Mixing
Price Calculation Paradox: How to cover tax and fees when these values depend upon one another
$begingroup$
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
edited Jan 5 at 17:53
Blue
47.9k870152
asked Jan 5 at 15:28
ptrcaoptrcao
155213
$endgroup$
add a comment |
$begingroup$
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
edited Jan 5 at 17:53
Blue
47.9k870152
asked Jan 5 at 15:28
ptrcaoptrcao
155213
$endgroup$
$begingroup$
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.
$endgroup$
– callculus
Jan 5 at 16:10
1
$begingroup$
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.
$endgroup$
– callculus
Jan 5 at 17:04
$begingroup$
But how do we formulate one single equation in which taxes and fees are both factored into the final price?
$endgroup$
– ptrcao
Jan 5 at 17:20
$begingroup$
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.
$endgroup$
– callculus
Jan 5 at 17:21
add a comment |
$begingroup$
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
edited Jan 5 at 17:53
Blue
47.9k870152
asked Jan 5 at 15:28
ptrcaoptrcao
155213
$endgroup$
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
edited Jan 5 at 17:53
Blue
47.9k870152
asked Jan 5 at 15:28
ptrcaoptrcao
155213
edited Jan 5 at 17:53
Blue
47.9k870152
asked Jan 5 at 15:28
ptrcaoptrcao
155213
edited Jan 5 at 17:53
Blue
47.9k870152
edited Jan 5 at 17:53
Blue
47.9k870152
edited Jan 5 at 17:53
Blue
47.9k870152
47.9k870152
asked Jan 5 at 15:28
ptrcaoptrcao
155213
asked Jan 5 at 15:28
ptrcaoptrcao
155213
asked Jan 5 at 15:28
ptrcaoptrcao
155213
155213
$begingroup$
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.
$endgroup$
– callculus
Jan 5 at 16:10
1
$begingroup$
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.
$endgroup$
– callculus
Jan 5 at 17:04
$begingroup$
But how do we formulate one single equation in which taxes and fees are both factored into the final price?
$endgroup$
– ptrcao
Jan 5 at 17:20
$begingroup$
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.
$endgroup$
– callculus
Jan 5 at 17:21
add a comment |
$begingroup$
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.
$endgroup$
– callculus
Jan 5 at 16:10
1
$begingroup$
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.
$endgroup$
– callculus
Jan 5 at 17:04
$begingroup$
But how do we formulate one single equation in which taxes and fees are both factored into the final price?
$endgroup$
– ptrcao
Jan 5 at 17:20
$begingroup$
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.
$endgroup$
– callculus
Jan 5 at 17:21
$begingroup$
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.
$endgroup$
– callculus
Jan 5 at 16:10
$begingroup$
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.
$endgroup$
– callculus
Jan 5 at 16:10
1
1
$begingroup$
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.
$endgroup$
– callculus
Jan 5 at 17:04
$begingroup$
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.
$endgroup$
– callculus
Jan 5 at 17:04
$begingroup$
But how do we formulate one single equation in which taxes and fees are both factored into the final price?
$endgroup$
– ptrcao
Jan 5 at 17:20
$begingroup$
But how do we formulate one single equation in which taxes and fees are both factored into the final price?
$endgroup$
– ptrcao
Jan 5 at 17:20
$begingroup$
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.
$endgroup$
– callculus
Jan 5 at 17:21
$begingroup$
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.
$endgroup$
– callculus
Jan 5 at 17:21
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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
$$G={Pover.78}$$
answered Jan 5 at 17:03
saulspatzsaulspatz
14.3k21329
$endgroup$
$begingroup$
Simple and elegant and offers clarifies the problem.
$endgroup$
– ptrcao
Jan 5 at 17:24
$begingroup$
@ptrcao But your equations still clarify nothing since they are still contradictory. I can only hope that you have understood the topic.
$endgroup$
– callculus
Jan 5 at 17:35
$begingroup$
@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.
