Coaching Centre
Break-even, profit and loss
Cost and revenue as a pair of simultaneous equations, the profit equation, break-even by algebra and by graph, target profit, how changes to price and costs move the break-even point, and a spreadsheet model.
Break-even is the most common use of simultaneous equations in the HSC. The cost line and the revenue line are the two equations, and their intersection is where the business stops losing money. This section goes further than Year 11: a single profit equation, targets, and what happens when the numbers change.
Revise: cost and revenue lines are introduced in Year 11 Linear models and their limits.
Cost, revenue and profit equations
Cost: $C = F + vn$ (fixed costs plus variable cost per item)
Revenue: $R = pn$ (selling price per item)
Profit: $P = R - C = (p - v)n - F$
Break-even: $R = C$, which happens when $n = \dfrac{F}{p - v}$
Why does $p - v$ matter so much? Each item brings in $p$ and costs $v$ to make, so each sale leaves $p - v$ to put towards the fixed costs. Once enough sales have paid off $F$, every further sale is profit. That is why $F \div (p - v)$ gives the break-even number.
A candle maker pays $\$600$ a month for a market stall and equipment. Each candle costs $\$4$ in wax and wicks and sells for $\$10$. Find the break-even point and the profit on $150$ candles.
Check Example 1 (fixed $\$600$, cost $\$4$, price $\$10$). Then predict, before you move anything, what happens to break-even if the stall rent doubles. Check it. Next put the rent back and raise the price to $\$13$: does break-even halve? Finally set the price equal to the cost per item and read the note.
Reading the graph
- The cost line starts at $F$ on the vertical axis: even with no sales, the fixed costs are paid.
- The revenue line starts at the origin: no sales, no money in.
- Left of the crossing, cost is above revenue: a loss. Right of it, revenue is above cost: a profit.
- The vertical gap between the lines at any $n$ is the size of the profit or loss.
Target profit and changing the numbers
The candle maker wants a profit of $\$1200$ a month. How many candles must be sold?
Wax prices rise, so each candle now costs $\$5$ to make. By how much must the selling price rise to keep the break-even point at $100$ candles?
The pattern behind these: break-even goes up when fixed costs or the cost per item go up, and goes down when the price goes up. If $p \le v$ the business can never break even, because every sale adds to the loss.
A spreadsheet model
A spreadsheet lets you test many sales levels at once. With the number of candles in column A, starting in A2, and the fixed cost, cost per item and price in cells F1, F2 and F3:
| Cell | Formula | Meaning |
|---|---|---|
| B2 | =$F$1+$F$2*A2 | cost |
| C2 | =$F$3*A2 | revenue |
| D2 | =C2-B2 | profit (negative means a loss) |
The dollar signs fix the reference, so the formula still points to F1, F2 and F3 when you fill it down. Break-even is the row where column D changes from negative to positive. Change F3 and the whole table updates, which is the spreadsheet version of moving the price slider.
Your turn
A food truck has fixed costs of $\$1800$ a week. Each meal costs $\$7$ to make and sells for $\$16$. Find the break-even number of meals and the profit on $300$ meals.
Show the working
Each meal contributes $16 - 7 = \$9$. Break-even: $1800 \div 9 = 200$ meals, with revenue $16 \times 200 = \$3200$. Profit on $300$ meals: $9 \times 300 - 1800 = \$900$.
$C = 600 + 4n$ and $R = 10n$. What is the break-even number of items?
$10n = 600 + 4n$, so $6n = 600$ and $n = 100$.
Using the same model, what is the profit, in dollars, on $150$ items?
$P = 6n - 600 = 6 \times 150 - 600 = 300$.
At the break-even point, the profit is:
At break-even, revenue equals cost, so $R - C = 0$.
A food truck has fixed costs of $\$1800$, each meal costs $\$7$ and sells for $\$16$. How many meals to break even?
$1800 \div (16 - 7) = 1800 \div 9 = 200$.
The candle maker has $P = 6n - 600$. How many candles give a profit of $\$1200$?
$6n - 600 = 1200$, so $6n = 1800$ and $n = 300$.
Fixed costs rise while the price and cost per item stay the same. The break-even point:
$n = F \div (p - v)$: a larger $F$ divided by the same $p - v$ gives a larger $n$.
A business sells each item for $\$8$ and each costs $\$9$ to make. Which is true?
Each sale contributes $8 - 9 = -\$1$, so selling more only makes the loss bigger.
Fixed costs are $\$500$, each item costs $\$3$ and sells for $\$18$. What is the smallest whole number of items needed to break even?
$500 \div (18 - 3) = 500 \div 15 = 33.\overline{3}$. $33$ items is not quite enough, so round up to $34$.
Exam style: A candle costs $\$5$ to make and fixed costs are $\$600$. What selling price, in dollars, gives a break-even point of $100$ candles?
Each candle must contribute $600 \div 100 = 6$. So $p - 5 = 6$ and $p = 11$.
Exam style: With fixed costs $\$1800$, cost per meal $\$7$ and price $\$16$, what is the profit, in dollars, on $150$ meals? (A loss is negative.)
$P = 9 \times 150 - 1800 = 1350 - 1800 = -450$, a loss of $\$450$, because $150$ is below the break-even of $200$.