I stop writing notes the moment a candidate starts multiplying before saying what they are computing. The sheet in my hand does not reward formula recall. It rewards a spoken setup, units that match, and rounding you announce before you use it.
That is the thesis for this page. Interviewers score the identity and the spoken rounding, not whether you memorized twenty equations the night before. After reading, you will have a usable reference by family, three side-by-side contrasts that show how the same number can sound pass or fail, five timed mixed reps, and a one-page sheet you can drill cold.
The full interview flow lives on the case interview cheat sheet. Arithmetic technique lives on how case interview math is scored. This article stays on the formulas themselves.
What the room actually scores
McKinsey's published practice case GlobaPharm tells candidates outright: "Remember that calculators are not allowed — you may write out your calculations on paper." Bain's case interview prep page tells you to "think aloud" and "share your reasoning and calculations, even if you're unsure." BCG's case interview preparation frames the same skill as performing "quick calculations" while you structure, ask, and decide. None of those pages ask you to recite a formula bank from memory.
In a typical first round of eight candidates, maybe two or three lose points on a clean number that arrived with no setup. The arithmetic was fine. The interviewer could not follow the trail, so the trail scored zero.
Here is the same answer, twice, for three formulas. Same digits. Different scores.
| Formula | Dives into arithmetic | States the identity first |
|---|---|---|
| Contribution margin | "So 48 minus 30 is 18, times 40,000 is… 720,000." | "Contribution is price minus variable cost: $48 − $30 = $18 per unit. At 40k units that is about $720k contribution before fixed costs." |
| Break-even units | "Two million divided by 25 is 80,000." | "Break-even units equal fixed cost over contribution per unit: $2M ÷ $25 = 80,000 units. Below that volume the launch loses money." |
| Rule of 72 | "72 divided by 9 is 8." | "Rule of 72: years to double ≈ 72 ÷ growth rate. Doubled in nine years, so CAGR ≈ 8%." |
The second column is slower by two seconds and cleaner by a full point on most sheets I have used. The first column sounds like homework. The second sounds like a workstream.
Ned's rule. Say the identity, the units, and the round you are taking before any multiplication. If you cannot say that sentence in one breath, you are not ready to calculate yet.
Profit and margins
| Formula | Meaning | Worked number | Under-10s shortcut |
|---|---|---|---|
| Profit = Revenue − Cost | Absolute dollars left after costs | Revenue $120M, cost $96M → profit $24M | Subtract lead digits, keep the M label |
| Profit margin = Profit ÷ Revenue | Profit as a share of sales | $24M ÷ $120M = 20% | 24/120 = 1/5 = 20% |
| Gross margin = (Rev − COGS) ÷ Rev | Markup after product cost only | $80M rev, $48M COGS → 40% | 32/80 = 0.4 |
| Revenue = Volume × Price | Top-line identity | 2M units × $15 = $30M | Multiply lead digits, add zeros after |
| Cost = Fixed + Variable | Cost split every profit tree needs | Fixed $8M + variable $12M = $20M | Add the two buckets; do not mix units |
Damodaran's January 2026 industry margins are the peer set I use before calling any margin "good."
Unit economics and contribution
| Formula | Meaning | Worked number | Under-10s shortcut |
|---|---|---|---|
| Contribution / unit = Price − VC / unit | Cash each unit adds before fixed costs | $50 − $30 = $20 | Subtract; keep $/unit |
| CM% = Contribution ÷ Price | Share of price that covers fixed + profit | $20 ÷ $50 = 40% | 20/50 = 2/5 |
| Total contribution = CM × Volume | Pool available for fixed costs | $20 × 50,000 = $1M | 20 × 50k = 1,000k |
| Variable cost total = VC/unit × Volume | Scale cost with volume | $30 × 50,000 = $1.5M | Same lead-digit × zeros |
If you only memorize one family beyond profit, make it this one.
Growth, CAGR, and the rule of 72
| Formula | Meaning | Worked number | Under-10s shortcut |
|---|---|---|---|
| Growth rate = (End − Start) ÷ Start | One-period change | $80M → $100M = +25% | Diff over start |
| CAGR ≈ (End/Start)^(1/n) − 1 | Compound annual rate over n years | Exact when asked; else approximate | Prefer Rule of 72 in the room |
| Rule of 72: years ≈ 72 ÷ r% | Years to double at rate r | 8% → ~9 years | 72 ÷ rate |
| Rule of 72: r% ≈ 72 ÷ years | Implied CAGR if you know the double | Doubled in 9 years → ~8% | Flip the same fraction |
Practice offline with the CAGR calculator. In the room, say Rule of 72 and move on.
Break-even, payback, and simple ROI
| Formula | Meaning | Worked number | Under-10s shortcut |
|---|---|---|---|
| Break-even units = Fixed ÷ CM/unit | Volume that covers fixed costs | $2M ÷ $25 = 80,000 | Divide; label units |
| Break-even revenue = Fixed ÷ CM% | Revenue that covers fixed costs | $2M ÷ 0.40 = $5M | Fixed / margin share |
| Payback (years) = Investment ÷ Annual cash | Years to recover cash outlay | $4M ÷ $1M/yr = 4 years | Straight division |
| Simple ROI = Gain ÷ Investment | Undiscounted return on spend | $1.2M gain on $4M = 30% | Gain over cost |
Check setups on the breakeven calculator while you drill. In the room you will not have it — that is McKinsey's GlobaPharm point.
