Case Interview Formulas, With Every Line Recomputed

The profit identity, and the two ways to break it

Profit equals revenue minus cost. That is the whole subject, and everything else is a way of opening one of the two terms.

Revenue opens into volume times price. Cost opens into fixed cost plus variable cost per unit times volume. Substituting gives the form that actually gets used:

Profit equals volume times price minus fixed cost minus volume times variable cost. Collect the volume terms and you get volume times unit contribution minus fixed cost, where unit contribution is price minus variable cost.

Worked once, on stated inputs. A business sells 400,000 units at 30 dollars against a variable cost of 25.50 dollars, with fixed costs of 1.2 million dollars.

• Revenue: 400,000 times 30 is 12,000,000 dollars.
• Unit contribution: 30.00 minus 25.50 is 4.50 dollars.
• Total contribution: 400,000 times 4.50 is 1,800,000 dollars.
• Profit: 1,800,000 minus 1,200,000 is 600,000 dollars.
• Check against the long form: variable cost is 400,000 times 25.50, which is 10,200,000. Then 12,000,000 minus 1,200,000 minus 10,200,000 is 600,000. The two routes agree.

That check matters more than it looks. Running the same profit two ways catches most arithmetic slips in one step, and it takes twenty seconds.

Operating margin on those numbers is 600,000 over 12,000,000, which is 5 percent exactly. Note which base that is: revenue. A percentage without its base is the most common failure on this page, and it gets its own section below.

Breakeven, and which fixed cost convention applies

Breakeven volume equals fixed cost divided by unit contribution. One published case gives clean inputs for it.

McKinsey's Electro-Light case, read through a text proxy at mckinsey.com, states that the drink "would launch in a 16-ounce format" at "a price of $2 to retailers", that "each bottle would cost $1.90 to produce and deliver in the new process", and that the company "needs to bear $40 million in total fixed costs, including marketing expenses and increased costs across its production and distribution network". Page checked 30 September 2026.

The breakeven arithmetic below is ours, derived from those published figures, and recomputed by hand.

• Unit contribution: 2.00 minus 1.90 is 0.10 dollars.
• Breakeven bottles: 40,000,000 over 0.10 is 400,000,000 bottles.
• Check: 400,000,000 times 0.10 dollars is 40,000,000 dollars, which covers the stated fixed cost exactly.
• In gallons, since the page describes the format as one eighth of a gallon: 400,000,000 over 8 is 50,000,000 gallons.

Two things about that result are worth saying out loud in a real case. Four hundred million bottles is a large number for a launch, which is what a 10 cent margin on a 2 dollar product produces. And the unit conversion at the end is where a candidate loses the thread: bottles and gallons are different bases, and the question may be asked in either.

One convention warning, because it changes answers. Bain's published coffee case charges the entire cost of opening the shop against the first year, so its threshold is a capital-recovery threshold rather than an operating one. McKinsey's case divides fixed cost by unit margin with no capital recovery in it. Same equation, different definition of fixed cost, and the answers differ by a factor. Ask which convention applies before you divide.

What one cent does to a threshold, and the general form

Thin margins make thresholds volatile, and this is where a candidate most often says the wrong thing about a right number.

Take the same 40 million dollars of fixed cost and move unit contribution from 10 cents to 9 cents.

Recomputed by hand. 40,000,000 over 0.09 is 444,444,444 bottles, so about 444.4 million. Against 400 million that is 444.444 over 400, which is 1.1111, so 11.1 percent more volume. And 40,000,000 over 0.08 is 500,000,000, which against 400 million is 25.0 percent more.

Now the sentence that goes with it, and this is the part that is easy to get wrong. The margin fell by 1 cent out of 10, which is a 10 percent reduction in margin. The volume needed rose by 11.1 percent. Those two numbers are different, and neither one is the other.

The same trap appears on wider margins. Take 6 cents falling to 5 cents on 160 million units. Contribution at 6 cents is 160,000,000 times 0.06, which is 9,600,000 dollars. To hold 9.6 million at 5 cents you need 9,600,000 over 0.05, which is 192,000,000 units, and 192 over 160 is 1.20, so 20 percent more volume. The share of margin lost is 1 over 6, which is 16.7 percent. A page that reported 17 percent as the volume increase would be quoting the wrong quantity, and this site made that exact error in September 2026.

