Consulting Math Formulas: The Essential Reference for Case Interviews

Consulting case interviews test 15 core math formulas across 8 categories — margins, growth rates, break-even, ROI, NPV, and unit economics. According to Hacking the Case Interview, these formulas appear in roughly 80% of quantitative case questions. The real challenge is not memorizing them but knowing which formula applies within the first 30 seconds of a math prompt and narrating the setup clearly as you work.

Margin Formulas: Gross, Operating, and Contribution

Margin questions appear in almost every profitability case. Gross margin = (Revenue − COGS) / Revenue measures what remains after direct production costs. Operating margin = EBIT / Revenue captures the full P&L including SG&A and depreciation. Contribution margin = (Revenue − Variable Costs) / Revenue isolates what each incremental unit contributes toward covering fixed costs.

According to Corporate Finance Institute, confusing gross margin percentage with gross profit dollars is the most common candidate error. If a product sells at $50 with $30 COGS, gross margin is 40% — not the $20 absolute dollar figure interviewers are asking about. Industry benchmarks matter: SaaS companies run 70–80% gross margins, consumer goods 40–60%, retail 25–40%, and manufacturing 20–35%.

Growth Rate Formulas: YoY and CAGR

Year-over-year growth is the simplest formula: (Current − Prior) / Prior. When an interviewer gives you "grew from X to Y over N years," they want compound annual growth rate (CAGR), not simple average growth. CAGR = (End Value / Start Value)^(1/n) − 1.

The Rule of 72 shortcut eliminates complex exponent calculations: if something doubled, CAGR ≈ 72 / n years. A market that doubled in 6 years grew at approximately 12% CAGR. According to Wall Street Prep, the most common CAGR application in interviews is comparing company growth to market growth — a company growing at 8% CAGR in a 12% CAGR market is losing share, a diagnostic signal for the rest of the case.

Interviewers design CAGR numbers to simplify. Look for doubling (CAGR ≈ 72/n), tripling, or values that are perfect squares before reaching for more complex approximation methods.

Break-Even Analysis: Units and Revenue

Break-even is the most commonly tested formula in profitability and new-product cases. Break-even units = Fixed Costs / Contribution Margin per Unit. Break-even revenue = Fixed Costs / Contribution Margin %.

The formula answers: "At what volume does this product stop losing money?" It appears in new product launches, new store evaluations, and price-change decisions. Corporate Finance Institute highlights margin of safety as a key extension: how far above break-even is actual volume? A company selling 100K units against a 40K break-even has a 60% margin of safety.

A worked price-change scenario: a subscription product at $120/year with $30 variable costs and $3.6M fixed costs has a break-even of 40,000 subscribers ($3.6M / $90 CM). If price drops to $100, CM falls to $70 and break-even rises to 51,429 — a 28% increase in volume needed.

ROI and Payback Period

ROI = (Gain − Cost) / Cost tells you return as a percentage. Payback Period = Investment Cost / Annual Net Cash Inflow tells you how quickly the investment is recovered. Both appear in capital allocation and "should we spend $X on Y" questions.

According to Corporate Finance Institute, the most common candidate mistake is forgetting to net out annual operating costs of the new system. A $2M automation system saving $800K/year yields 40% ROI with 2.5-year payback. But if annual maintenance costs $100K, net inflow drops to $700K and payback extends to 2.9 years.

In a consulting context, benchmark ROI against the company's cost of capital (often 10–15%). A 40% ROI with a 2.5-year payback is typically attractive relative to alternative uses of capital.

NPV: Net Present Value

NPV is the most conceptually demanding formula. The simplified perpetuity version — NPV = Annual Cash Flow / Discount Rate — is what most case interviews use. With growth: NPV = Annual Cash Flow / (Discount Rate − Growth Rate).

According to PrepLounge, interviewers most often test NPV to see whether candidates can identify if a project's present value exceeds its upfront cost. If a project generates $4M/year growing at 2% with a 12% discount rate, NPV = $4M / 10% = $40M. Subtract the $20M investment and net NPV is $20M positive — a clear "go" recommendation.

