Need help with making different price items maintain a set average!

[TomOsiris]

Stuck, please help

I have 3 grades of product; High (h), Middle (m), Low (l), and Scrap (s)
The average (mean) price of these items must be $3.25/lb
Scrap price (s) must $0.50/lb. This is only set price, the rest are variable.

The price ratios I need to maintain are;
m=1.375*(l)
h=1.75*(l)

I need to solve for (l), which will give me prices for (m) and (h)

all of the above grades will have various amounts. for example,

(s)* 40,000 lb 0.4(t)
(l)* 30,000 lb 0.3(t)
(m)* 15,000 lb 0.15(t)
(h)* 15,000 lb 0.15(t)

In this example, we have 100,000 total lb. = (t)
The total price for these items is $325,000 $3.25(t)

since (s) = $0.50/lb, we can determine that we paid $305,000 for the rest of the three grades. $325,000 - (40,000*0.5) = $305,000
since we have twice as much (l) as the 2 other grades, we can simplify that to 2(l)+1.375(l)+1.75(l) = 305,000

This where I'm getting a bit lost. I need to know how to formulate an equation so that I can change any/all of the amounts per grade, and extrapolate the correct ratio prices for (l), (m), and (h); and the average (mean) price of (l),(m),(h), and (s) always is $3.25

Please send help, I can't figure it out

 

[Cubist]

You effectively have one unknown quantity, l, since...

s=0.5
m=1.375*l
h=1.75*l

Therefore the average price requirement is enough to determine a value for l...

(s+l+m+h)/4 = 3.25

0.5+l+1.375*l+1.75*l = 3.25*4

0.5 + l*(1+1.375+1.75) = 13

l*4.125 = 13 - 0.5

l = 12.5/4.125 = (approx) 3.030303

To check, plug this value into (s+l+m+h)/4 and you'll get the required average of 3.25

=======

The assumption in your example that the total price will be the (average price * total amount purchased) is flawed because the purchased quantities of s,l,m,h are different.

Consider an extreme example where the customer buys 100,000lb of scrap ONLY - the total weight bought is 100,000lb so would the cost be 325,000 ? No, obviously the cost is 0.5*100,000 = 50,0000 since he didn't buy any of the more expensive options.

 

[Cubist]

IF you wanted the WEIGHTED average to be 3.25, then

Let qs,ql,qm,qh=be the quantity of s,l,m,h purchased, respectively

Then...

( qs*s + ql*l + qm*m + qh*h ) / (qs+ql+qm+qh) = 3.25

0.5*qs + l*(ql + 1.375*qm + 1.75*qh) = 3.25 * (qs+ql+qm+qh)

l*(ql + 1.375*qm + 1.75*qh) = (3.25*(qs+ql+qm+qh) - 0.5*qs)

l = (3.25*(qs+ql+qm+qh) - 0.5*qs) / (ql + 1.375*qm + 1.75*qh)

Using the figures from your example

qs=40000
ql=30000
qm=15000
qh=15000

Plugged into the equation above gives l = (approx) 3.9675

 

[TomOsiris]

so perhaps I could have been more clear in my in original question

These are grades of product I will be purchasing, a "crop". I will be be paying $3.25 flat rate per lb, no matter how many lbs I get.
Once I have received the "crop" it will be divided into the various grades, with a cost assigned each grade.
There will always be some amount of each of the 4 grades within the crop.
I do not know how to incorporate the percentage of the total by grade into my equation.
For example if 40% if $0.50/lb, how do I calculate what the other 60% needs to be to equal make the average $3.25?

 

[JeffM]

As cubist implied, you need a weighted average

[MATH]\dfrac{40 * 0.50}{100} + \dfrac{60 * x}{100} = 3.25 \implies 20 + 60x = 325 \implies x = \dfrac{325}{60} \approx 5.42.[/MATH]
Let's see why that works. Say we purchase 100 pounds for 325 dollars.

[MATH](40 * 0.50) + (60 * 5.42) = 325.20.[/MATH]
Of course you make no profit on this, but you have a good heart.

Wait. You still have to divide the 60% up. You use the same technique with it.

Clear now?

 

[TomOsiris]

Jeff, this helped me find the correct formula!
Thank you very much. This is work-related and had me stuck for a few days