Calculate break-even selling price based on total lost, there are margins and maximum values

Imagine a magazine which has different formats in which an advertising party can advertise, the % after each format is the price of the 1/1 page times the given %:

1/1 page: 100%, 2/1 page: 180%, 1/2 page: 80%, 1/4 page: 40%.


Example:

1/1 page: 100$, 2/1 page: 180$, 1/2 page: 80$, 1/4 page 40$.


After substracting each costs of the magazine without setting a price for the formats (let's say each format's price is 0$, no other revenues), the total costs are 100.000 dollars. I want to know what each page should be sold to the advertisers to make it break-even (no other costs). To make things more complicated (the part which I am stuck at), is the maximum amount of page per format availiable. This are the facts: 1/1 page: 12, 2/1 page: 1, 1/2 page: 6, 1/4 page: 9  So there are in total 19,25 advertising pages available. Each full advertising page is build up like this: 62,34% are 1/1 pages, 10,39% are 2/1 pages, 15,58% are 1/2 pages and 11,69% are 1/4 pages.  I need to know for each format how much it should cost to play break-even. I find it hard because of the margins each format has and the maximum amount of pages it can contain for each format. Thanks! 2 Answers I left out the rest of the calculations in the previous answer because I was hoping you would be interested in doing the rest. I don't know if this is homework, but I'll explain you all of the steps so you can do problems like this on your own in the future :) I'll set up this for you: If x is the pages sold at 100% margin, y is sold at 180% (1.8x), z is sold at 80% (.8x) and p is sold at 40% of x (.4x), we have the equation: 1x + 1.8x + .8x +.4x = 100 This equals 25, which means we should sell 25 pages of each to break even. However, we don't have that many pages, right? So we have to use what we have. Using this equation: 12pages(1x)+ 1page(1.8x) + 6pages(.8x) + 9pages(.4x) = 100 Solve for x, and this will give 1 solution to your problem. So multiply 12 by 1, 1 by 1.8, 6 by .8, 9 by .4, add those all up and do 100/your equation and you have a solution. :) hint, the answer is a decimal number in between 3 and 7. :) this solution assumes you sell all of your ad space. X=4.504504504504 This gives you the price of the 100% page, assuming the total cost was 100 dollars. You will want to probably round up to$4.51.

But here we go, our first ads sell at 100% of 4.504504, our next ads sell at 180% of 4.504504 = 1.8(4.504504)= 8.108 or \$8.11, our smaller ads sell at 80% of the price and 40% of the price, so do .8(4.504504) and .4(4.504504) and you have the cost, per ad, of the remaining ads (assuming you want to sell every ad in your magazine)! These prices obviously are the break-even prices---selling these ads for more, assuming all ads sold would generate a real profit. Selling under these prices OR not selling all of the ads successfully will result in a loss.

Unless I'm mistaken, I believe you have a 4 variable equation here, defined by 12x + y +6z + 9p = 100.000 (I don't know if you mean 100 or 100,000 but you get the idea). I'm using the 1/1 page as x, the 2/1 page as y, the 1/2 page as z and the 1/4 page as p.

This problem looks hard, but there is some easy substituton that turns this into an algebra problem.

I think you're saying that there are different Ad sizes and those sizes have a percent of the cost of the standard 1/1 ad. If this is what you mean, then you can use substitution for x---for example, if we know that 1/4 page sells for 40% of x, that means p=.4x

Which leads to these equalities: x=x y= 1.8x z= .8x p= .4x

Now you can literally plug these into the 4 variable equation I set for above, and solve for x.

• Thanks for giving an answer. However, this does not fully cover everything I need. The amount of maximum pages is not covered here. I hope you can help me out. – J. Doe Jul 1 '17 at 17:54