Break-Even Analysis (Cost vs Revenue)
Learn break-even analysis for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-01). A business's cost and revenue (income) can both be drawn as straight lines against the number of items sold, and where they cross shows when the business starts to make money.
You will write a pair of equations for cost and revenue, find the break-even point where income equals cost, and read from the graph whether the business makes a profit or loss — useful for a school car wash, a market stall or a small business of your own.
Theory
Break-even analysis compares a business’s income with its cost. This Year 12 Standard 1 (NSW) guide shows how to graph both lines, find the break-even point where income equals cost, and work out the profit or loss for any number of items.
Break-even analysis compares a business’s income (revenue) with its cost, both graphed against the number of items \(n\).
The cost line starts above 0, at the fixed cost, and rises by the cost per item. The income line starts at the origin: sell nothing, earn nothing.
The break-even point is where the lines cross: income equals cost and the profit is \(\$0\). When the income line is below the cost line the business makes a loss; when it is above, it makes a profit.
Cost and income for \(n\) items:
Quick method — each item adds \((\text{price}-\text{cost per item})\) towards the fixed cost:
How to find the break-even point
- Write the cost rule (fixed cost \(+\) cost per item \(\times n\)) and the income rule (price \(\times n\)).
- Find where they are equal: read the crossing on the graph, or use \(\text{fixed cost}\div(\text{price}-\text{cost per item})\).
- Read the number of items on the horizontal axis — that is the break-even point.
- Check by working out income and cost at that number: they should be equal.
(i) The cost line starts at \(\$30\). (ii) The lines cross at \(n=6\). (iii) Find the gap at \(n=10\).
| \(\text{Income}\) | \(=\) | \(8(6)=\$48\) |
| \(\text{Cost}\) | \(=\) | \(30+3(6)=\$48\) |
| \(\text{Profit}\) | \(=\) | \(8(10)-(30+3(10))\) |
| \(\) | \(=\) | \(80-60=\$20\) |
Fixed cost \(\$30\); break-even at 6 items; profit \(\$20\).
Each brownie adds \(\$2-\$0.50=\$1.50\) towards the fixed cost.
| \(n\) | \(=\) | \(45\div 1.50\) |
| \(\) | \(=\) | \(30\) |
| \(\text{Income}\) | \(=\) | \(2(30)=\$60\) |
| \(\text{Cost}\) | \(=\) | \(45+0.50(30)=\$60\ \checkmark\) |
30 brownies must be sold.
Profit \(=\) income \(-\) cost.
(a) \(n=15\)
| \(\text{Income}\) | \(=\) | \(7(15)=\$105\) |
| \(\text{Cost}\) | \(=\) | \(40+3(15)=\$85\) |
| \(\text{Profit}\) | \(=\) | \(105-85=\$20\) |
(b) \(n=6\)
| \(\text{Income}\) | \(=\) | \(7(6)=\$42\) |
| \(\text{Cost}\) | \(=\) | \(40+3(6)=\$58\) |
| \(\text{Profit}\) | \(=\) | \(42-58=-\$16\) |
(a) a profit of \(\$20\); (b) a loss of \(\$16\) (break-even is at \(40\div 4=10\) items).
With no sales the income is \(\$0\), so the income line goes through the origin. The cost line starts at the fixed cost. Past the crossing, income is above cost.
Income is the line through \((0,0)\); the fixed cost is \(\$50\); after 20 items the business starts to make a profit.
Common pitfalls
Frequently asked questions
What is the break-even point?
It is the number of items where income equals cost, so there is no profit and no loss. On a graph it is where the income line and the cost line cross.
How do you calculate the break-even point?
Divide the fixed cost by the amount each item contributes, which is the selling price minus the cost per item. For example, 24 dollars divided by (6 minus 2) gives 6 items.
Which line is income and which is cost on a break-even graph?
The income line starts at the origin, because selling nothing earns nothing. The cost line starts higher up, at the fixed cost.
How do you find profit from a break-even graph?
At the number of items you want, read the income and the cost and subtract: profit equals income minus cost. It is the gap between the two lines. A negative answer is a loss.
Does a business make a profit at the break-even point?
No. At break-even the profit is exactly 0 dollars. The business only makes a profit when it sells more items than the break-even number.