PartnerRoundBy the end you can
- Write the right formula for profit, margin, growth, breakeven, payback, ROI, share or value within seconds of hearing the question
- Multiply and divide large numbers in your head with rounding, powers of ten and fraction shortcuts
- Talk through a calculation so the interviewer can follow it, see you catch slips and hear the so-what
- Sanity-check any answer against a handful of benchmark numbers
On this page
01Seven formula families
Case maths is rarely hard maths. It is arithmetic on large numbers, under time pressure, while someone watches. Almost every calculation belongs to one of seven formula families, sometimes two chained together.
Know them cold, so your working memory goes on the numbers rather than on recalling the formula.
Price minus variable cost is contribution per unit: the number breakeven runs on.
Listen for the family in the question before you touch the numbers: 'will this pay for itself?' is payback, 'how many do we need to sell?' is breakeven, 'what is it worth?' is valuation. Naming the family out loud buys thinking time and shows structure.
02Profit, margins, growth and share
Margins
Gross margin is revenue minus cost of goods sold, divided by revenue. Operating margin also takes off overheads such as sales, marketing and admin. Net margin takes off interest and tax as well. If a case says 'margin' without a qualifier, ask which one or say which you are assuming.
Margin and markup trip people up. A product that costs $60 and sells for $90 makes $30: a 50% markup on cost, but a 33% margin on price. Retailers often talk in markup, finance teams in margin.
Percent vs percentage points
If margin moves from 10% to 12%, it rose 2 percentage points, or 20 percent. Use 'points' for the gap between two percentages. Interviewers notice.
Growth and CAGR
Growth over one year is new ÷ old - 1. Over several years, use the compound annual growth rate: CAGR = (end ÷ start)^(1 ÷ years) - 1. You will not take fifth roots in your head, so work backwards from compounding you know: 1.1^5 ≈ 1.61, 1.05^10 ≈ 1.63, 1.07^10 ≈ 1.97.
Growth is asymmetric. A 20% fall needs a 25% rise to recover, because 1 ÷ 0.8 = 1.25. A 50% fall needs a 100% rise.
Market share
Share is company sales ÷ market sales, in the same units and the same year. Relative market share divides your sales by the largest rival's: below 1.0x means you are not the leader. A client selling $150m in a $1.2bn market has 12.5% share. If the leader sells $300m, the client's relative share is 0.5x.
Check yourself
Operating margin rose from 8% to 10%. Which statement is correct?
03Breakeven, payback, ROI and valuation
Investment questions ask one of three things: how much must we sell, how fast do we get our money back, and what is it worth? Each has a one-line formula.
Breakeven
Breakeven volume = fixed costs ÷ contribution per unit, where contribution is price minus variable cost. Then compare breakeven with realistic demand. The verdict lives in the gap between the two.
Payback and ROI
Payback = upfront investment ÷ annual cash flow, if the cash flow is steady. If it ramps up, add year by year until the running total covers the investment.
ROI = (total return - investment) ÷ investment. It means nothing without a period: 60% over two years and 60% over ten are very different deals. Simple ROI and payback ignore the time value of money, so say so when the horizon is long.
Valuation
A flat cash flow that lasts forever is worth cash flow ÷ discount rate. If it grows at g a year, it is worth next year's cash flow ÷ (r - g). So $10m a year at a 10% discount rate is worth $100m. Growing at 2%, it is worth $10m ÷ 8% = $125m.
Multiples are the shortcut bankers use: enterprise value = EBITDA × a multiple taken from comparable companies or deals. A multiple is shorthand for growth and risk. A higher multiple means investors expect more growth or less risk.
Worked example
Worked example: a new product line
A client is weighing a new product. Price $40 a unit, variable cost $25, fixed costs $6m a year, upfront investment $9m. Expected sales: 600,000 units a year for five years. Does it make sense?
- Contribution per unit: $40 - $25 = $15.
- Breakeven: $6m ÷ $15 = 400,000 units a year. Expected volume of 600,000 is 50% above breakeven.
- Annual profit: 600,000 × $15 = $9m of contribution, minus $6m of fixed costs = $3m a year.
- Payback: $9m ÷ $3m = 3 years.
- Five-year ROI: 5 × $3m = $15m back on $9m invested, so ($15m - $9m) ÷ $9m ≈ 67% over five years.
So what: the launch pays back in three years, and volume can fall a third short of plan before it stops making money. Next, test how confident we are in 600,000 units.
Check yourself
A business will generate $20m of free cash flow next year, growing 3% a year forever. The discount rate is 8%. Roughly what is it worth?