NPV and perpetuity shortcuts
| Formula | Meaning | Worked number | Under-10s shortcut |
|---|---|---|---|
| PV of perpetuity = CF ÷ r | Value of flat cash forever | $4M ÷ 0.10 = $40M | Cash over rate |
| Growing perpetuity = CF ÷ (r − g) | Flat cash growing at g forever | $4M ÷ (0.10 − 0.02) = $50M | Cash over (r − g) |
| NPV ≈ −Investment + PV of benefits | Value created after discounting | −$30M + $40M = +$10M | Subtract invest from PV |
| Discount factor (1 period) ≈ 1 − r | Rough one-year haircut | $10M at 10% ≈ $9M today | Haircut by the rate |
Cases want the perpetuity shortcut with the rate you assumed, not a five-year DCF.
Market sizing ratios and per-capita anchors
| Formula | Meaning | Worked number | Under-10s shortcut |
|---|---|---|---|
| Top-down market = Pop × Pen × Freq × Price | Narrow a population into spend | 10M × 20% × 4 × $25 = $200M | Multiply left to right; round each step |
| Bottom-up = Units × Spend/unit × Count | Scale one unit of demand | 500 stores × $400k = $200M | Store × revenue/store |
| Per-capita spend = Market ÷ Population | Sanity check on a size | $200M ÷ 10M = $20/person | Market over people |
| Share = Segment ÷ Total | Relative size | $40M ÷ $200M = 20% | Segment over total |
Bain's Coffee Shop Co. case scores the assumption chain, not a memorized GDP figure.
Elasticity and price-volume trade-offs
| Formula | Meaning | Worked number | Under-10s shortcut |
|---|---|---|---|
| %Δ Profit ≈ %Δ Price + %Δ Volume (rough) | Directional mix check | +10% price, −5% volume → still up | Add signed % moves on contribution |
| Max volume loss for price rise | Keep contribution flat | Price +10%, CM was 40% → can lose ~20% volume before profit falls (order-of-magnitude) | New CM × new vol ≥ old CM × old vol |
| Price elasticity (arc) = %ΔQ ÷ %ΔP | Sensitivity of volume to price | −8% vol / +10% price = −0.8 | %Q over %P |
Skip the formal elasticity label. Keep the inequality that holds total contribution flat.
Five mixed timed reps
Set a phone timer. Ten seconds to state the identity and round; twenty more to finish. Answers below.
- Revenue $90M, cost $72M. Profit and profit margin?
- Price $60, VC $36, fixed $1.8M. Break-even units?
- Market doubles from $40B to $80B in 8 years. Implied CAGR via Rule of 72?
- Project pays $5M per year forever; discount rate 10%; invest $35M. NPV?
- City of 2M people, 25% buy once a year at $40. Market size?
Answers: (1) profit $18M, margin 20%; (2) CM $24, break-even 75,000 units; (3) 72 ÷ 8 ≈ 9%; (4) PV $50M, NPV +$15M; (5) 2M × 0.25 × $40 = $20M.
One-page memorisation table
| Family | Core identity to say first |
|---|---|
| Profit | Profit = Rev − Cost; Rev = V × P; Cost = F + V |
| Contribution | CM = P − VC; BE units = F ÷ CM |
| Growth | Rule of 72: double time ≈ 72 ÷ r% |
| Investment | Payback = Invest ÷ annual cash; ROI = gain ÷ invest |
| NPV | Perpetuity = CF ÷ r; NPV = −I + PV |
| Market size | Pop × pen × freq × price (or store × $/store) |
| Price-volume | New CM × new vol ≥ old CM × old vol |
Recite the left column, say the identity, invent two numbers, finish.
Drill the setups today
Ten minutes on /drills/quick-math (free forever), identity out loud before every answer, then /drills/math for mixed scored sets. For a full case under voice pressure: /start or /interview. Everything is open for seven days, no card; then $120 for a recruiting season or $49 a month.
CoachNed is not affiliated with McKinsey, BCG, Bain, Deloitte, or any firm named above.
Frequently asked questions
Do I need a calculator in a case interview?
No for live MBB-style cases. McKinsey's GlobaPharm practice case states calculators are not allowed and that you may write on paper. Some online screens differ by office; prepare as if you have only pen and paper.
Which formulas should I memorize first?
Contribution margin and break-even units. They appear in pricing, launch, and investment prompts more often than NPV or formal CAGR. Add the Rule of 72 and the four-factor market-size chain next.
How precise does case math need to be?
Precise enough that your recommendation does not flip if you clean the rounding. Announce the round ("call it $50M"), keep units, and check order of magnitude. Firms publish that they want quick, reasoned numbers, not spreadsheet digits.
Is NPV required for every firm?
No. Flat and growing perpetuity shortcuts show up most on investment and PE-flavored cases. Know them; do not force a DCF when a payback answer closes the question.
How is this different from a consulting math formulas list?
This page is the interview set: identities you speak under time. Broader on-the-job formulas and unit discipline sit in related guides linked below.
Sources
- McKinsey GlobaPharm practice case — firm statement that calculators are not allowed and paper calculations are expected. Checked 2026-09-24.
- Bain case interview preparation — firm tip to think aloud and share reasoning and calculations. Checked 2026-09-24.
- BCG case interview preparation — firm description of structuring, questions, data, and quick calculations. Checked 2026-09-24.
- Bain Coffee Shop Co. practice case — official top-down and bottom-up sizing with assumptions. Checked 2026-09-24.
- Damodaran current data (NYU Stern) — industry margin and return datasets for peer context (updated January 2026). Checked 2026-09-24.