$endgroup$
– ptrcao
Jan 5 at 17:57
add a comment |
$begingroup$
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 $0.88ldots$.
answered Jan 5 at 15:35
Ethan BolkerEthan Bolker
42.3k548111
$endgroup$
$begingroup$
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.)
$endgroup$
– ptrcao
Jan 5 at 15:44
$begingroup$
@ptrcao: you are correct. That makes the disagreement worse.
$endgroup$
– Ross Millikan
Jan 5 at 16:38
$begingroup$
I have fixed my error, just to keep the record straight. @saulsplatz 's answer fills in the missing data.
$endgroup$
– Ethan Bolker
Jan 5 at 22:32
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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
$$G={Pover.78}$$
answered Jan 5 at 17:03
saulspatzsaulspatz
14.3k21329
$endgroup$
$begingroup$
Simple and elegant and offers clarifies the problem.
$endgroup$
– ptrcao
Jan 5 at 17:24
$begingroup$
@ptrcao But your equations still clarify nothing since they are still contradictory. I can only hope that you have understood the topic.
$endgroup$
– callculus
Jan 5 at 17:35
$begingroup$
@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.
$endgroup$
– ptrcao
Jan 5 at 17:57
add a comment |
$begingroup$
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
$$G={Pover.78}$$
answered Jan 5 at 17:03
saulspatzsaulspatz
14.3k21329
$endgroup$
$begingroup$
Simple and elegant and offers clarifies the problem.
$endgroup$
– ptrcao
Jan 5 at 17:24
$begingroup$
@ptrcao But your equations still clarify nothing since they are still contradictory. I can only hope that you have understood the topic.
$endgroup$
– callculus
Jan 5 at 17:35
$begingroup$
@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.
$endgroup$
– ptrcao
Jan 5 at 17:57
add a comment |
$begingroup$
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
$$G={Pover.78}$$
answered Jan 5 at 17:03
saulspatzsaulspatz
14.3k21329
$endgroup$
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
$$G={Pover.78}$$
answered Jan 5 at 17:03
saulspatzsaulspatz
14.3k21329
edited Jan 5 at 17:19
answered Jan 5 at 17:03
saulspatzsaulspatz
14.3k21329
answered Jan 5 at 17:03
saulspatzsaulspatz
14.3k21329
answered Jan 5 at 17:03
saulspatzsaulspatz
14.3k21329
14.3k21329
$begingroup$
Simple and elegant and offers clarifies the problem.
$endgroup$
– ptrcao
Jan 5 at 17:24
$begingroup$
@ptrcao But your equations still clarify nothing since they are still contradictory. I can only hope that you have understood the topic.
$endgroup$
– callculus
Jan 5 at 17:35
$begingroup$
@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.
$endgroup$
– ptrcao
Jan 5 at 17:57
add a comment |
$begingroup$
Simple and elegant and offers clarifies the problem.
$endgroup$
– ptrcao
Jan 5 at 17:24
$begingroup$
@ptrcao But your equations still clarify nothing since they are still contradictory. I can only hope that you have understood the topic.
$endgroup$
– callculus
Jan 5 at 17:35
$begingroup$
@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.
$endgroup$
– ptrcao
Jan 5 at 17:57
$begingroup$
Simple and elegant and offers clarifies the problem.
$endgroup$
– ptrcao
Jan 5 at 17:24
$begingroup$
Simple and elegant and offers clarifies the problem.
$endgroup$
– ptrcao
Jan 5 at 17:24
$begingroup$
@ptrcao But your equations still clarify nothing since they are still contradictory. I can only hope that you have understood the topic.
$endgroup$
– callculus
Jan 5 at 17:35
$begingroup$
@ptrcao But your equations still clarify nothing since they are still contradictory. I can only hope that you have understood the topic.
$endgroup$
– callculus
Jan 5 at 17:35
$begingroup$
@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.
$endgroup$
– ptrcao
Jan 5 at 17:57
$begingroup$
@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.