04Mental maths that saves minutes
Interviewers do not want a human calculator. They want quick, reliable arithmetic with no lost zeros. Six techniques cover nearly everything.
1. Round, then correct
Round one number to something friendly, calculate, then correct for the rounding. 98 × 37 = 100 × 37 - 2 × 37 = 3,700 - 74 = 3,626.
If you round both numbers, round them in opposite directions so the errors partly cancel. 47 × 62 ≈ 50 × 60 = 3,000, against an exact 2,914: about 3% out, which is fine for most case questions.
2. Tame the zeros
Strip the zeros, do the core sum, then put them back. With powers of ten: 3,000 × 450,000 = 3 × 10^3 × 4.5 × 10^5 = 13.5 × 10^8 = 1.35bn.
With units: thousand × thousand = million, and thousand × million = billion. So 3k × 450k = 1,350m = 1.35bn. Division works the same way: $2.4bn ÷ 80,000 = 2.4 × 10^9 ÷ (8 × 10^4) = 0.3 × 10^5 = $30,000.
3. Swap percentages for fractions
So 37.5% of 640 is 3/8 × 640 = 240, and dividing by 12.5% is the same as multiplying by 8. Percentages also flip: x% of y = y% of x, so 16% of 25 is 25% of 16, which is 4.
4. Rule of 72
Years to double ≈ 72 ÷ annual growth rate in percent. At 8% a year a market doubles in about 9 years, at 12% in about 6. It runs backwards too: a business that doubled in 6 years grew about 12% a year. It is close enough for any rate from about 2% to 20%.
5. Multiply by 5, 15, 25 and 50
× 5 is × 10 ÷ 2. × 25 is × 100 ÷ 4, so 25 × 368 = 36,800 ÷ 4 = 9,200. × 15 is × 10 plus half as much again, so 15 × 68 = 680 + 340 = 1,020. × 50 is × 100 ÷ 2, and × 125 is × 1,000 ÷ 8.
6. Divide by splitting
Break an awkward divisor into factors: 1,872 ÷ 24 = 1,872 ÷ 4 ÷ 6 = 468 ÷ 6 = 78. Or break the top number into chunks the divisor goes into cleanly: 1,872 = 1,680 + 192, and 1,680 ÷ 24 = 70 while 192 ÷ 24 = 8, so 78.
Check yourself
A market has 40m customers, each spending $250 a year. What is it worth a year?
05Doing maths out loud
The interviewer is scoring the process as much as the number. Silent maths looks like panic, and a wrong answer with no visible working gives them nothing to credit. Narrate the structure, not the arithmetic.
- 01ApproachSay the formula before any numbers
- 02Set upWrite each input with its unit
- 03ComputeNarrate milestones, not every carry
- 04Sanity-checkIs the size plausible?
- 05So whatLink the result to the question
Say the approach before the numbers. It lets the interviewer correct a wrong method in five seconds, not after two minutes. Label units on every line: 'm units', '$ per unit', '$m a year'. Unit slips cause most large errors.
Finish with a sanity check against something you know: a benchmark, the client's revenue or a per-person figure. Then say what the number means for the decision.
Weak
(Silence for 40 seconds.) About three years. Actually, hold on, I think I dropped a zero. Can I start again?
Strong
Payback is investment divided by annual profit. Investment is $9m. Annual profit is 600,000 units at $15 of contribution, so $9m, less $6m of fixed costs: $3m a year. $9m divided by $3m is three years. If the client's hurdle is four years, the launch clears it.
Why: The strong answer states the formula first, labels every number and ends on what the result means for the decision. Even if one input were wrong, the interviewer could see exactly where, and would still credit the method.
06Benchmark numbers worth knowing
Benchmarks are for sanity checks, not for showing off. Know them to a round number and say 'roughly'. If the interviewer gives you a different figure, use theirs.
Ranges vary by company, year and definition, so use them to catch answers that are out by a factor of three, not to quote. A grocer with a 20% operating margin, or a mature software company making 3%, should send you back to the maths or prompt the question 'what is unusual here?'
07Three worked examples
Each is the kind of question that comes up mid-case. Cover the working and try it first.
Worked example
Example 1: a price cut
A client sells 2m units a year at $25. Variable cost is $15 a unit and fixed costs are $15m. A competitor forces a 10% price cut. What happens to profit, and how much extra volume would restore it?
- Today: contribution is $25 - $15 = $10 a unit. Profit = 2m × $10 - $15m = $5m, a 10% margin on $50m of revenue.
- After the cut: price $22.50, contribution $7.50. Profit = 2m × $7.50 - $15m = $0.
