Beginning Algebra With Applications, Chapter 3, 3.2, Section 3.2, Problem 188

A manufacturing engineer determines that the cost per unit for a compact disc is $\$ 3.35$ and that the fixed cost is $\$ 6180$. The selling price for the compact disc is $\$ 8.50$. Find the break-even point.

To determine the break-even point, or the number of units that must be sold so that the profit or no loss occurs, and an economist uses the equation $Px = Cx + F$, where $P$ iis the selling price per unit, $x$ is the number of units sold, $C$ is the cost to make each unit and $F$ is the fixed cost.


Solving for the break-even point $x$,


$
\begin{equation}
\begin{aligned}

Px =& Cx + F
&& \text{Given equation}
\\
\\
Px - Cx =& F
&& \text{Subtract } Cx
\\
\\
x(P-C) =& F
&& \text{Factor out } x
\\
\\
x =& \frac{F}{P-C}
&& \text{Divide by } P-C
\\
\\
x =& \frac{6180}{8.5-3.35}
&& \text{Substitute $F = 6,180, P = 8.5$ and $C = 3.35$}
\\
\\
x =& 1,200
&&

\end{aligned}
\end{equation}
$



In other words, the engineer should sell $1,200$ units of compact disc in order to break-even.