A new machine for the crackers: Lena's crackers sell to cafés at $2.50 a packet. The flour, butter and packet cost $0.60, so each packet leaves $1.90 towards the fixed costs of $7,280 a year. In 5.5.3 that gave a break-even of 3,832 packets, and with 18,000 sold in 2025, a margin of safety of 14,168.
Now Priti, the head baker, wants a packing machine for the cracker line. It is leased for $1,520 a year, paid whatever number of packets they make: a fixed cost.
Nothing about a single packet has changed. It still sells for $2.50 and still leaves $1.90. But there is now more to cover before the line makes a profit, so more packets must be sold.
The same month Valley Grain puts up the price of flour, adding 10 cents to each packet. This time the fixed costs stay put, but every packet leaves less: $1.80 instead of $1.90.
Break-even quantity = fixed costs ÷ contribution per unit. Change either side of the division and the answer moves.
| Before | Machine: fixed costs +$1,520 | Flour: variable cost +$0.10 | |
|---|---|---|---|
| Fixed costs a year | $7,280 | $8,800 | $7,280 |
| Variable cost a packet | $0.60 | $0.60 | $0.70 |
| Contribution a packet | $1.90 | $1.90 | $1.80 |
| Break-even quantity | 7,280 ÷ 1.90 = 3,832 | 8,800 ÷ 1.90 = 4,632 | 7,280 ÷ 1.80 = 4,045 |
| Profit on 18,000 packets | $26,920 | $25,400 | $25,120 |
| Margin of safety | 14,168 packets | 13,368 packets | 13,955 packets |
The rule for costs: A cost goes up: the break-even quantity rises, profit falls and the margin of safety shrinks. A cost comes down: all three move the other way.
A fixed cost works on the top of the division. A variable cost works through contribution, the bottom: each packet covers less, so more packets are needed.
Any change, in three steps
Contribution per unit
Price minus variable cost per unit, with the new figures. If only a fixed cost changed, it stays the same: $1.90.
The new break-even
New fixed costs ÷ new contribution: 8,800 ÷ 1.90 = 4,631.6. Round up to 4,632, because 4,631 packets would still leave a small loss.
What changed
Take one from the other: 4,632 − 3,832 = 800 more packets. Then profit and the margin of safety at the forecast sales: $25,400 and 13,368.
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Lena wants to charge the cafés more: Lena thinks $2.50 is too cheap for a hand-made cracker. She plans to charge the cafés $2.80 a packet from next month. The costs stay as they were: $0.60 a packet and $7,280 a year.
Every packet now leaves $2.80 − $0.60 = $2.20 instead of $1.90 (a higher contribution per unit). Each one pays off more of the fixed costs, so fewer are needed: 7,280 ÷ 2.20 = 3,309.1, rounded up to 3,310 packets. That is 522 fewer than before.
The division, though, only tells you how many packets must be sold. It says nothing about how many the cafés will buy at the new price.
| At $2.50 now | At $2.80, cafés still buy 18,000 | At $2.80, cafés buy 16,000 | |
|---|---|---|---|
| Contribution a packet | $1.90 | $2.20 | $2.20 |
| Break-even quantity | 3,832 | 3,310 | 3,310 |
| Profit | $26,920 | $32,320 | $27,920 |
| Margin of safety | 14,168 packets | 14,690 packets | 12,690 packets |
A lower break-even is not the whole story: If two cafés switch to a cheaper supplier and sales fall to 16,000, break-even has still fallen, but the margin of safety has shrunk and most of the extra profit has gone.
A price cut works the other way. At $2.20 each packet leaves only $1.60, and break-even rises to 4,550. At the same 18,000 packets profit would fall to $21,520, so the cut only pays if it wins a lot of new orders.
So a price change moves break-even in the opposite direction to the price: up with a cut, down with a rise. Whether profit and the margin of safety follow depends on how many customers stay, which is the question 3.3.4 asked about revenue.
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On the cracker chart from 5.5.2 the fixed cost line is flat at $7,280. The total cost line starts at $7,280 and climbs $0.60 for every packet, reaching $18,080 at 18,000 packets. The total revenue line starts at zero and climbs $2.50 a packet, reaching $45,000. Break-even is where revenue crosses total cost; at 18,000 packets the gap between them, $26,920, is the profit.