Unit Economics: CAC, LTV, and Payback

Unit economics formulas appear in tech, subscription, and marketplace cases. CAC = Total Sales & Marketing Spend / New Customers Acquired. LTV = (Monthly Revenue x Gross Margin %) / Monthly Churn Rate. LTV:CAC ratio benchmarks: below 1:1 destroys value, 3:1 is healthy SaaS, above 5:1 may signal under-investment in growth.

CAC Payback Period = CAC / (Monthly Revenue x Gross Margin %). Industry benchmark for B2B SaaS: under 18 months is healthy; over 24 months signals capital efficiency problems. A SaaS company with $150 monthly ARPU, 70% gross margin, and 2% monthly churn has LTV of $5,250 and at $2,000 CAC yields a 2.6:1 ratio — below the 3:1 benchmark, flagging a need to improve retention or cut acquisition costs.

According to Corporate Finance Institute, LTV:CAC and CAC payback together tell you whether a business model is sustainable at scale.

Worked Example: Chaining Multiple Formulas

A B2B software company has $80M revenue, 75% gross margin, 2,000 new customers last year at $8M sales and marketing spend, $3,000 monthly ARPU, and 1.5% monthly churn. They are considering a $15M investment expected to generate $5M annual profit growing at 5%, with a 15% hurdle rate.

Step 1 — Unit economics: CAC = $8M / 2,000 = $4,000. Monthly gross profit = $3,000 x 75% = $2,250. LTV = $2,250 / 1.5% = $150,000. LTV:CAC = 37.5:1 (exceptional). CAC payback = $4,000 / $2,250 = 1.8 months.

Step 2 — Investment NPV: NPV = $5M / (15% − 5%) = $50M. Net NPV = $50M − $15M = $35M positive.

Step 3 — Synthesis: "Strong unit economics with 37.5:1 LTV:CAC provide headroom for growth investment. The $15M enterprise expansion generates $35M net NPV at a 15% hurdle rate. Key risk: enterprise churn may exceed the current 1.5% — if it doubles to 3%, LTV halves and the economics change materially."

Related Guides

• Case Interview Math Practice — 30 timed drills using these exact formulas
• Free case interview math practice page — drill-ready format with instant scoring
• Mental Math Shortcuts for Case Interviews — rounding, anchoring, and approximation techniques
• Profitability Framework — how to structure a profitability diagnosis before reaching for formulas
• Market Sizing Step-by-Step — where growth rate and market share formulas dominate
• Pricing Strategy Cases — where break-even formulas appear most often
• Case Interview Prep Guide — complete preparation roadmap

Test yourself

Test yourself

A company has revenue of $200M and COGS of $130M. What is the gross margin?

• A35%
• B45%
• C65%
• D30%

Fixed costs are $4.5M. Price per unit is $50. Variable cost per unit is $20. How many units are needed to break even?

• A90,000 units
• B150,000 units
• C225,000 units
• D75,000 units

A SaaS company's monthly ARPU is $200, gross margin is 80%, and monthly churn is 2%. A customer costs $3,200 to acquire. What is the LTV:CAC ratio?

• A4:1
• B6:1
• C2.5:1
• D8:1

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Sources and Further Reading (checked March 20, 2026)

• Hacking the Case Interview — 26 essential case interview formulas: https://www.hackingthecaseinterview.com/pages/case-interview-formulas
• IGotAnOffer — Case interview maths guide: https://igotanoffer.com/blogs/mckinsey-case-interview-blog/case-interview-maths
• Wall Street Prep — CAGR: Compound Annual Growth Rate: https://www.wallstreetprep.com/knowledge/cagr-compound-annual-growth-rate/
• Corporate Finance Institute — Break-Even Analysis: https://corporatefinanceinstitute.com/resources/accounting/break-even-analysis/
• Corporate Finance Institute — ROI Formula: https://corporatefinanceinstitute.com/resources/accounting/return-on-investment-roi-formula/
• PrepLounge — Net Present Value (NPV) in case interviews: https://www.preplounge.com/en/case-interview-basics/net-present-value-npv
• Management Consulted — Case Interview Formulas: https://managementconsulted.com/case-interview-formulas/