Calculate discount needed in order to achieve required profit margin. (algebra with fractions)

Question

This problem seems to require algebra with fractions. This is basic high school (or even middle school?) stuff, but embarrassingly, I seem to have forgot it.

Background

I own a small business and was trying to calculate what kind of discounts I would offer if I wanted to achieve a desired profit margin.

I'm calculating profit margin in this way:

Original price:           200   (p)
Cost to complete service:  80   (c)
Discount:                  20   (d)

Profit margin (m) = (p-d-c) / (p-d) = .555  (55.5%)

I put together a spreadsheet trying to figure the different discounts that I'd need in order to achieve a desired profit margin. This means I need to solve for d, unfortunately, I completely forgot how to isolate d in this circumstance (fraction with arithmetic on the top and bottom).

Question

How do you solve for d in:

(p-d-c) / (p-d) = m

(m is for profit margin)

Answer 1

First, multiply on both sides by the denominator to eliminate the fraction: $$p-d-c = m(p-d) = mp - md$$

Then move all the terms containing $d$ to one side, and everything else to the other side: $$md - d = mp -p + c$$

Now factor out the $d$. $$(m-1)d = mp - p + c$$

Finally, divide by the factor to get an expression for $d$. $$d = \frac{mp - p + c}{m-1}$$