Case Math Drills: 12 Worked Examples to Build Fluency
Drill articles only work if you actually do them. Twelve case-math problems with worked solutions across the patterns that show up most: percentage shifts, big-number scaling, profitability math, and 90-second market sizing.
Most case-math advice tells you which shortcuts to use. This article gives you the actual drills with worked solutions. Read the prompt, attempt it out loud against a timer, then check your work against the solution. If your answer matches and your timing is in range, move on. If it doesn't, re-attempt the same drill the next day before adding new ones.
How to use these drills
- Out loud, always. The bottleneck in real interviews is verbal pacing — narrating math while computing. Silent practice does not train this.
- Time yourself. Each drill has a target time. Faster is fine; slower means re-do.
- Show your work verbally. The interviewer grades reasoning, not just the answer. Practice narrating the steps you would take.
- Sanity-check at the end. Order of magnitude. Direction. Units. Three quick checks before you state the answer.
Drill set 1: Percentage shifts and unit economics
Drill 1.1 (target: 30 seconds)
A retailer's product line has 40% gross margin on $80 average price. They are considering a 10% price cut they believe will drive 15% volume increase. What happens to gross profit per unit and total gross profit?
New price: $80 × 0.90 = $72. Cost was $80 × 0.60 = $48. New gross profit per unit: $72 − $48 = $24, vs old $32. That's a 25% drop in per-unit gross profit. Volume up 15%, so total gross profit = 0.75 × 1.15 = 0.86, a 14% decline overall. The price cut destroys value despite the volume increase.
Drill 1.2 (target: 30 seconds)
A SaaS company has $20M ARR, churning at 8% annually, growing new bookings at 30% annually. What is the net ARR growth rate?
Net = new bookings − churn. New bookings = 30% × $20M = $6M. Churn = 8% × $20M = $1.6M. Net add = $4.4M, so net ARR growth = $4.4M / $20M = 22%.
Drill 1.3 (target: 45 seconds)
A restaurant chain raised wages 12% (impacts 30% of their cost base). Same-store revenue is flat. Their margin was 8%. What is the new margin?
Total cost base was 92% of revenue. Labor (30% of cost base) = 27.6% of revenue. After 12% wage increase, labor becomes 27.6% × 1.12 = 30.9% of revenue, an increase of ~3.3 percentage points. New margin = 8% − 3.3% =~4.7%. Margin is nearly halved without a revenue or efficiency response.
Drill 1.4 (target: 30 seconds)
A subscription business has $50 monthly ARPU, 18-month average customer lifetime, $400 customer acquisition cost. What is the LTV:CAC ratio? Is this a healthy business?
LTV = $50 × 18 = $900 (gross; if asked for contribution- margin LTV, multiply by gross margin). LTV:CAC = $900:$400 =2.25:1. The benchmark is 3:1+. So the business is below healthy threshold — either CAC needs to come down or retention needs to improve.
Drill set 2: Big-number scaling and contribution margin
Drill 2.1 (target: 45 seconds)
A manufacturer produces 4.5M units per year at a variable cost of $32 per unit and fixed costs of $80M. They sell at $58 per unit. What is their EBITDA margin?
Revenue = 4.5M × $58 = $261M. Variable cost = 4.5M × $32 = $144M. Contribution = $117M. EBITDA = $117M − $80M = $37M. EBITDA margin = $37M / $261M ≈ 14%.
Drill 2.2 (target: 60 seconds)
The same manufacturer is considering automation that would cut variable cost per unit by 25% and increase fixed costs by $40M. At what unit volume does the new structure beat the old?
New variable cost = $32 × 0.75 = $24/unit. Variable cost savings per unit = $8. Additional fixed cost = $40M. Breakeven units = $40M / $8 = 5M units/year. They currently produce 4.5M, so the automation is marginally below breakeven at current volume — only worth it if expected volume rises ~10%+ above current.
Drill 2.3 (target: 60 seconds)
A pharma company invests $1.2B in R&D for a drug. Trial success rate to launch is 25%. If the drug launches, expected peak annual revenue is $800M with 35% net margin and 7-year exclusivity. Is the project NPV-positive at a 12% discount rate?
