In a typical first-round case, the interviewer marks math wrong less often for arithmetic and more often for the wrong identity with the right digits. Candidates reach for "margin" when they mean contribution dollars, call arithmetic growth a CAGR, or divide fixed cost by a percent. This page is the sheet: the twenty formulas that actually show up in cases, each with units, a two-sentence worked example, and the trap it hides.
Sibling posts cover how to practice case math, mental math setup, and shortcuts. Keep this tab open as the reference. When you need a machine check on the two identities candidates ask for most, use CoachNed's CAGR calculator and breakeven calculator.
What the sheet is for
Formulas do not choose the issue tree. They convert a named quantity into another named quantity once you already know which branch you are on. If profit fell, you still start with revenue versus cost; only then do you pull contribution, margin, or breakeven.
I stop listening when a candidate writes letters without units. "BE = F / CM" is fine until CM is stored as 34% and F is $490k. That path invents 1,440 units instead of about 3,500. Convert percent contribution into dollars per unit before you divide.
Ned's rule. Write the answer unit on the same line as the formula before you compute. If you cannot say whether the answer is dollars, units per year, or a pure percent, you picked the wrong identity.
The twenty formulas
Each row is one identity. The example is two sentences you can reproduce. The trap is what I have seen blow the mark with the "right" formula on the page.
| # | Identity | Formula (with units) | Worked example | Trap |
|---|---|---|---|---|
| 1 | Profit | Profit ($) = Revenue ($) − Cost ($) | HarborCoat paint division books $48m revenue and $41m cost. Profit is $7m. | Mixing EBITDA with net income mid-sentence without saying which profit you mean. |
| 2 | Revenue | Revenue ($) = Price ($/unit) × Volume (units) | Price $12.50 and volume 2.4m cans. Revenue is $30m. | Using list price when the case already gave net realized price after discounts. |
| 3 | Total cost | Cost ($) = Fixed ($) + Variable ($) | Fixed $9m plus variable $22m. Total cost is $31m. | Parking depreciation in variable cost because "it feels like manufacturing." |
| 4 | Variable cost total | Variable ($) = VC ($/unit) × Volume (units) | VC $4.20/can × 2.4m cans = $10.1m. | Adding a one-time setup fee into per-unit VC and then scaling it with volume forever. |
| 5 | Gross margin | Gross margin (unitless) = (Revenue − COGS) / Revenue | Revenue $30m, COGS $18m. Gross margin is 40%. | Calling absolute gross profit "$12m margin" without the percent word, then treating it as a rate later. |
| 6 | Contribution per unit | Contribution ($/unit) = Price ($/unit) − Variable cost ($/unit) | Price $12.50, VC $4.20. Contribution is $8.30 per can. | Subtracting allocated fixed overhead from price and calling the remainder contribution. |
| 7 | Contribution margin % | CM% (unitless) = Contribution ($/unit) / Price ($/unit) | $8.30 / $12.50 = 66.4%. Say about 66%. | Dividing fixed cost by CM% when you meant contribution dollars (see row 8 vs 9). |
| 8 | Breakeven volume | Units = Fixed ($/period) / Contribution ($/unit) | Fixed $9m/year ÷ $8.30/can ≈ 1.08m cans/year. | Using a one-time capex as "fixed" without saying you switched to payback. Check with the breakeven calculator. |
| 9 | Breakeven revenue | Revenue ($) = Fixed ($) / CM% | $9m / 0.664 ≈ $13.6m revenue to cover fixed. | Applying CM% to units instead of dollars, or using gross margin when variable cost differs from COGS. |
| 10 | Operating margin | Op. margin (unitless) = Operating profit ($) / Revenue ($) | Op. profit $7m on $48m revenue → about 15%. | Comparing a firm's EBITDA margin to a peer's EBIT margin as if they matched. |
| 11 | CAGR | CAGR (1/year) = (End / Start)^(1/n) − 1 | $80m → $120m over 4 years: 1.5^(1/4) − 1 ≈ 11%, not 50%/4 = 12.5%. Confirm on the CAGR calculator; CFI's CAGR note states the same geometric form. | Using one-year growth or an arithmetic average of yearly rates and calling it CAGR; that overstates multi-year compounding. |
| 12 | Rule of 72 | Years to double ≈ 72 / (growth % per year) | At 8% CAGR, 72/8 ≈ 9 years to double. Wolfram MathWorld presents the Rule of 72 as the standard doubling-time approximation to ln(2)/ln(1+r). | Treating Rule of 72 as exact CAGR math, or using it on margins instead of growth rates. |
| 13 | NPV | NPV ($) = Σ CF_t / (1+r)^t − Initial investment ($) | Capex $10m; years 1–3 cash $4m each; r = 10%. PV ≈ 4/1.1 + 4/1.21 + 4/1.331 ≈ $9.95m; NPV ≈ −$0.05m. HBR's NPV refresher frames the same time-value idea. | Forgetting to subtract (or include) year-0 outlay, or discounting uneven cash as if payback decided value. |
| 14 | Payback | Payback (years) = Investment ($) / Annual cash ($/year) | $10m capex / $4m per year = 2.5 years. | Calling a short payback a good project when NPV at the hurdle rate is negative. |
| 15 | Price elasticity | Elasticity (unitless) = (% change in quantity) / (% change in price) | Price +10%, quantity −4% → elasticity −0.4 (inelastic). OpenStax Principles of Economics defines it as percent change in quantity over percent change in price. | Dropping the sign and then raising price on elastic demand because "elasticity is 1.5 so we win." |
| 16 | Market share | Share (unitless) = Firm sales / Total market sales (same units) | Firm $120m in a $800m market → 15% share. | Mixing units sold in the numerator with dollars in the denominator. |
| 17 | Penetration | Penetration (unitless) = Customers / Addressable population | 2.1m subscribers in 30m households → 7% penetration. | Using total population when the case defined addressable as adults with smartphones. |
| 18 | Utilization | Utilization (unitless) = Actual output / Rated capacity (same units) | Kiln rated 800k t/year, ran 720k → 90% utilization. | Reporting the rate (t/day) as annual capacity without multiplying by days and utilization. |
| 19 | Little's Law | Inventory (units) = Throughput (units/time) × Flow time (time) | Line ships 40 jobs/day; average 5 days in process → WIP ≈ 200 jobs. John Little's 1961 result in Operations Research is the classic L = λW statement (see the INFORMS / RePEc abstract). | Raising WIP to "increase throughput" past the bottleneck; flow time rises instead. |
| 20 | CLV (case form) | CLV ($) ≈ Margin per period ($) / (1 + d − r) when cash is end-of-period | Margin $40/year, retention r = 0.80, discount d = 0.10 → CLV ≈ 40 / (1.10 − 0.80) ≈ $133. Wharton-linked work by Fader and Hardie stresses that constant-r formulas are teaching tools, not full empirical CLV. | Capitalizing acquisition spend with a retention rate that already churned the same cohort twice; name whether you include the first payment. |
One-year change is just (new − old) / old; use row 11 only when n is greater than one. In the room I care that you name which CLV convention you used more than that you memorized every textbook variant.
