FC ÷ CM per unit; FC ÷ CM ratio; round units up; prove it

Set the equation to zero

Break-even is the volume at which operating income is zero — contribution margin exactly equals fixed costs. Set the profit equation of L10.1 to zero and solve:

$$21\times \text{units}-315,000=0\Rightarrow \text{Break-even units}=\frac{\text{Fixed costs}}{\text{CM per unit}}=\frac{315,000}{21}=15,000\text{ skier-days}$$

Each skier-day contributes 21 toward the 315,000; it takes 15,000 of them. The commonest wrong answer on the paper divides by the price: 315,000 ÷ 30 = 10,500 — the volume at which revenue reaches 315,000, with the variable costs still unpaid. The denominator is what each unit leaves after its own variable cost.

In dollars

The same point in sales dollars can be reached two ways. Multiply the units by the price — 15,000 × 30 = 450,000 — or, for a business with no single unit, divide fixed costs by the CM ratio (L9.5):

$$\text{Break-even sales}=\frac{\text{Fixed costs}}{\text{CM ratio}}=\frac{315,000}{0.70}=450,000$$

Seventy cents of every sales dollar is contribution; it takes 450,000 of dollars to yield 315,000 of it. Units use the CM per unit; dollars use the CM ratio — the two formulas are the same idea with the unit chosen differently, and 315,000 ÷ 21 gives units, never dollars.

Prove it

A formula answer earns the method mark only when the statement at that volume shows a zero:

Sales                    15,000 × 30     450,000
Less: variable costs     15,000 × 9      135,000
Contribution margin      15,000 × 21     315,000
Less: fixed costs                        315,000
Operating income                               0

Contribution margin at break-even equals fixed costs — 315,000, not zero. The statement is the proof, and it is also the check: if the foot is not zero, the division was wrong.

Round up — Cedar Ridge Golf Club Ltd.

Green fee 55, variable cost 3, fixed costs 335,000 (L9.3): CM per round 52, and 335,000 ÷ 52 = 6,442.3 rounds. A club cannot sell three-tenths of a round. At 6,442 rounds contribution is 6,442 × 52 = 334,984, which is 16 short, so the club is still at a loss; the 6,443rd round is the one that crosses. Break-even units round up, always — =ROUNDUP(335000/52, 0). In dollars, 6,443 × 55 = 354,365. (Dividing 335,000 by the ratio 0.9455 gives 354,327 — the unrounded point; a case that asks for dollars from a rounded unit count wants the 354,365.)

A company with only a ratio

Bramble Lane-style: fixed costs 84,000 and a CM ratio of 0.35, nothing per unit. Break-even sales = 84,000 ÷ 0.35 = 240,000. No unit figure exists and none is asked for; the ratio route is the whole answer.