A formula without units is a slogan. In a case you write what the number is measured in on the same line as the equation. This page is the sheet: contribution, break-even, CAGR, margin, growth, and capacity. It is not a mental-math course and not a profitability framework. Use it when you know which identity you need and you refuse to drop a thousand or a year.
If you cannot name the unit of the answer before you compute, you picked the wrong identity.
The identities (write them this way)
| Identity | Formula | Answer unit |
|---|---|---|
| Contribution | price − variable cost | $ / unit |
| Break-even volume | fixed cost ÷ contribution per unit | units / period |
| Margin | profit ÷ revenue (or contribution ÷ price) | dimensionless (say %) |
| Growth | (new − old) / old | 1 / period (say % per year) |
| CAGR | (end / start)^(1/n) − 1 | 1 / year |
| Capacity (output) | rate × time × utilization | units / year |
Contribution ($/unit) = P − V. P and V must be per the same unit (per scan, per tonne, per seat-month). Mixing $/year price with $/unit cost is how MRI cases die.
Break-even (units/year) = F / (P − V). F is $ / year. If F is a one-time capex, you are doing payback, not break-even — see break-even versus ROI / payback.
Margin (unitless). Gross margin = (P − COGS) / P. EBITDA margin = EBITDA / revenue. Never add a “margin of $4m” unless you mean dollars of profit. Say “EBITDA margin 11%” or “EBITDA $4m.”
Growth (1/year). (120 − 100) / 100 = 0.20 = 20% per year if the gap is one year. If the gap is four years, that 20% is not CAGR.
CAGR (1/year) = (end/start)^(1/n) − 1. Shortcut: 80 → 120 in 4 years is 1.5× in 4 years. 1.5^(1/4) is a bit under 11% (1.1^4 ≈ 1.46). Say “about 11%.” Do not use (50%/4) = 12.5% and call it CAGR. That is average arithmetic growth and it overstates.
Capacity. Output = (units / time) × time × utilization. Utilization is unitless (0–1). Time’s unit must cancel the denominator of the rate.
Worked example: cement kiln capacity
Prompt. A 2,400 tonne / day clinker kiln, scheduled 330 days / year, runs at 91% of scheduled days. Energy is $38 / tonne. Selling price net of non-energy variable cost is $79 / tonne. What is annual output, and what is contribution after energy?
Output (t / year) = 2,400 t/day × 330 day/year × 0.91.
2,400 × 330 = 792,000 t / year at 100% of schedule.
792,000 × 0.91 = 720,720 t / year. Say 721 thousand tonnes per year.
Contribution after energy ($ / year) = output × (79 − 38) $/t = 721,000 t/year × 37 $/t.
721 × 37: 700 × 37 = 25,900; 21 × 37 = 777; $26.7m / year.
If someone quotes “capacity 2,400 tonnes” as the answer, they reported a rate, not annual output. The interviewer asked for a year. Units caught it.
If utilization is omitted, you overstate 792 / 721 − 1 ≈ 10% too much output and ~$2.6m too much contribution. That is the whole reason the identity includes utilization.
Formulas people mix up in the room
Payback (years) = investment ($) / annual cash ($ / year). It is not a return. A 2.7-year payback can still be a bad NPV at 15%.
ROI (unitless) = annual profit or cash / investment. State whether it is year-1 cash / capex or average. 0.375 = 37.5% per year only if the numerator is per year.
Rule of 72 is a doubling-time shortcut (years ≈ 72 / percent growth), not a margin identity. Do not put it on this sheet as if it were CAGR.
Weighted average. Mix price = Σ (share_i × price_i). Shares must sum to 1. If commercial is 34% and you treat 34 as dollars, you are done.
When this sheet is the wrong tool
You do not know what question you are answering. Formulas do not pick the tree. If profit fell, start with profitability, then grab contribution or margin when a branch needs a number.
Discounting a 15-year uneven stream. Use NPV. This sheet will not save a WACC debate.
Market sizing. That is a chain of assumptions, not one identity. See step-by-step sizing.
The mistake that is unique to formula sheets
Memorizing letters without the unit row. Candidates write BE = F / CM and then divide $490k by 0.34 because they stored “contribution margin” as a percent. If CM is 34% of $410, contribution is $139 / unit, not 0.34. Break-even would be 490,000 / 139 ≈ 3,500, not 1,440. Always convert percent contribution into $ / unit before dividing into $ fixed.
What to write on the page
output [t/y] = 2400 t/d × 330 d/y × 0.91
contrib [$/y] = output [t/y] × (P − V − energy) [$/t]
If the brackets cancel, you are safe. If they do not, you are about to present a rate as a stock.
Drill identities with the units spoken
Quick math is where contribution, break-even, and capacity get automatic. Keep the unit in the sentence.