Need equation to find break-even point for products with fixed costs

Question

This is probably highschool-level stuff, but I haven't been in highschool for a long time...

Anyway, I have a product with known costs for a single production run. It will be sold by someone making 25% commission. In addition to the commission, we will have to pay payroll taxes amounting to 9% of the commission paid.

I need to figure out how many dollars' worth of products need to be sold in order to break even on the combined total of the product run, the commission to that point, and the payroll tax to that point.

In plain English, here's the equation I'm working with:

$ProductionRun + ($TotalSales * 25%) + ($TotalCommission * 9%) = $TotalCosts

However, what I'm after is the dollar amount where TotalSales matches TotalCosts.

How would one go about this?

Answer 1

If $s$ is sales and $p$ is production cost, your commission is $0.25s$ and your payroll tax is $0.09 \cdot 0.25s=0.0225s$. The total costs are then $p+0.2725s$ and you break even when this is $s$. You need $$p+0.2725s=s\\s=\frac p{1-0.2725}\approx 1.3746p$$

Answer 2

If I understand what you want correctly, here's how I'd solve the problem

Since commission is 25% of sales, we multiply total sales by 0.25. Moreover, due to taxes, we multiply the commission by 1.09

So:

$$\textrm{Production_Costs} + \textrm{Total_Sales}\cdot0.25\cdot1.09=\textrm{Total_Sales}$$ $$\textrm{Production_Costs} = 0.7275*\textrm{Total_Sales}$$

Hence, sales need to be $\frac 1 {0.7275} \cdot100\%-100\%=37.457\%$ higher than production costs.