Consulting Math: Every Relation a Case Borrows, Recomputed

What consulting math is, and the two places it comes from

Consulting math is a small, closed set of relations. The whole of it divides by where the numbers arrive from, and the division decides how you prepare for each half.

The left column you can derive at the table from the words in front of you, and every line of it is worked on case interview formulas. Getting it right is a question of speed, and speed is trained on mental math for case interviews.

The right column is the harder half, and it is the one worked line by line below, because it behaves differently in a way that decides whether you get it right: the convention is not in the prompt.

Nobody will tell you that receivables run on revenue while inventory and payables run on cost of goods. Nobody will tell you whether the payback bar is three years or five, because the bar belongs to the client and the arithmetic has no opinion about it. Nobody will tell you whether the perpetuity denominator carries a growth term. Each of those is a decision you make out loud, and each one is a place an interviewer can watch you choose a base.

So the one habit that covers all of consulting math is to name the convention before you divide, which is what the six worked sections below do six times.

The conditions consulting math is tested under

Two published pages from the same firm set out the two regimes, and they contradict each other on purpose.

Bain publishes, on its digital assessment page, that a calculator is allowed during that assessment and ruled out in the interviews that follow. So the same relations get tested twice under opposite constraints. In the assessment, the arithmetic is not the test and the reading of a chart is. In the interview, the arithmetic is audible, and a long division performed out loud is part of what the interviewer is listening to.

That changes what is worth drilling. For the assessment half, practice extracting the right two numbers from an exhibit, since the division itself is free. For the interview half, practice the relations that are long enough to lose your place in: a working capital build has six lines, a perpetuity with growth has a denominator you can invert by accident, and a weighted average has two products before it has a quotient.

Bain also publishes a case that shows the third condition, which is neither speed nor accuracy. Its coffee case folds the entire cost of opening a shop into the first-year threshold. That makes its breakeven a capital-recovery bar rather than an operating one, and no amount of arithmetic recovers the difference. You have to ask which bar applies, and the asking is scored.

One more condition is published nowhere, and it is worth saying plainly. No firm in our corpus publishes how many calculations a case contains, how long you get for one, or what counts as an acceptable error. The absence is consistent across every firm we have read.

Weighted averages, and the trap in the simple mean

The most common finance error in a case is averaging two rates without weighting them. It arrives whenever a business has two segments.

Stated inputs: a Standard segment selling 300,000 units at 4 dollars of contribution, and a Premium segment selling 100,000 units at 10 dollars.

Recomputed by hand. 300,000 times 4 is 1,200,000. 100,000 times 10 is 1,000,000. Together 2,200,000 over 400,000 units gives 5.50 dollars of weighted average contribution.

Now the naive version. Averaging 4 and 10 gives 7 dollars, which is 1.50 dollars per unit too high. Across 400,000 units that is 600,000 dollars of imaginary contribution, and 600,000 over 2,200,000 is 27.3 percent, rounded from 27.27.

So the unweighted average overstates the whole contribution pool by more than a quarter. A quarter is far past a rounding problem. It describes a different business.

The check that prevents it takes one sentence: which segment has the volume? Whenever the larger volume sits on the lower rate, the weighted average is closer to the low number, and a simple mean will always be too optimistic.

One published case asks for exactly this. Deloitte's LeadAuto case, extracted from its page data at usrecruiting.deloitte.com, asks for operating margin per bus on Base and Luxury configurations. Its data sheet is an image, so no figure from it is reproduced here, and what is citable is the shape of the question: two configurations, one blended answer.

Mix effects with volume held flat

Once you have a weighted average, the second move is worth more than the first. A mix shift changes profit with no change in volume at all.

Take the same business and move Premium from 25 percent of units to 30 percent, with total volume held at 400,000.

• Premium: 30 percent of 400,000 is 120,000 units, times 10 dollars is 1,200,000 dollars.
• Standard: 280,000 units times 4 dollars is 1,120,000 dollars.
• New total: 1,200,000 plus 1,120,000 is 2,320,000 dollars.
• Gain: 2,320,000 minus 2,200,000 is 120,000 dollars, which is 120,000 over 2,200,000, so 5.45 percent, rounded from 5.4545.

Read the sentence that goes with it carefully. The gain is 5.45 percent of contribution, achieved with zero additional units. The new weighted average contribution is 2,320,000 over 400,000, which is 5.80 dollars, up 30 cents from 5.50.

This is the calculation that answers the question a client actually asks in a margin case: can we make more money on the same volume. And it has a bound worth knowing. If the whole 400,000 units were Premium, contribution would be 4,000,000 dollars, so the maximum available from mix alone is 4,000,000 minus 2,200,000, which is 1,800,000 dollars. Quoting the ceiling alongside the increment tells the client how much room the lever has.

Perpetuity, the one-line valuation

A recurring saving or a recurring gain has a value, and it is one division.

Value equals annual cash flow divided by the discount rate. The 120,000 dollar mix gain, if it persists:

• At a 10 percent discount rate: 120,000 over 0.10 is 1,200,000 dollars.
• At 8 percent: 120,000 over 0.08 is 1,500,000 dollars.
• At 12 percent: 120,000 over 0.12 is 1,000,000 dollars.

Recomputed by hand. Notice the sensitivity. Two points off the discount rate adds 300,000 dollars of value, which is 25 percent. In a case you will rarely be given a rate, so state the one you are assuming and say which way the answer moves if it is wrong.

