**Methods for calculating the break-even point**

The break-even point is when total revenues and total costs are equal, that is, there is no profit but also no loss made. There are three methods for ascertaining this break-even point:

**(1) The equation method**

A little bit of simple maths can help us answer numerous different cost‑volume-profit questions.

We know that total revenues are found by multiplying unit selling price (USP) by quantity sold (Q). Also, total costs are made up firstly of total fixed costs (FC) and secondly by variable costs (VC). Total variable costs are found by multiplying unit variable cost (UVC) by total quantity (Q). Any excess of total revenue over total costs will give rise to profit (P). By putting this information into a simple equation, we come up with a method of answering CVP type questions. This is done below continuing with the example of Company A above.

Total revenue – total variable costs – total fixed costs = Profit

(USP x Q) – (UVC x Q) – FC = P

(50Q) – (30Q) – 200,000 = P

Note: total fixed costs are used rather than unit fixed costs since unit fixed costs will vary depending on the level of output.

It would, therefore, be inappropriate to use a unit fixed cost since this would vary depending on output. Sales price and variable costs, on the other hand, are assumed to remain constant for all levels of output in the short-run, and, therefore, unit costs are appropriate.

Continuing with our equation, we now set P to zero in order to find out how many items we need to sell in order to make no profit, i.e. to break even:

(50Q) – (30Q) – 200,000 = 0

20Q – 200,000 = 0

20Q = 200,000

Q = 10,000 units.

The equation has given us our answer. If Company A sells less than 10,000 units, it will make a loss. If it sells exactly 10,000 units it will break-even, and if it sells more than 10,000 units, it will make a profit.

**(2) The contribution margin method**

This second approach uses a little bit of algebra to rewrite our equation above, concentrating on the use of the ‘contribution margin’. The contribution margin is equal to total revenue less total variable costs. Alternatively, the unit contribution margin (UCM) is the unit selling price (USP) less the unit variable cost (UVC). Hence, the formula from our mathematical method above is manipulated in the following way:

(USP x Q) – (UVC x Q) – FC = P

(USP – UVC) x Q = FC + P

UCM x Q = FC + P

Q = FC + P

UCM

So, if P = 0 (because we want to find the break-even point), then we would simply take our fixed costs and divide them by our unit contribution margin. We often see the unit contribution margin referred to as the ‘contribution per unit’.

Applying this approach to Company A again:

UCM = 20, FC = 200,000 and P = 0.

Q = FC

UCM

Q = 200,000

20

Therefore, Q = 10,000 units

The contribution margin method uses a little bit of algebra to rewrite our equation above, concentrating on the use of the ‘contribution margin’.

**(3) The graphical method**

With the graphical method, the total costs and total revenue lines are plotted on a graph; $ is shown on the y axis and units are shown on the x axis. The point where the total cost and revenue lines intersect is the break-even point. The amount of profit or loss at different output levels is represented by the distance between the total cost and total revenue lines.

Hence, it is the difference between the variable cost line and the total cost line that represents fixed costs. The advantage of this is that it emphasizes contribution as it is represented by the gap between the total revenue and the variable cost lines.