Formula to determine value with 3 variables

Hendersonc241

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I am trying to figure out a formula to determine the value of an item with an unknown amount of platinum, palladium, and rhodium in it. I have previous values and the dates of those values. I want to figure out a formula with multiple values on multiple dates and the prices of those metals on those dates, based on the change of value of the item compared to the value of those 3 metals, to then be able to input the current value of the 3 metals and get the current value of the item. Hopefully that made sense. Basically a formula that can determine the amount of metals in the item based on the change of 3 variables at different times. Thank you for any help.
 

Dr.Peterson

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I am trying to figure out a formula to determine the value of an item with an unknown amount of platinum, palladium, and rhodium in it. I have previous values and the dates of those values. I want to figure out a formula with multiple values on multiple dates and the prices of those metals on those dates, based on the change of value of the item compared to the value of those 3 metals, to then be able to input the current value of the 3 metals and get the current value of the item. Hopefully that made sense. Basically a formula that can determine the amount of metals in the item based on the change of 3 variables at different times. Thank you for any help.
This isn't clear to me. First you say you want to "determine the value of an item", then later you say you want to "determine the amount of metals in the item". Also, some things look like they might be impossible to determine, though I'm too confused to be sure!

How about giving us a simple example of what you would be given, and what you want to find?
 

Hendersonc241

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ok sorry, the item has platinum, palladium and rhodium in it, but I do not know how much of each. On Feb 23rd it's value is $253 and the price per ounce of platinum is $1231, palladium is $2224, and rhodium is $21000. On Mar 10th, the value of the item is $265 and Platinum is $1186, palladium is $2262, and Rhodium is $20000. On Mar 17th the value is $268 and Platinum is 1204, Palladium is $2489, and Rhodium is $23500. Basically I am trying to find a formula to compare the changes in the metal prices and the changes in the item prices to determine the item price at any given time if I plug in the metal prices. I guess I am trying to figure out how much of each metal is in the item maybe? I have around 40 different price dates to compare if that helps at all.
 

JeffM

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It makes a sort of sense

\(\displaystyle v(t) = x(t) + p_1(t)m_1 + p_2(t)m_2 + p_3(t)m_3.\)

What that means is that the value of the item at time t is some -non-negative value x greater than the sum of the values of the three constituent metals.

If you assume that x is constant over time, you can figure out what the quantities of metals are if you have numerical data for v, p_1, p_2, and p_3 on four different dates.

But that may not be a very good assumption.
 

Hendersonc241

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Ok, I get that, but the value would not be greater than the sum of the 3, because the price of the metals are price per ounce, and the item will have less than an ounce of each metal. Maybe a gram or 2. So will that still work? If I am looking at this right, x would be the constant, ie the amount of metals in the item?
 

Dr.Peterson

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ok sorry, the item has platinum, palladium and rhodium in it, but I do not know how much of each. On Feb 23rd it's value is $253 and the price per ounce of platinum is $1231, palladium is $2224, and rhodium is $21000. On Mar 10th, the value of the item is $265 and Platinum is $1186, palladium is $2262, and Rhodium is $20000. On Mar 17th the value is $268 and Platinum is 1204, Palladium is $2489, and Rhodium is $23500. Basically I am trying to find a formula to compare the changes in the metal prices and the changes in the item prices to determine the item price at any given time if I plug in the metal prices. I guess I am trying to figure out how much of each metal is in the item maybe? I have around 40 different price dates to compare if that helps at all.
Is this a real-life problem, or an assignment?

If it's the latter, and you don't have to think about issues like whether the value depends on other things besides the values of the three metals (such as manufacturing costs or other materials), then you can just write and solve a system of three linear equations using the three sets of numbers you gave.

If it's real-life, then you'll have to make further assumptions, such as JeffM's, which adds a fourth variable. You can use additional values beyond four to test for consistency of your assumptions.

Ok, I get that, but the value would not be greater than the sum of the 3, because the price of the metals are price per ounce, and the item will have less than an ounce of each metal. Maybe a gram or 2. So will that still work? If I am looking at this right, x would be the constant, ie the amount of metals in the item?
I'm not sure of your objection. Are you assuming that the value of the item is affected only by the three metals, and nothing else? The suggested "x" is not the amount of metals, but the "overhead", in some sense, which I'd think would have to exist. But then, I know nothing about the real "item".
 

JeffM

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Ok, I get that, but the value would not be greater than the sum of the 3, because the price of the metals are price per ounce, and the item will have less than an ounce of each metal. Maybe a gram or 2. So will that still work? If I am looking at this right, x would be the constant, ie the amount of metals in the item?
Of course the weight of the unit exceeds the weight of some of its materials. But that is not what the equation is about.

You seem to be assuming that the price at which an item is exchanged equals the cost of the materials. This is generally not even close to correct. Do you think that the price of a car is simply the sum of the prices of the parts. In fact, the greatest component of the price of any physical good is almost always labor cost, not cost of materials.

Cost of materials and of labor place a floor under the sustainable price of any item. They do not impose a ceiling.

