Break Even¶
A business breaks even when their \(net\;profit\) is equal to \(0\), or perhaps more simply, their \(revenue\) is the same as their \(total\;costs\) The units a business needs to sell to break even is regarded to as its \(break\;even\;point\).
Equation¶
The equation to establish the break even point is the following:
\[ break\;even\;point = \frac{fixed\;costs}{price\;per\;unit - cost\;per\;unit} \]
Where:
- break even point, is the number of units the business needs to sell to break even
- fixed costs, is the fixed costs of the business
- price per unit, is the price of one unit
- cost per unit, is the cost of one unit
Example¶
For sake of example, lets say a business has fixed costs of £102, and their price per unit is £5, with manufacturing costs of £2. What is their break even point?
\[
\begin{align}
break\;even\;point & = \frac{fixed\;costs}{price\;per\;unit - cost\;per\;unit} \\
& = \frac{102}{5 - 2} \\
& = 34
\end{align}
\]
Deriving the equation¶
Through use of careful maths, the break even equation can be derived simply from \(net\;profit = 0\).
Fully-qualify the equation¶
First, the equation needs to be fully-qualified, as in make as less abstract as it can possibly be.
\[
\begin{align}
net\;profit & = 0 \\
gross\;profit - expenditure & = 0 \\
revenue - cost\;of\;sales - expenditure & = 0 \\
units\;sold * price\;per\;unit - units\;sold * costs\;per\;unit - fixed\;costs & = 0 \\
units\;sold * (price\;per\;unit - costs\;per\;unit) - fixed\;costs & = 0
\end{align}
\]
In BIDMAS, we trust¶
Now for the fun part, rearranging the current equation.
\[
\begin{align}
units\;sold * (price\;per\;unit - costs\;per\;unit) - fixed\;costs & = 0 \\
units\;sold * (price\;per\;unit - costs\;per\;unit) & = fixed\;costs
\end{align}
\]
and finally...
\[
\begin{align}
units\;sold & = \frac{fixed\;costs}{price\;per\;unit - costs\;per\;unit} \\
break\;even\;point & = \frac{fixed\;costs}{price\;per\;unit - costs\;per\;unit}
\end{align}
\]
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