- To get back to $5m: ($15m + $5m) ÷ $7.50 ≈ 2.67m units, a 33% volume increase.
So what: a 10% price cut wipes out all the profit, because contribution per unit falls by a quarter. Matching the competitor only works if volume rises by a third, which is unlikely in a mature market.
Worked example
Example 2: growth and doubling
Revenue grew from $200m to $320m over five years. What was the CAGR, and if that growth continues, when does revenue pass $640m?
- Total growth: $320m ÷ $200m = 1.6x.
- CAGR: 1.1^5 ≈ 1.61, so the CAGR is just under 10% a year.
- $640m is double $320m. Rule of 72 at about 10%: 72 ÷ 10 ≈ 7 years.
So what: at roughly 10% a year, revenue passes $640m about seven years from now, twelve years after the start. Ask whether the market is growing that fast or the client is taking share, because that decides whether it can last.
Worked example
Example 3: two ways to value a business
A private equity client wants a quick value for a target with EBITDA of $80m and free cash flow of $50m expected next year. Comparable companies trade at about 8x EBITDA. Assume a 10% discount rate and 2% long-run growth.
- Multiple: $80m × 8 = $640m.
- Growing perpetuity: $50m ÷ (10% - 2%) = $50m ÷ 0.08 = $625m.
- The two methods land within 3% of each other.
So what: a fair value is about $600m to $650m. If the seller wants $800m, the client needs a growth or cost story worth an extra $150m to $200m to justify the price.
Cheat sheet
The lines worth rereading the night before.
- 1Name the formula family before touching numbers: profit, margin, growth, breakeven, payback and ROI, share or value.
- 2Breakeven volume = fixed costs ÷ (price - variable cost). Payback = investment ÷ annual cash flow.
- 3Growing perpetuity = next year's cash flow ÷ (r - g). Cross-check it with an EBITDA multiple.
- 4Margin is on price, markup is on cost. The gap between two percentages is in points.
- 5Tame zeros with units: thousand × thousand = million, thousand × million = billion.
- 6Know 1/8 = 12.5%, 1/7 ≈ 14.3%, 1/6 ≈ 16.7%. Rule of 72 for doubling. 1.1^5 ≈ 1.6.
- 7Say the approach, label units, sanity-check the size, then give the so-what.
Now drill it
Reading builds the model; drills make it automatic. Start with the first topic, easiest drills first.
- Quick mathsDaily warm-up, new every day · Ten timed questions on raw arithmetic: big multiplications, percentages and fractions.
- Case maths sprintDaily warm-up, new every day · Ten timed questions in the sentences an interviewer says: growth, breakeven, margin and share.
- Percentages & ratios8 drills · Percent change, points versus percent, reverse percentages and weighted mixes: the arithmetic under every case.
- Revenue & growth8 drills · Revenue as price times volume, compounding, CAGR and the rule of 72, and splitting growth into its drivers.
- Profit & margins8 drills · Gross, operating and contribution margin, margin moves in points, and what a cost or price change does to profit.
- Breakeven & payback8 drills · Breakeven volume, price and market share, payback periods, and the volume you need to justify a price cut.
- Investment & returns8 drills · ROI, NPV, perpetuities and valuation multiples, and how to pick between two investments.
Read the full guide
The same ground on the blog, with more worked examples and practice on the page.
Questions candidates ask
Can I use a calculator in a consulting case interview?
Generally not in a live case interview at McKinsey, BCG or Bain: you work with pen and paper while the interviewer watches. Some online assessments do allow a calculator, so check the instructions for yours. Practise on paper so live cases feel normal.
How fast does my case maths need to be?
Accuracy and clarity matter more than speed. A two-step calculation, such as profit and then margin, should take under a minute including the set-up. Interviewers will wait for a candidate who is structured and right; they mark down one who is fast and loses a zero.
What maths comes up most in case interviews?
Percentages, growth rates, profit and margins, breakeven, payback and market sizing make up the bulk. Valuation usually stops at a perpetuity or a multiple, and full discounted cash flow models are rare. Anything beyond arithmetic, such as statistics, is unusual.
How should I practise mental maths for consulting interviews?
Little and often: 10 to 15 minutes of timed drills a day beats a weekend cram. Practise out loud with paper, mixing percentages, large multiplications, division and growth. Then practise inside full cases, where the hard part is choosing the right formula under pressure.
What is the difference between margin and markup?
Margin is profit as a share of the selling price; markup is profit as a share of cost. A product that costs $60 and sells for $90 has a 50% markup but a 33% margin. Confusing the two is a common slip in retail and pricing cases.