Break-Even Analysis (Profit & Loss)
Learn break-even analysis for NSW Year 12 Mathematics Standard 2. You model a business's cost and revenue as two straight lines and find the break-even point — the number of items where revenue exactly equals cost, so there is no profit and no loss.
This topic shows how to set up the cost and revenue equations, solve where revenue equals cost to find the break-even point, and calculate profit and loss for any level of output — a practical Standard 2 skill for small-business and spreadsheet problems.
Theory
Break-even analysis finds the point where a business's revenue just covers its cost. This Year 12 Standard 2 (NSW) guide models cost and revenue as straight lines, finds the break-even point where \(R=C\), and shows how to calculate profit and loss — including in a spreadsheet.
In break-even analysis a business's cost \(C\) and revenue (income) \(R\) are written as straight lines against the number of items \(n\). Cost is a fixed cost plus a cost per item; revenue is the selling price times the number sold.
The break-even point is where the two lines cross — the value of \(n\) at which \(R=C\), so income exactly covers cost and there is neither a profit nor a loss.
Profit is revenue minus cost, \(P=R-C\). When \(R>C\) (to the right of break-even) the business makes a profit; when \(R
Write cost and revenue as straight lines in the number of items \(n\):
The break-even point is where revenue equals cost:
Profit is revenue minus cost:
A spreadsheet makes the profit and loss easy to see (muffins at \(\$3.50\), costs \(\$80+\$1\times n\)):
| \(n\) | Revenue \(R=3.5n\) | Cost \(C=80+n\) | Profit \(R-C\) |
|---|---|---|---|
| 0 | $0 | $80 | -$80 (loss) |
| 20 | $70 | $100 | -$30 (loss) |
| 32 | $112 | $112 | $0 (break-even) |
| 100 | $350 | $180 | $170 (profit) |
How to solve a break-even problem
- Write the cost equation \(C=\) fixed cost \(+\) (cost per item)\(\times n\).
- Write the revenue equation \(R=\) (selling price)\(\times n\).
- Set \(R=C\) and solve for \(n\) — the break-even number of items (or read it where the lines cross on the graph).
- Interpret: for a chosen \(n\), find profit \(P=R-C\); a positive \(P\) is a profit, a negative \(P\) is a loss.
Break-even is where the lines cross, \(R=C\).
| \(6n\) | \(=\) | \(40+2n\) |
| \(4n\) | \(=\) | \(40\) |
| \(n\) | \(=\) | \(10\) |
| \(R=C\) | \(=\) | \(\$60\) |
Washing 10 cars breaks even at \(\$60\).
Write revenue and cost, then set \(R=C\).
| \(R\) | \(=\) | \(8n\) |
| \(C\) | \(=\) | \(150+3n\) |
| \(8n\) | \(=\) | \(150+3n\) |
| \(5n\) | \(=\) | \(150\) |
| \(n\) | \(=\) | \(30\) |
Selling 30 jars breaks even (revenue = cost = \(\$240\)).
Break-even at \(R=C\); then profit \(=R-C\).
| \(3.5n\) | \(=\) | \(80+n\) |
| \(2.5n\) | \(=\) | \(80\) |
| \(n\) | \(=\) | \(32\) |
| \(n=100:\ R-C\) | \(=\) | \(350-180=\$170\) |
| \(n=20:\ R-C\) | \(=\) | \(70-100=-\$30\) |
32 muffins break even; \(100\) gives a \(\$170\) profit, \(20\) gives a \(\$30\) loss.
Set profit \(P=R-C\) equal to \(\$1000\).
| \(13n-(600+5n)\) | \(=\) | \(1000\) |
| \(8n-600\) | \(=\) | \(1000\) |
| \(8n\) | \(=\) | \(1600\) |
| \(n\) | \(=\) | \(200\) |
Selling 200 cases makes a \(\$1000\) profit (break-even is \(75\)).
Common pitfalls
Frequently asked questions
What is the break-even point?
It is the number of items a business must sell so that its revenue exactly equals its cost. At that point there is no profit and no loss. On a graph it is where the revenue line and the cost line cross.
How do you calculate the break-even point?
Write revenue as the selling price times the number of items, and cost as the fixed cost plus the cost per item times the number of items. Set revenue equal to cost and solve for the number of items.
How do you work out profit and loss?
Profit equals revenue minus cost. If the result is positive the business makes a profit; if it is zero it breaks even; if it is negative it makes a loss of that amount.
What is the difference between the profit region and the loss region on a break-even graph?
To the right of the break-even point the revenue line is above the cost line, so the business makes a profit. To the left of it the cost line is above the revenue line, so the business makes a loss.
Why is the cost line above the origin?
Because a business usually has fixed costs, such as rent or equipment, that must be paid even when no items are sold. The fixed cost is the vertical intercept of the cost line, and the cost per item is its gradient.
How is a spreadsheet used for break-even analysis?
You list the number of items in one column and use formulas to work out revenue, cost and profit in the next columns. The break-even point is the row where profit changes from negative (a loss) to positive (a profit).