Rewrite a function to get c*

Oct 2013
25
0
I've got the following function:

(c*/c)*(ProfitD - c*) + (1 -(c*/c))*(ProfitM - c*)=0

I need to derive c* as a function of the other parameters.
Furthermore, I know the answer should be:
c*= (c * ProfitM) / (c + ProfitM - ProfitD)

However, I'm not able to solve c* by myself. I do not know which intermediate steps to use in order to derive this answer. Can someone show, step by step, how to get this answer?
It would really help a lot!
 
Jul 2010
12,211
522
St. Augustine, FL., U.S.A.'s oldest city
I'm going to write \(\displaystyle c^{*}\) as \(\displaystyle c'\)...we are given:

\(\displaystyle \frac{c'}{c}\(\text{Profit}_D-c'\)+\(1-\frac{c'}{c}\)\(\text{Profit}_M-c'\)=0\)

If we distribute, we obtain:

\(\displaystyle \frac{c'}{c}\text{Profit}_D-\frac{c'^2}{c}+\text{Profit}_M-c'-\frac{c'}{c}\text{Profit}_M+\frac{c'^2}{c}=0\)

Collect like terms:

\(\displaystyle \frac{c'}{c}\text{Profit}_D+\text{Profit}_M-c'-\frac{c'}{c}\text{Profit}_M=0\)

Move all terms not involving \(\displaystyle c'\) to the right side:

\(\displaystyle \frac{c'}{c}\text{Profit}_D-c'-\frac{c'}{c}\text{Profit}_M=-\text{Profit}_M\)

Factor out \(\displaystyle c'\) on the left side:

\(\displaystyle c'\(\frac{\text{Profit}_D}{c}-1-\frac{\text{Profit}_M}{c}\)=-\text{Profit}_M\)

Combine terms within parentheses:

\(\displaystyle c'\(\frac{\text{Profit}_D-c-\text{Profit}_M}{c}\)=-\text{Profit}_M\)

Multiply through by \(\displaystyle \frac{c}{\text{Profit}_D-c-\text{Profit}_M}\):

\(\displaystyle c'=-\frac{c\text{Profit}_M}{\text{Profit}_D-c-\text{Profit}_M}=\frac{c\text{Profit}_M}{c+\text{Profit}_M-\text{Profit}_D}\)
 
Oct 2013
25
0
Thank you for your answer.

However, there is 1 small step which is still slightly unclear.
After you combined the terms within parantheses, you say you 'multiply trough by c/profitD - c - ProfitM'. Why do we do this and do not see how we get the result one line below.

Sorry for the inconvenience. I hope you can help me with this last part.
 
Oct 2013
25
0
Is it that you want to get ProfitD-c-ProfitM/c to the other side of the equation, which is possible by change the numerator and the denominator and multiplying with that part you already had on the other side?
 
Jul 2010
12,211
522
St. Augustine, FL., U.S.A.'s oldest city
if we have:

\(\displaystyle ab=c\)

and we want to solve for \(\displaystyle a\), we need to multiply through by \(\displaystyle \frac{1}{b}\) (or equivalently divide through by \(\displaystyle b\)):

\(\displaystyle \frac{a\cancel{b}}{\cancel{b}}=\frac{c}{b}\)

\(\displaystyle a=\frac{c}{b}\)

Doing this isolates \(\displaystyle a\), and we have thereby solved for \(\displaystyle a\).
 

Denis

Math Team
Oct 2011
14,592
1,026
Ottawa Ontario, Canada
Pvunderink said:
(c*/c)*(ProfitD - c*) + (1 -(c*/c))*(ProfitM - c*)=0

I need to derive c* as a function of the other parameters.
Furthermore, I know the answer should be:
c*= (c * ProfitM) / (c + ProfitM - ProfitD)
To save yourself "writing out stuff" time, here's how I attack these (I'm lazy!):

Let c* = k, profitD = d, profitM = m ; then:

(d - k)k/c + (m - k)(1 - k/c) = 0

(d - k)k/c + (m - k)[(c - k)/c) = 0

(d - k)k + (m - k)(c - k) = 0

kd - k^2 + cm - km - kc + k^2 = 0

kc + km - kd = cm

k(c + m - d) = cm

k = cm / (c + m - d)