Consulting Math Formulas: The Essential Reference Sheet for Case Interviews

Consulting case interviews test 15 core math formulas across 8 categories: gross margin, operating margin, contribution margin, year-over-year growth, CAGR, break-even (units and revenue), ROI, payback period, NPV, and unit economics (CAC, LTV, LTV:CAC, and CAC payback). According to Hacking the Case Interview, these formulas appear in approximately 80% of quantitative case questions. The challenge is not memorizing the formulas — it is knowing which one applies in the first 30 seconds of a math question and narrating the calculation setup clearly as you work.

Why Formula Fluency Is a Separate Skill From Math Ability

You can be fast at arithmetic and still freeze when an interviewer says "what's the payback period on this investment?" The gap isn't computation — it's not having the formula immediately accessible without thinking about it.

According to Hacking the Case Interview's formula guide, the 26 formulas that appear in case interviews reduce to about 8 core concepts. The rest are variations or combinations. This reference sheet focuses on that core set — the formulas that appear in 80% of quantitative case problems.

The Formula Reference Sheet

Here is the complete reference organized by category. Definitions, formula, when to use, and the mistake that trips up most candidates.

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Margin Formulas

Margin questions appear in almost every profitability case. You need three margin types cold.

Gross Margin

Formula:

Gross Margin % = (Revenue − COGS) / Revenue × 100

Or equivalently: Gross Margin % = 1 − (COGS / Revenue)

What it measures: How much revenue remains after paying the direct costs of producing a good or delivering a service. Cost of goods sold (COGS) includes raw materials, direct labor, and manufacturing overhead — not marketing, G&A, or R&D.

When to use it:

• Diagnosing why a product line is underperforming
• Comparing margin across business units or geographies
• Evaluating a pricing decision's impact on profitability

Worked example: A consumer goods company sells a product at $50/unit. COGS is $30/unit.

Gross Margin = ($50 − $30) / $50 = 40%

If the interviewer then says COGS rises 20% due to input cost inflation, COGS becomes $36. New gross margin = ($50 − $36) / $50 = 28%. Margin dropped 12 percentage points — that's the impact you'd communicate.

Common trap: Candidates confuse gross margin percentage with gross profit dollars. If the interviewer asks "what happened to gross margin?" they usually mean the percentage, not the absolute dollar change.

Operating Margin (EBIT Margin)

Formula:

Operating Margin % = EBIT / Revenue × 100

Where EBIT = Earnings Before Interest and Taxes = Revenue − COGS − Operating Expenses

When to use it: When you need to evaluate operational efficiency across the full P&L, including SG&A, R&D, and depreciation — but before financing structure (debt/interest) distorts the picture. Useful for comparing companies with different capital structures.

Worked example: Revenue = $500M, COGS = $300M, SG&A = $100M. EBIT = $500M − $300M − $100M = $100M. Operating margin = $100M / $500M = 20%.

Contribution Margin

Formula:

Contribution Margin % = (Revenue − Variable Costs) / Revenue × 100

Per-unit version: CM per unit = Price − Variable Cost per Unit

When to use it: Break-even analysis, pricing decisions, and evaluating whether an additional unit of volume is profitable. Contribution margin is the foundation of break-even calculation — you'll always need it there.

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Growth Rate Formulas

Year-Over-Year Growth Rate

Formula:

YoY Growth % = (Current Period Value − Prior Period Value) / Prior Period Value × 100

When to use it: Any time you're comparing two time periods. Revenue growth, unit growth, market growth. The simplest formula in the toolkit.

Worked example: Revenue went from $120M to $138M. YoY growth = ($138M − $120M) / $120M = $18M / $120M = 15%.

CAGR (Compound Annual Growth Rate)

Formula:

CAGR = (End Value / Start Value)^(1/n) − 1

Where n = number of years between start and end.

When to use it: Multi-year trend analysis, market growth rates, portfolio performance. When an interviewer gives you "grew from X to Y over N years," they want CAGR, not simple average growth.

The Rule of 72 shortcut: To estimate CAGR when something doubles, use 72 / n years ≈ CAGR %. If a market doubled in 6 years: 72 / 6 = 12% CAGR. This works well for 6–15% growth rates and saves time in interviews.

Worked example: A market grew from $4B to $6.25B over 4 years.

CAGR = ($6.25B / $4B)^(1/4) − 1 = (1.5625)^0.25 − 1

In an interview, you'd recognize 1.5625 ≈ 1.25^2, so (1.25^2)^0.5 = 1.25^(0.5) ≈ 1.118 — approximately 12% CAGR. Or use the Rule of 72: market grew 56% in 4 years, roughly 12% annually. Close enough for a case answer.

According to Wall Street Prep's CAGR guide, the most common interview application is comparing company growth to market growth — a company growing at 8% CAGR in a 12% CAGR market is losing share, which is a diagnostic signal for the rest of the case.

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Break-Even Analysis

Break-even is the most commonly tested formula in profitability and new product cases. It answers the question: "At what volume does this product or investment stop losing money?"