With growth, the denominator becomes the rate minus the growth rate. At 10 percent discount and 3 percent growth: 120,000 over 0.07 is 1,714,286 dollars, so about 1.71 million. The formula breaks when growth approaches the discount rate, and that is worth flagging out loud rather than producing an enormous number.

The reverse use is the more common one in an interview. If a client will pay 1.2 million dollars for a stream, and the stream is 120,000 dollars a year, the implied rate is 120,000 over 1,200,000, which is 10 percent. That lets you sanity-check an offer price in one division.

Payback against a published hurdle

Payback equals capital divided by annual cash flow. The interesting part is never the division, it is the bar.

On the mix example: if capturing the shift requires 600,000 dollars of sales retraining, payback is 600,000 over 120,000, which is 5 years exactly.

Whether 5 years is good depends entirely on a rule the client sets. One published case shows what that looks like when the rule wins.

Oliver Wyman's oil and gas case at oliverwyman.com runs a water distribution entry with 250,000 kiloliters a day across 300 operating days, at 10 rupees per kiloliter, against variable cost of 5 rupees and 20 crore of annual fixed cost.

• Volume: 250,000 times 300 is 75,000,000 kiloliters.
• Revenue: 75,000,000 times 10 is 750,000,000 rupees, which is 75 crore. The page states "Annual revenue: Rs. 750,000,000, assuming 300 operational days per year".
• Variable cost: 75,000,000 times 5 is 375,000,000 rupees, so 37.5 crore.
• Total cost: 37.5 plus 20 is 57.5 crore. The page states "Total annual costs: Rs. 575,000,000 or 57.5 crore".
• Profit before tax: 75 minus 57.5 is 17.5 crore.

All three of the firm's lines reconcile, recomputed by hand, on a page checked 30 September 2026. And the published recommendation is still no: the entry "does not align with the client's investment criteria for a five-year payback period."

A profitable business, refused on a rule. That is the lesson of the whole section. Bain's coffee case takes the same idea from the other side, folding the entire cost of opening the shop into the first-year threshold, so its breakeven is a capital-recovery bar rather than an operating one. Ask which bar applies before you divide.

Working capital, and the cash a growth plan eats

Growth consumes cash before it produces any, and one calculation shows how much.

Stated inputs: 240 million dollars of revenue, cost of goods at 70 percent, 45 days of receivables, 60 days of inventory, 30 days of payables.

• Cost of goods: 240 times 0.70 is 168 million dollars.
• Receivables: 240 million times 45 over 365 is 29.59 million dollars.
• Inventory, on cost: 168 million times 60 over 365 is 27.62 million dollars.
• Payables, on cost: 168 million times 30 over 365 is 13.81 million dollars.
• Working capital: 29.59 plus 27.62 minus 13.81 is 43.40 million dollars.
• Cash conversion cycle: 45 plus 60 minus 30 is 75 days.

Recomputed by hand, each figure rounded to two decimals and the rounding stated. 240 over 365 is 0.657534, times 45 gives 29.589. 168 over 365 is 0.460274, times 60 gives 27.616 and times 30 gives 13.808.

Now the growth consequence. If revenue grows 20 percent and the day counts hold, working capital grows in proportion: 43.40 times 0.20 is 8.68 million dollars of additional cash absorbed. A plan that adds 20 percent to revenue and 8.68 million to working capital has to fund the second number before it collects the first.

Two notes on bases, because both are easy to get wrong. Receivables run on revenue, and inventory and payables run on cost of goods. Using revenue for all three would give 240 times 60 over 365, or 39.45 million of inventory, which overstates it by 11.83 million.

Share arithmetic, and the two bases it can use

Market share is two different numbers depending on the base, and a case will sometimes hand you one and ask about the other.

Stated inputs: a client sells 900,000 units at 40 dollars. The market is 6,000,000 units worth 216 million dollars.

Recomputed by hand. 36 over 216 is 0.16667, so 16.67 percent, which is one sixth exactly. 40 over 36 is 1.1111, so an 11.1 percent premium.

The gap between 15.0 and 16.67 is the whole insight, and it has one cause: the client prices 11.1 percent above the market average. Saying that in one sentence is worth more than either share figure alone.

One naming caution, because it is a real error. 16.67 percent is one in six. A figure like 15.8 percent is not, since one over 0.158 is 6.33. Calling a threshold one in six when it is 15.8 percent misstates it by almost a point, and this site has made that error before.

Where these consulting math formulas land in our weights

The four lines below are ours, read from the grading code in src/lib/grading. No consulting firm in our corpus publishes a marking scheme.

Drilling the finance lines

1. Weight every average you compute. Total over total, never the mean of two rates.
2. Run one mix shift per case with volume held flat. Then quote the ceiling as well as the increment.
3. Value one recurring line as a perpetuity, at two rates. State which rate you assumed.
4. Ask for the hurdle before dividing for payback. Five years is good or bad only against a rule.
5. Build one working capital figure from days. Receivables on revenue, inventory and payables on cost of goods.
6. Compute both shares whenever you are given prices. Volume and revenue, then explain the gap with the price premium.

Our breakeven calculator and CAGR calculator will check these lines once you have done them by hand, and the math drill reports Answer, Working, Units and Interpretation separately. All three are free and none needs an account. None of them will tell you which hurdle the client applies, which is the input that decides the recommendation.

For the return relations in a dedicated form, see ROI and payback. For the drills that time the left column of the map above, see case interview math practice.