Model practical problems with linear cost and revenue functions and find the break-even point

What this dot point is asking

NESA wants you to set up linear cost and revenue equations for a small business or stall, find the break-even quantity where cost equals revenue, and calculate profit at any production level. This is one of the most predictable Section II questions in the entire paper.

The answer

Cost, revenue and profit

For a linear model with $n$ units produced or sold:

• Cost $C = F + v n$, where $F$ is fixed cost and $v$ is variable cost per unit.
• Revenue $R = p n$, where $p$ is selling price per unit.
• Profit $P = R - C = (p - v) n - F$.

The quantity $(p - v)$ is the contribution margin per unit. It is what each sold unit contributes towards covering fixed costs and then towards profit.

Break-even

Break-even is the quantity at which profit is zero, that is $C = R$:

Below the break-even quantity the business loses money. Above it, the business is in profit.

If the answer is not a whole number, round up to the next integer (you cannot sell a partial item, and at the rounded-down value the business is still in loss).

Graphical interpretation

Plot $C$ and $R$ against $n$ on the same axes. Both are straight lines through $(0, F)$ and $(0, 0)$ respectively. The break-even quantity is where they cross. To the right of the crossing, the gap $R - C$ is the profit. To the left, $C - R$ is the loss.

Reading off questions

Standard phrasings: "find the number of units needed to break even", "find the profit if $x$ are sold", "by how much does the profit increase per extra unit sold" (this is just $p - v$).