The variables \(\displaystyle m_1, \ m_2, \ m_3\) represent weights of metals used in each item expressed in a consistent unit of weight, say ounces of metal used per item.

The functions \(\displaystyle p_1(t), \ p_2(t), \ p_3(t)\) represent units of money per unit of weight of metal, say dollars per ounce of metal.

And v(t) represents the price of one of the items. Everything then is expressed in monetary terms, and x does not represent a weight. It represents how much greater the price of the item is than the cost of the metals used in manufacturing the item.

It is about finding how much greater is the price of the item than the cost of those three materials. If that differential in value is constant in time, and only then, you can determine the weights of metal used by solving a system of four simultaneous equations.

P.S. After reading Dr. Peterson's response, if your observations are close in time and the percentage differences in the price of the item are small relative to the percentage changes in the costs of the metal, then the assumption that x is constant will be a very plausible assumption. Did you understand his comment about extra data to check the validity of the assumption?
 
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Hendersonc241

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Yes, the value of the item is based on the value of the 3 metals. The item is salvage, now it won't be the exact percent of the value of the metals because of course you can never recycle 100%, but the price rises and drops based on the rise and drop of those 3 metals. The items are catalytic converters. There is no overhead or labor cost because the item is being recycled. Maybe this will help, I have about 40 lists, each a week apart or near there, and each list has the price at that date of about 7000 different converters. What I am trying to do is determine how the change in the 3 metal prices on those dates changed the value of the converters so I can, in the future, get the current price by inputting the current value of the 3 metals.
 

JeffM

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On the assumption that the value of these scrapped converters depends solely on the prices of the recoverable precious metals, then the reccoverable amount of the different metals in each type of converter can be found from just three data points.

\(\displaystyle v_1 = m_1p_{1,1} + m_2p_{2,1} + m_3p_{3,1}.\)

\(\displaystyle v_2 = m_1p_{1,2} + m_2p_{2,2} + m_3p_{3,2}.\)

\(\displaystyle v_3 = m_1p_{1,3} + m_2p_{2,3} + m_3p_{3,3}.\)

Solve that system of simultaneous equations for each type of converter and check it against some other points.
 

Hendersonc241

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Yes, the value of the item is based on the value of the 3 metals. The item is salvage, now it won't be the exact percent of the value of the metals because of course you can never recycle 100%, but the price rises and drops based on the rise and drop of those 3 metals. The items are catalytic converters. There is no overhead or labor cost because the item is being recycled. Maybe this will help, I have about 40 lists, each a week apart or near there, and each list has the price at that date of about 7000 different converters. What I am trying to do is determine how the change in the 3 metal prices on those dates changed the value of the converters so I can, in the future, get the current price by inputting the current value of the 3 metals.
Ok I will try working that out and if it only requires 3 points and I have about 40, I should be able to use the others to check and make sure the numbers match and it is working correctly! Thank you very much.
 

Hendersonc241

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On the assumption that the value of these scrapped converters depends solely on the prices of the recoverable precious metals, then the reccoverable amount of the different metals in each type of converter can be found from just three data points.

\(\displaystyle v_1 = m_1p_{1,1} + m_2p_{2,1} + m_3p_{3,1}.\)

\(\displaystyle v_2 = m_1p_{1,2} + m_2p_{2,2} + m_3p_{3,2}.\)

\(\displaystyle v_3 = m_1p_{1,3} + m_2p_{2,3} + m_3p_{3,3}.\)

Solve that system of simultaneous equations for each type of converter and check it against some other points.
I guess I jumped the gun a little, I don’t fully understand the equation. The V is the value of item? And the m and p’s are metal price? What is the 1,2,and 3 at the end of the mp? Like on m1p1,1 what is the one after the comma? And where is the variable I am solving for, the x so to speak? I’ll try to do some research and figure it out. Thank you again
 

JeffM

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I guess I jumped the gun a little, I don’t fully understand the equation. The V is the value of item? And the m and p’s are metal price? What is the 1,2,and 3 at the end of the mp? Like on m1p1,1 what is the one after the comma? And where is the variable I am solving for, the x so to speak? I’ll try to do some research and figure it out. Thank you again
The v’s are the value of the item at three different times. The p’s are the prices of the three different metals at times corresponding to the times when you priced the item. The m’s are your three unknowns, namely the weight of recoverable metal in a single item.

So \(\displaystyle m_3 p_{3, 2}\) is the unknown weight of metal 3 recoverable times the unit price of the third metal at the second pricing time.

Do you know how to solve a system of 3 simultaneous equations for three unknowns?
 
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Hendersonc241

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I think so, just wasn’t sure what I was solving for there, but looks like m is my x so I’ll try solving each one for m. Thank you 😊
 

JeffM

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I think so, just wasn’t sure what I was solving for there, but looks like m is my x so I’ll try solving each one for m. Thank you 😊
There are three different m’s.
 

Hendersonc241

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Would I be able to use Cramer's rule to solve this?
 

JeffM

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That is what it’s for.
 

Hendersonc241

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ok wasn't sure if i should use Cramer's rule or the making 2 pairs and solving for a single variable. Thank you.
 

JeffM

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