Formula (units):

Break-Even Units = Fixed Costs / Contribution Margin per Unit

Formula (revenue):

Break-Even Revenue = Fixed Costs / Contribution Margin %

When to use it:

• New product launch: how many units must we sell before the product is profitable?
• New store/location: what revenue is needed to cover the incremental fixed costs?
• Price decrease decision: if we drop price by 10%, how much more volume is needed to break even?

Worked example — New product launch:

A company wants to launch a subscription product at $120/year. Variable costs (support, hosting, fulfillment) are $30 per customer per year. Fixed development and marketing costs are $3.6M.

• Contribution margin per unit = $120 − $30 = $90
• Break-even units = $3.6M / $90 = 40,000 subscribers

If the company's market has 2M potential customers and expects a 3% conversion rate, that's 60,000 customers — above break-even. The launch is financially viable.

Worked example — Price change:

If the company drops price to $100 to drive volume:

• New CM per unit = $100 − $30 = $70
• New break-even = $3.6M / $70 = ~51,429 subscribers (+28% more needed)

Is the 20% price decrease justified by demand elasticity? That's the next question in the case.

Corporate Finance Institute's break-even analysis guide 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 — it can absorb significant volume decline before losing money.

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ROI (Return on Investment)

Formula:

ROI % = (Gain from Investment − Cost of Investment) / Cost of Investment × 100

Or equivalently: ROI = Net Profit / Investment Cost

When to use it: Capital allocation decisions, technology investments, marketing spend evaluation, any "should we spend $X on Y" question.

Extended version — payback period:

Payback Period (years) = Investment Cost / Annual Net Cash Inflow

Worked example:

A retailer is evaluating a $2M warehouse automation system. The system will save $600K per year in labor costs and generate $200K per year in additional throughput capacity.

• Annual net inflow = $600K + $200K = $800K
• ROI = $800K / $2M = 40% per year
• Payback period = $2M / $800K = 2.5 years

In a consulting context, you'd benchmark this against the company's cost of capital (often given as 10–15%) and compare to alternative uses of the $2M. A 40% ROI with a 2.5-year payback is typically attractive.

Common trap: Candidates often forget to net out the annual cost of running the new system (maintenance, licensing, retraining). If the automation system costs $100K/year to maintain, annual net inflow drops to $700K, and payback extends to ~2.9 years.

According to Corporate Finance Institute's ROI formula guide, the most sophisticated ROI questions layer in time value of money — which takes us to NPV.

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NPV (Net Present Value)

NPV is the most conceptually demanding formula in the consulting math toolkit. Most case interviews use a simplified version — you're rarely asked to discount 10 years of cash flows individually.

Full formula:

NPV = Σ [Cash Flow_t / (1 + r)^t] − Initial Investment

Simplified for perpetuities (most common in cases):

NPV = Annual Cash Flow / Discount Rate

Use this when cash flows are assumed to continue indefinitely (or for a very long time). It's the Gordon Growth Model without growth: if a project generates $5M/year and the discount rate is 10%, the NPV of the perpetuity is $50M.

With constant growth:

NPV = Annual Cash Flow / (Discount Rate − Growth Rate)

When to use it:

• Investment decisions with multi-year cash flows
• Market entry or acquisition justification
• Infrastructure or IT spending with long payback horizons
• Any question that gives you a discount rate and asks "is this worth it?"

Worked example:

A consulting client is deciding whether to invest $20M to enter a new market. The market is expected to generate $4M in annual profit, growing at 2% per year, in perpetuity. The company's discount rate (hurdle rate) is 12%.

• NPV = $4M / (12% − 2%) = $4M / 10% = $40M
• Net NPV = $40M − $20M initial investment = $20M positive NPV

The project creates $20M of value above the investment — a clear "go" recommendation.

Two-period simplification: If the case has a 2–3 year forecast, discount each year individually:

• Year 1: $4M / 1.12 ≈ $3.57M
• Year 2: $4M / 1.12² ≈ $3.19M
• Year 3: $4M / 1.12³ ≈ $2.85M + terminal value

PrepLounge's NPV case interview guide notes that interviewers most often test NPV to see if candidates can identify whether a project's present value of returns exceeds its upfront cost — the conceptual test, not the arithmetic.

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Unit Economics

Unit economics formulas appear in tech, subscription, and marketplace cases. They measure profitability at the level of a single customer or transaction.

Customer Acquisition Cost (CAC)

Formula:

CAC = Total Sales & Marketing Spend / Number of New Customers Acquired

When to use it: Evaluating marketing efficiency, comparing channels, assessing whether a growth rate is sustainable.

Worked example: A SaaS company spent $6M on sales and marketing last quarter and acquired 3,000 new customers. CAC = $6M / 3,000 = $2,000 per customer.

Customer Lifetime Value (LTV)

Formula:

LTV = Average Revenue Per Customer × Gross Margin % × Average Customer Lifetime

Or for subscription businesses with known churn:

LTV = (Average Monthly Revenue × Gross Margin %) / Monthly Churn Rate

Worked example (churn version): Monthly ARPU = $150, gross margin = 70%, monthly churn = 2%.