A change in a cost or the price moves one line, and the crossing point moves with it.
| The change | Which line moves | How it moves | Break-even point | Margin of safety |
|---|---|---|---|---|
| Fixed costs rise | Fixed cost and total cost | Both shift up, parallel: same slope, higher start ($8,800) | Moves right (4,632) | Shrinks |
| Variable cost per unit rises | Total cost only | Pivots up from the same start: steeper ($0.70 a packet) | Moves right (4,045) | Shrinks |
| Price rises | Total revenue only | Pivots up from zero: steeper ($2.80 a packet) | Moves left (3,310) | Grows, if sales hold |
| Price falls | Total revenue only | Pivots down from zero: flatter ($2.20 a packet) | Moves right (4,550) | Shrinks |
Showing a change on the chart
Keep the old lines
Leave the original fixed cost, total cost and total revenue lines in place, each labelled.
Draw the new line
With the packing machine, the new total cost line starts at $8,800 and reaches 8,800 + 0.60 × 18,000 = $19,600 at 18,000 packets. Label it clearly as the new total cost.
Mark the new break-even
Where total revenue crosses the new total cost line, drop down to the output axis: about 4,632 packets, to the right of the old 3,832.
Show both margins of safety
Along the output axis, from each break-even point to the 18,000 sold: 14,168 before, 13,368 after. The gap has narrowed by 800 packets.
Three lines, three levers: Fixed costs decide where the cost line starts. Variable cost decides how steeply it climbs. The price decides how steeply revenue climbs.
Anything that moves the crossing point to the left lowers break-even and widens the margin of safety; anything that moves it right does the opposite.
A neat answer: 3,832 packets: The chart says the cracker line breaks even at 3,832 packets. Priti is less sure: "Only if every packet sells, every café pays $2.50, and flour stays where it is."
She has put her finger on it. Break-even is a model: it reaches a neat answer by simplifying the business, and each simplification (assumption) is a limit on what it can tell Lena.
The limitations of break-even
It assumes everything made is sold
No unsold stock. But crackers go stale in about ten days, and packets baked and never ordered bring in no money while their costs are already paid.
It assumes one price at every output
Revenue is drawn as a straight line. But Lena gives her biggest café a 10% discount, so real revenue climbs less steeply at high output.
It assumes costs are straight lines
Variable cost is $0.60 on every packet. But Valley Grain charges less per sack for large orders, and night overtime costs more, so the cost per packet moves with output.
It assumes fixed costs stay fixed
True only up to a point. If orders outgrow the oven, a second oven adds a step to the fixed costs, and the whole calculation changes.
It looks at one product at a time
Mill Lane bakes loaves as well as crackers. How much of the rent counts as the crackers' fixed cost is a judgement, and a different split gives a different break-even.
It rests on forecasts and stands still
18,000 is a forecast, and the chart is a snapshot. Every change in a cost or the price needs a new chart, and it cannot say how many customers will actually buy.
Why Lena still uses it
- Quick and cheap. Three figures and a division give a clear target for Priti's team.
- It tests a change before it is made. The packing machine, the flour rise and the new price were all checked on paper first.
- Lenders expect it. A bank reading a plan for the Station Road shop wants to know how many sales it needs to cover its costs.
Where it can mislead her
- Forecasts can be wrong. A break-even built on hopeful sales or costs gives false comfort.
- It ignores customers. It says how many must be sold, not how many will be, least of all at a new price.
- Real lines bend. Discounts, bulk buying and a second oven break the straight lines it relies on.
Name the limitation, then show it: "It assumes all output is sold" names a limitation. "The crackers go stale in ten days, so packets baked for orders that never come bring in no money and the real break-even is higher" shows what it means for this business.
Always the second: the assumption, then what breaks when it is not true here.
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How this comes up: With a table of costs and a price, a short calculation: the change in the break-even quantity if fixed costs are 10% higher, or the new break-even quantity after a 20% price rise, always with the working shown.
Or in words: comment on the effect on the break-even quantity if the selling price rises to a given figure. Say which way it moves, then why.
The limitations come in where a decision rests on a break-even figure: as the balance in a longer answer about pricing, costs or a new product.
The calculation pattern
- The new figure that changed. Fixed costs 10% higher: 7,280 × 1.10 = $8,008.
- Contribution per unit. Unchanged here: $2.50 − $0.60 = $1.90. Recalculate it whenever the price or the variable cost changes.
- The new break-even. 8,008 ÷ 1.90 = 4,214.7, rounded up to 4,215 packets.
- The change. 4,215 − 3,832 = a rise of 383 packets, with the unit.
Two traps: New is not change. Stopping at 4,215 answers a different question. When the question asks for the change, finish with the difference, and say whether it is a rise or a fall.
Read-off is not working. A new line drawn on a chart and read off shows no calculation. Write the division, and give the answer in units (packets), rounded up.
Lena's crackers sell to cafés for $2.50 a packet. The variable cost is $0.60 a packet and the fixed costs are $7,280 a year. Calculate the change in the break-even quantity if the fixed costs rise by 10% (show all your working).
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