Operating income = (price − VC) × units − FC, drawn

The whole of Module 9 in one line

Northlake Nordic Centre Inc.'s CM statement (L9.6) is sales less variable costs less fixed costs. Write each line as a function of skier-days and the statement collapses into the profit equation:

$$\text{Operating income}=(\text{Price}-\text{VC per unit})\times \text{Units}-\text{Fixed costs}$$

$$\text{Operating income}=(30-9)\times \text{skier-days}-315,000=21\times \text{skier-days}-315,000$$

Evaluate it anywhere inside the relevant range:

At 10,000 skier-days the contribution does not cover the fixed costs and the season loses 105,000. At 15,000 it covers them exactly — the break-even point, where operating income is zero. At 20,000 the L9.6 statement reappears: 105,000. Each row is a CM statement; the equation is the statement with the arithmetic done in advance.

The chart

The cost-volume-profit chart draws three lines over skier-days from 0 to 30,000:

• Fixed costs — flat at 315,000 at every volume.
• Total cost — starts at 315,000 (not at the origin) and rises 9 per skier-day: 315,000 + 9 × units, the cost equation of L9.3.
• Revenue — starts at the origin and rises 30 per skier-day.

Revenue starts below total cost and climbs faster. Where the two lines cross is break-even: 15,000 skier-days, 450,000 of revenue (15,000 × 30 = 450,000 = 315,000 + 9 × 15,000). Left of the crossing the total-cost line is on top and the gap between the lines is the loss; right of it revenue is on top and the gap is operating income. At 20,000 the gap is 600,000 − 495,000 = 105,000.

Reading the slopes

The revenue line's slope is the price, 30. The total-cost line's slope is the variable cost per unit, 9. The gap between them therefore widens by 30 − 9 = 21 per skier-day — the contribution margin — which is why every 5,000 skier-days moves operating income by 105,000. A common misreading is that the revenue line's slope is the CM; it is not — the CM is the difference of the two slopes. Another is that break-even is where revenue meets the fixed-cost line; that crossing (at 10,500 skier-days) means only that revenue has reached 315,000, while variable costs are still unpaid.

One fact, three ways

The equation, the CM statement and the chart are the same model. The equation answers a question fast; the statement proves the answer line by line, and is what the marker wants to see; the chart shows the whole range at once and is what an owner understands. Module 10 uses all three: compute with the equation, prove with the statement, explain with the chart.