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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