# Rewrite a function to get c*

#### Pvunderink

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!

#### MarkFL

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 $$\displaystyle \frac{c}{\text{Profit}_D-c-\text{Profit}_M}$$:

$$\displaystyle c'=-\frac{c\text{Profit}_M}{\text{Profit}_D-c-\text{Profit}_M}=\frac{c\text{Profit}_M}{c+\text{Profit}_M-\text{Profit}_D}$$

#### Pvunderink

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.

#### Pvunderink

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?

#### MarkFL

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 $$\displaystyle a$$.

Thank you!

#### Denis

Math Team
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)