One multi-step walk-through
Prompt. Northpine Timber sells fence posts at $18 each. Variable cost is $7. Fixed cost is $2.2m per year. Last year volume was 280k posts; this year the team wants 350k. Capex of $1.5m would cut variable cost to $6. Discount rate for a quick NPV screen is 12%. Annual cash improvement if the project runs is the incremental contribution after the VC cut, on the new volume.
Step 1 — contribution now. $18 − $7 = $11/post. At 350k posts, contribution = 350k × $11 = $3.85m. Breakeven today = $2.2m / $11 ≈ 200k posts. They are already above breakeven.
Step 2 — contribution after capex. $18 − $6 = $12/post. Contribution at 350k = $4.20m. Incremental contribution versus the $11 case = $0.35m per year.
Step 3 — payback versus NPV. Payback on $1.5m at $0.35m/year ≈ 4.3 years. Rough three-year NPV of the increment: 0.35/1.12 + 0.35/1.254 + 0.35/1.405 ≈ $0.84m, so NPV ≈ 0.84 − 1.5 = −$0.66m. Short story for the interviewer: payback looks merely long; NPV at 12% says do not spend unless volume or the VC save is higher.
That is the point of the sheet. Identities 6, 8, 13, and 14 sit on one page; the judgment is which one answers the question the partner asked.
How to use this without becoming a formula tourist
Say the unit aloud. "Contribution eight-thirty per post" beats "CM is good."
Convert percent to dollars before breakeven. If someone hands you "34% contribution margin" and a $22 price, contribution is about $7.50/unit, not 0.34.
CAGR when n > 1. One-year growth is just growth. Multi-year needs the root, or the calculator.
Payback is not a return. It is a clock. NPV is the value screen.
Operations cases. Utilization and Little's Law belong together: capacity sets the ceiling; WIP and flow time explain the queue. Pair this sheet with operations and cost cases when the prompt is a plant, not a P&L.
Practice this today
Copy the twenty formulas onto one notebook page with a blank "unit" column. Then run ten timed problems on quick math (free forever) and force yourself to speak the unit before the number. For a full case that will demand contribution and margin under time, take the five-minute first rep at /start or the live CopperCup interview at /interview. Everything on CoachNed is open for seven days with no card; then $120 for a recruiting season or $49 a month.
Frequently asked questions
How many math formulas do I need for case interviews?
About twenty cover nearly every first- and second-round quantitative ask: profit and margin identities, breakeven, growth and CAGR, Rule of 72, NPV and payback, elasticity, share and penetration, utilization, Little's Law, and a simple CLV. Memorize identities with units, not letter soup.
What is the difference between contribution margin and gross margin?
Contribution margin strips variable cost from price and is the right input to breakeven. Gross margin strips COGS from revenue; COGS may include items that do not move one-for-one with the next unit. If the case gives a clean P and V, use contribution.
Is the Rule of 72 accurate enough in a case?
Yes for a sanity check ("at ~9% we double in eight years"). No as a substitute for CAGR when the interviewer gave start, end, and n. Use Rule of 72 to talk; use CAGR to compute.
Should I calculate NPV in every investment case?
Only when cash spans multiple years and a discount rate is given or clearly implied. If the partner asks "how long until we get the money back," lead with payback, then offer NPV if the numbers support it.
Where can I check CAGR or breakeven quickly while practicing?
Use CoachNed's /tools/cagr-calculator and /tools/breakeven-calculator, then re-do the same problem on paper so the identity sticks without the tool. CoachNed is not affiliated with McKinsey, BCG, Bain, Deloitte, or any firm named in interview prep.
Sources
- CAGR (Compound Annual Growth Rate) (Corporate Finance Institute) — geometric CAGR definition used in row 11. Checked 2026-09-24.
- Rule of 72 (Wolfram MathWorld) — doubling-time approximation for row 12. Checked 2026-09-24.
- A Refresher on Net Present Value (Harvard Business Review) — time value of money framing for row 14. Checked 2026-09-24.
- Price Elasticity of Demand (OpenStax Principles of Economics 3e) — %ΔQ / %ΔP definition for row 16. Checked 2026-09-24.
- A Proof for the Queuing Formula: L = λW (Little, Operations Research, via RePEc) — primary citation for Little's Law in row 20. Checked 2026-09-24.