LTV = ($150 × 70%) / 2% = $105 / 0.02 = $5,250

LTV:CAC Ratio

Formula:

LTV:CAC = LTV / CAC

Benchmarks:

• < 1:1 — Destroying value on every customer acquired. Unsustainable.
• 1:1 – 3:1 — Marginal. Not enough return for the risk.
• 3:1 — Industry rule of thumb for healthy SaaS businesses.
• > 5:1 — Strong unit economics, may be under-investing in growth.

Using the examples above: LTV = $5,250, CAC = $2,000. LTV:CAC = 2.6:1. Below the 3:1 benchmark — the company needs to either improve retention (reduce churn), raise ARPU, or cut CAC.

CAC Payback Period

Formula:

CAC Payback Period = CAC / (Monthly Revenue × Gross Margin %)

Worked example: CAC = $2,000, monthly revenue = $150, gross margin = 70%.

Payback = $2,000 / ($150 × 70%) = $2,000 / $105 = 19 months

Industry benchmark for B2B SaaS: under 18 months is healthy; over 24 months signals capital efficiency problems. This company at 19 months is borderline — a case worth probing in an interview.

According to Corporate Finance Institute's analysis of unit economics, the LTV:CAC ratio and CAC payback period together tell you whether a business model is sustainable at scale, and interviewers use them to test whether candidates understand SaaS or marketplace economics beyond surface level.

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Complete Formula Reference Table

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Worked Case: Applying Multiple Formulas in Sequence

The scenario: Your client is a B2B software company with $80M in annual revenue. Revenue is growing at 20% per year. Gross margin is 75%. They acquired 2,000 new customers last year, spending $8M on sales and marketing. Monthly churn is 1.5%. Average monthly revenue per customer is $3,000. The company is considering a $15M investment in a new enterprise sales team, expected to generate $5M in additional annual profit, growing at 5% per year. The company's hurdle rate is 15%.

Step 1 — Assess current unit economics:

• CAC = $8M / 2,000 = $4,000
• Monthly gross margin per customer = $3,000 × 75% = $2,250
• LTV = $2,250 / 1.5% = $150,000
• LTV:CAC = $150,000 / $4,000 = 37.5:1 — exceptional
• CAC payback = $4,000 / $2,250 = 1.8 months — extremely fast

Step 2 — Evaluate the investment with NPV:

• NPV = $5M / (15% − 5%) = $5M / 10% = $50M
• Net NPV = $50M − $15M = $35M positive

Step 3 — Synthesize:

"The company's current unit economics are strong — 37.5:1 LTV:CAC with under 2-month payback suggests significant headroom to invest in growth. The proposed $15M enterprise sales investment generates $35M in net present value at a 15% hurdle rate, representing a compelling return. I'd recommend proceeding, with a focus on maintaining gross margin as the enterprise segment often requires more professional services support. Key risk to monitor is whether enterprise churn is higher than the current 1.5% — if it rises to 3%, LTV halves and the economics change materially."

That's the complete arc: formula application → quantified output → structured recommendation.

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Related Skills to Build Alongside Formula Fluency

Formula recall is only useful if the surrounding skills are sharp. For case interview math practice — the drills that build raw speed — work through timed calculations using these exact formulas.

For the mental shortcuts that make formula execution faster under pressure, the case interview math mental shortcuts guide covers rounding, anchoring, and approximation techniques that pair directly with the formulas above.

If you want to understand how these formulas live inside a complete profitability analysis, the profitability framework guide shows how to structure the diagnosis before you ever reach for a calculation.

For market sizing cases — where growth rates and market share formulas dominate — the market sizing framework and market sizing step-by-step guides walk through how to set up and execute a complete sizing.

Break-even formulas come up most often in pricing strategy cases — the pricing guide covers exactly when and how to apply them in context.

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Formula Prep Checklist

Execution checklist

• Memorize all 15 formulas in the reference table above

  Cold recall under interview pressure — not looking them up mid-case

• Practice each formula with 3 numeric examples until setup is automatic

  Knowing the formula isn't enough; the setup (identifying inputs correctly) is where candidates lose points

• Learn the Rule of 72 for CAGR approximation

  Saves 30–60 seconds on any doubling/halving calculation

• Practice linking formulas in sequence (like the worked case above)

  Real cases chain 2–3 formulas; candidates who can only apply them individually get stuck on multi-step problems

• Drill break-even with price change scenarios

  Interviewers love 'what if we cut price by X%?' — you need contribution margin and break-even available simultaneously

• Understand LTV:CAC benchmarks by industry

  Knowing 3:1 is the SaaS benchmark lets you immediately contextualize any unit economics answer

• Practice communicating math aloud, not just calculating

  Interviewers score how you explain your setup, not just whether the number is right

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Test Your Formula Knowledge

Test yourself

Question 1 of 3

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

• A35%
• B45%
• C65%
• D30%

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Practice Drills: Formula Execution Under Pressure

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Sources and Further Reading (checked March 15, 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/