Break-Even Quantity

Fixed costs arrive whether you sell anything or not: rent, insurance, the shop truck payment, the phone. Every unit you do sell contributes its price minus its own variable cost toward that pile, and break-even is simply the point where the contributions finish covering it. With $12,000 of monthly fixed cost and units that sell for $40 while costing $25 in material and labour, each one contributes $15, and $12000/15 = 800$ units clears the month.

The quantity $p - v$ is called the contribution margin, and it deserves the attention rather than the price. Raising the price by $5 lifts the contribution from $15 to $20 and drops break-even from 800 units to 600, a 25% cut in the work needed to survive. Shaving $5 off the variable cost does exactly the same thing. Neither is intuitive until you see that both act on a denominator.

If the price does not exceed the variable cost, no quantity ever breaks even and the solver refuses the question. This is not a technicality. A shop losing money on every job cannot make it up in volume, and the formula is the most honest way to demonstrate that to someone who believes otherwise.