$endgroup$
– ptrcao
Jan 5 at 17:57
add a comment |
$begingroup$
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 $0.88ldots$.
answered Jan 5 at 15:35
Ethan BolkerEthan Bolker
42.3k548111
$endgroup$
$begingroup$
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.)
$endgroup$
– ptrcao
Jan 5 at 15:44
$begingroup$
@ptrcao: you are correct. That makes the disagreement worse.
$endgroup$
– Ross Millikan
Jan 5 at 16:38
$begingroup$
I have fixed my error, just to keep the record straight. @saulsplatz 's answer fills in the missing data.
$endgroup$
– Ethan Bolker
Jan 5 at 22:32
add a comment |
$begingroup$
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 $0.88ldots$.
answered Jan 5 at 15:35
Ethan BolkerEthan Bolker
42.3k548111
$endgroup$
$begingroup$
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.)
$endgroup$
– ptrcao
Jan 5 at 15:44
$begingroup$
@ptrcao: you are correct. That makes the disagreement worse.
$endgroup$
– Ross Millikan
Jan 5 at 16:38
$begingroup$
I have fixed my error, just to keep the record straight. @saulsplatz 's answer fills in the missing data.
$endgroup$
– Ethan Bolker
Jan 5 at 22:32
add a comment |
$begingroup$
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 $0.88ldots$.
answered Jan 5 at 15:35
Ethan BolkerEthan Bolker
42.3k548111
$endgroup$
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 $0.88ldots$.
answered Jan 5 at 15:35
Ethan BolkerEthan Bolker
42.3k548111
edited Jan 5 at 22:33
answered Jan 5 at 15:35
Ethan BolkerEthan Bolker
42.3k548111
answered Jan 5 at 15:35
Ethan BolkerEthan Bolker
42.3k548111
answered Jan 5 at 15:35
Ethan BolkerEthan Bolker
42.3k548111
42.3k548111
$begingroup$
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.)
$endgroup$
– ptrcao
Jan 5 at 15:44
$begingroup$
@ptrcao: you are correct. That makes the disagreement worse.
$endgroup$
– Ross Millikan
Jan 5 at 16:38
$begingroup$
I have fixed my error, just to keep the record straight. @saulsplatz 's answer fills in the missing data.
$endgroup$
– Ethan Bolker
Jan 5 at 22:32
add a comment |
$begingroup$
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.)
$endgroup$
– ptrcao
Jan 5 at 15:44
$begingroup$
@ptrcao: you are correct. That makes the disagreement worse.
$endgroup$
– Ross Millikan
Jan 5 at 16:38
$begingroup$
I have fixed my error, just to keep the record straight. @saulsplatz 's answer fills in the missing data.
$endgroup$
– Ethan Bolker
Jan 5 at 22:32
$begingroup$
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.)
$endgroup$
– ptrcao
Jan 5 at 15:44
$begingroup$
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.)
$endgroup$
– ptrcao
Jan 5 at 15:44
$begingroup$
@ptrcao: you are correct. That makes the disagreement worse.
$endgroup$
– Ross Millikan
Jan 5 at 16:38
$begingroup$
@ptrcao: you are correct. That makes the disagreement worse.
$endgroup$
– Ross Millikan
Jan 5 at 16:38
$begingroup$
I have fixed my error, just to keep the record straight. @saulsplatz 's answer fills in the missing data.
$endgroup$
– Ethan Bolker
Jan 5 at 22:32
$begingroup$
I have fixed my error, just to keep the record straight. @saulsplatz 's answer fills in the missing data.
$endgroup$
– Ethan Bolker
Jan 5 at 22:32
add a comment |
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$begingroup$
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.
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– callculus
Jan 5 at 16:10
1
$begingroup$
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.
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– callculus
Jan 5 at 17:04
$begingroup$
But how do we formulate one single equation in which taxes and fees are both factored into the final price?
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– ptrcao
Jan 5 at 17:20
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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.
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– callculus
Jan 5 at 17:21