Annual profit if launched = $800M × 0.35 = $280M. 7 years of cash flow at 12% discount rate roughly equals ~4.5× annual = $280M × 4.5 = $1.26B (using discount factor approximation). Risk-adjusted: $1.26B × 0.25 = $315M expected NPV of cash flows. R&D cost: $1.2B. Expected NPV ≈ $315M − $1.2B = −$885M. Looks bad on these inputs alone — but real pharma deals factor in option value, follow-on indications, and partial-success monetization that can flip the math.
Drill 2.4 (target: 45 seconds)
A retailer's stores generate $4.2M average revenue with 18% contribution margin. Each store has $480K of fixed operating cost. How many stores break even per quarter?
Annual contribution per store = $4.2M × 0.18 = $756K. Annual fixed = $480K. So per-store annual EBITDA = $276K — the store is profitable. The question is asking breakeven in *quarterly* terms: quarterly contribution = $189K, fixed = $120K, profit = $69K. Soall stores break even quarterly at this revenue. The question setup tests whether you flag the mismatch (quarterly question, annual data).
Drill set 3: Market sizing in 90 seconds
Drill 3.1 (target: 90 seconds)
What is the annual US market for residential solar panels in dollars?
~130M US households. Solar penetration ~5% (~6.5M installations cumulative). Annual new installations roughly 800K (industry data; ~1% household add per year in growth states). Average system: ~$25K installed (8 kW × $3/W). Annual market = 800K × $25K =~$20B.
Drill 3.2 (target: 90 seconds)
How many cups of coffee are sold daily in NYC coffee shops?
NYC population ~8.3M. Coffee-shop coffee drinkers (commuters + tourists + locals): ~40% of population = ~3.3M people. Of those, half buy coffee from a shop on any given workday: ~1.6M daily coffee buyers. Average 1.2 cups per buyer = ~2M cups/day. At ~$5 per cup, ~$10M daily revenue across NYC coffee shops.
Drill 3.3 (target: 90 seconds)
What is the annual US market for commercial dishwashers (the units that go in restaurants)?
~660K restaurants in the US (BLS data; 1 per 500 people rough estimate). Each has ~1 commercial dishwasher, replaced every 8–12 years. Annual replacement = 660K / 10 = ~66K units. New restaurant openings: ~5% growth net of closures = ~33K new units. Total annual demand: ~100K units. Average commercial dishwasher: ~$5K. Market =~$500M. Plus ~$200M in service / installation = ~$700M total addressable.
Drill 3.4 (target: 90 seconds)
How many electric vehicles will be sold in the US in 2030?
Total US new vehicle sales: ~17M annually. EV penetration in 2024: ~9%, projected to reach 30–40% by 2030 (range from current industry forecasts; depends heavily on policy and incentives). Midpoint ~35%. So 17M × 0.35 =~6M EV sales in 2030. Frame the answer with the policy-uncertainty range: 4M (low) to 8M (high).
Common stumbling points
- Forgetting to convert annual to quarterly (or vice versa). Drill 2.4 specifically tests this — most candidates miss the unit mismatch on first attempt.
- Multiplying out percentages instead of using fractions. 25% = ¼. 33% = ⅓. 12.5% = ⅛. Whenever you see one of these, switch.
- Sanity-checking the answer in the wrong units. A market sizing in dollars but a calculation that ended in units. State the units at the end of every step.
- Computing precise numbers when the question wanted a range. "About 6 million" is the expected phrasing — not "6.0 million" or "5,950,000."
The drill cadence that works
15–20 minutes per day for two weeks beats 3 hours once a week. The fluency we are building is reflexive, not analytical — the same way you internalize multiplication tables, not the way you learn calculus. By the end of two weeks of consistent drilling, your brain stops "thinking" through these patterns and starts pattern-matching to known solutions, freeing your working memory for the structural reasoning the case actually tests.
Mental Math for Consulting: 12 Shortcuts You Can't Afford to Miss
The shortcuts behind these worked examples — round-and-correct, fraction conversion, scientific notation, and order-of-magnitude checks.
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