulate one single equation in which taxes and fees are both factored into the final price?
– ptrcao
2 days ago






But how do we formulate one single equation in which taxes and fees are both factored into the final price?
– ptrcao
2 days ago














I have to correct my equations. From the additional information you gave we can derive the following two equations: $color{blue}{0.9cdot textrm{ sales price}= textrm{ sales price-taxes}}$ and $color{blue}{0.88cdot textrm{ sales price=} textrm{ sales price - fee}}$. If you know the sales price you can derive the taxes from the first equation and the fee from the second equation.
– callculus
2 days ago






I have to correct my equations. From the additional information you gave we can derive the following two equations: $color{blue}{0.9cdot textrm{ sales price}= textrm{ sales price-taxes}}$ and $color{blue}{0.88cdot textrm{ sales price=} textrm{ sales price - fee}}$. If you know the sales price you can derive the taxes from the first equation and the fee from the second equation.
– callculus
2 days ago












2 Answers
2






active

oldest

votes


















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 $$begin{align}G&= P+T+F\
T&=.1G\F&=.12Gend{align}$$



We get
$$G={Pover.78}$$






share|cite|improve this answer























  • Simple and elegant and offers clarifies the problem.
    – ptrcao
    2 days ago










  • @ptrcao But your equations still clarify nothing since they are still contradictory. I can only hope that you have understood the topic.
    – callculus
    2 days ago










  • @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.
    – ptrcao
    2 days ago





















1














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 $0.88ldots$.






share|cite|improve this answer























  • 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.)
    – ptrcao
    2 days ago












  • @ptrcao: you are correct. That makes the disagreement worse.
    – Ross Millikan
    2 days ago










  • I have fixed my error, just to keep the record straight. @saulsplatz 's answer fills in the missing data.
    – Ethan Bolker
    2 days ago











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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes









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 $$begin{align}G&= P+T+F\
T&=.1G\F&=.12Gend{align}$$



We get
$$G={Pover.78}$$






share|cite|improve this answer























  • Simple and elegant and offers clarifies the problem.
    – ptrcao
    2 days ago










  • @ptrcao But your equations still clarify nothing since they are still contradictory. I can only hope that you have understood the topic.
    – callculus
    2 days ago










  • @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.
    – ptrcao
    2 days ago


















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 $$begin{align}G&= P+T+F\
T&=.1G\F&=.12Gend{align}$$



We get
$$G={Pover.78}$$






share|cite|improve this answer























  • Simple and elegant and offers clarifies the problem.
    – ptrcao
    2 days ago










  • @ptrcao But your equations still clarify nothing since they are still contradictory. I can only hope that you have understood the topic.
    – callculus
    2 days ago










  • @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.
    – ptrcao
    2 days ago
















1












1








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 $$begin{align}G&= P+T+F\
T&=.1G\F&=.12Gend{align}$$



We get
$$G={Pover.78}$$






share|cite|improve this answer














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 $$begin{align}G&= P+T+F\
T&=.1G\F&=.12Gend{align}$$



We get
$$G={Pover.78}$$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 2 days ago

























answered 2 days ago









saulspatzsaulspatz

14k21329




14k21329












  • Simple and elegant and offers clarifies the problem.
    – ptrcao
    2 days ago










  • @ptrcao But your equations still clarify nothing since they are still contradictory. I can only hope that you have understood the topic.
    – callculus
    2 days ago










  • @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.
    – ptrcao
    2 days ago




















  • Simple and elegant and offers clarifies the problem.
    – ptrcao
    2 days ago










  • @ptrcao But your equations still clarify nothing since they are still contradictory. I can only hope that you have understood the topic.
    – callculus
    2 days ago










  • @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.
    – ptrcao
    2 days ago


















Simple and elegant and offers clarifies the problem.
– ptrcao
2 days ago




Simple and elegant and offers clarifies the problem.
– ptrcao
2 days ago












@ptrcao But your equations still clarify nothing since they are still contradictory. I can only hope that you have understood the topic.
– callculus
2 days ago




@ptrcao But your equations still clarify nothing since they are still contradictory. I can only hope that you have understood the topic.
– callculus
2 days ago












@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.
– ptrcao
2 days ago






@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.
– ptrcao
2 days ago













1














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 $0.88ldots$.






share|cite|improve this answer























  • 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.)
    – ptrcao
    2 days ago












  • @ptrcao: you are correct. That makes the disagreement worse.
    – Ross Millikan
    2 days ago










  • I have fixed my error, just to keep the record straight. @saulsplatz 's answer fills in the missing data.
    – Ethan Bolker
    2 days ago
















1














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 $0.88ldots$.






share|cite|improve this answer























  • 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.)
    – ptrcao
    2 days ago












  • @ptrcao: you are correct. That makes the disagreement worse.
    – Ross Millikan
    2 days ago










  • I have fixed my error, just to keep the record straight. @saulsplatz 's answer fills in the missing data.
    – Ethan Bolker
    2 days ago














1












1








1






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 $0.88ldots$.






share|cite|improve this answer














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 $0.88ldots$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 2 days ago

























answered 2 days ago









Ethan BolkerEthan Bolker

41.8k547110




41.8k547110












  • 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.)
    – ptrcao
    2 days ago












  • @ptrcao: you are correct. That makes the disagreement worse.
    – Ross Millikan
    2 days ago










  • I have fixed my error, just to keep the record straight. @saulsplatz 's answer fills in the missing data.
    – Ethan Bolker
    2 days ago


















  • 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.)
    – ptrcao
    2 days ago












  • @ptrcao: you are correct. That makes the disagreement worse.
    – Ross Millikan
    2 days ago










  • I have fixed my error, just to keep the record straight. @saulsplatz 's answer fills in the missing data.
    – Ethan Bolker
    2 days ago
















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.)
– ptrcao
2 days ago






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.)
– ptrcao
2 days ago














@ptrcao: you are correct. That makes the disagreement worse.
– Ross Millikan
2 days ago




@ptrcao: you are correct. That makes the disagreement worse.
– Ross Millikan
2 days ago












I have fixed my error, just to keep the record straight. @saulsplatz 's answer fills in the missing data.
– Ethan Bolker
2 days ago




I have fixed my error, just to keep the record straight. @saulsplatz 's answer fills in the missing data.
– Ethan Bolker
2 days ago


















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