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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]?

Wilf
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1 Answers1

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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.

Jaap Scherphuis
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  • Thanks, you've answered the question perfectly. Unfortunately I've realised I asked the wrong question...

    The equation I wrote basically says that a minimum of 66p per product must be made in order for a sale to go ahead, but I was trying to write an equation that gave him a higher margin if the sale quantity was small and a lower margin if the sale quantity was low. Any idea how I might do that?

     – Wilf Jul 31 '17 at 12:01