I need to make a guide for a sales manager letting him know what the lowest he is allowed to sell a product for based on the quantity of the sale.
Right now I can only think of a cumbersome equation, as follows:
(([sale price]-[cost])-([sale price]*0.05))*[sale quantity]=[profit], profit must be equal to or above 0.66*[sale quantity]
In order to effectively use this equation, the sales manager will have to guess what the lowest price is, run the equation, then see if it is equal to 0.66*[quantity]. Is there an equation that will allow him to find the sales price that will return a profit of exactly 0.66*[quantity]?
I'll use these variables:
$p$ = sale price
$c$ = cost
$q$ = sale quantity
$f$ = profit
Your equation is $$( (p-c)-(p*0.05))*q = f$$ subject to $$f \ge 0.66*q$$
So you want to find the smallest $p$ such that $$( (p-c)-(p*0.05))*q \ge 0.66*q\\ (p-c)-(p*0.05) \ge 0.66\\ p - p*0.05 \ge 0.66 + c\\ 0.95*p \ge 0.66 + c\\ p \ge (0.66 + c)/0.95$$
So $(0.66 + c)/0.95$ gives you the lowest price, where c is the cost.