The general form is worth memorizing because it removes the guesswork. If margin falls from m to m minus d, the volume multiple is m divided by m minus d. At 6 and 1 that is 6 over 5, so 1.20. At 10 and 1 it is 10 over 9, so 1.111.

Subscription arithmetic, and 12.5 percent to stand still

Recurring revenue cases run on a small set of relations, and they all follow from monthly churn.

Inputs, stated: 240,000 subscribers, 18 dollars a month each, 2.5 percent monthly churn, 70 percent contribution margin, 90 dollars to acquire a subscriber.

1. Monthly revenue. 240,000 times 18 is 4,320,000 dollars.
2. Annual revenue. 4,320,000 times 12 is 51,840,000 dollars.
3. Monthly losses. 240,000 times 0.025 is 6,000 subscribers.
4. Average life. One over the monthly churn rate, so 1 over 0.025 is 40 months.
5. Monthly contribution per subscriber. 18 times 0.70 is 12.60 dollars.
6. Lifetime value. 40 times 12.60 is 504 dollars, and 504 over 90 gives a ratio of 5.6 against acquisition cost.

One derived line does more work than any of the six. Holding the base flat requires replacing 6,000 subscribers a month at 90 dollars each, which is 540,000 dollars. That is 540,000 over 4,320,000, so 12.5 percent of monthly revenue spent before a single net addition.

Say that percentage rather than the dollar figure. Twelve and a half percent of revenue to stand still is a sentence a partner can act on, and 540,000 dollars is a sentence that needs a second question.

The life formula deserves one caution. One over churn gives the average life only when churn is constant, and real cohorts decay unevenly. Say the assumption out loud.

Margin against markup, and percent against points

Two pairs of quantities get confused, and both confusions come from choosing the wrong denominator.

Margin against markup. Cost 80 dollars, price 100 dollars, so 20 dollars of gross profit. Margin is 20 over 100, which is 20 percent. Markup is 20 over 80, which is 25 percent. Same pair of prices, two different figures, and the gap widens as margins rise. A 50 percent margin is a 100 percent markup.

Percent against percentage points. A margin moving from 8 percent to 6 percent has fallen 2 percentage points, and 2 over 8 is a 25 percent fall in the margin. Both statements are true about the same event.

Here is why the second pair matters more than it sounds. Two points of gross margin measured on revenue arrive as a much bigger number when measured on profit, because profit is a smaller base. Take 50.4 million dollars of revenue at a 38 percent gross margin and 16.5 million of fixed cost.

• Gross margin: 50.4 times 0.38 is 19.152 million dollars.
• Operating profit: 19.152 minus 16.5 is 2.652 million dollars.
• Drop gross margin to 36 percent: 50.4 times 0.36 is 18.144 million dollars, so profit is 1.644 million.
• The fall: 2.652 minus 1.644 is 1.008 million dollars, which is 1.008 over 2.652, so 38.0 percent of profit.

Recomputed by hand. Two points on revenue, 38 percent of profit, and the only way to say that correctly is to state the profit base in the same breath. Both numbers describe the same 1.008 million dollars.

Growth, doubling, and the drift above 15 percent

Three growth relations come up, and one of them is an approximation whose error is worth knowing.

Compound growth. A quantity growing at r per period for n periods multiplies by one plus r to the power n. At 9 percent for 8 years: 1.09 squared is 1.1881, to the fourth is 1.41158, to the eighth is 1.99256. So 8 years at 9 percent very nearly doubles.

The rule of 72. Doubling time is roughly 72 divided by the percentage rate. At 9 percent, 72 over 9 is 8 years, which matches the exact 1.99 above almost perfectly.

Where it breaks. At 24 percent the rule gives 72 over 24, which is 3 years. The exact figure is 1.24 squared, which is 1.5376, times 1.24, which is 1.9066. So the true multiple after 3 years is 1.91 rather than 2.00, and the rule overstates by 0.0934 over 2, which is 4.7 percent.

The reverse direction has the same character. If something doubled over 5 years, the exact annual rate is 2 to the one fifth minus 1, which is 14.87 percent. The rule of 72 gives 72 over 5, or 14.4 percent. The gap is 0.47 percentage points, which is small enough to state as roughly 14 to 15 percent and move on.

The practical rule: the shortcut is reliable below about 15 percent and starts drifting above it. Above 20 percent, say the number is approximate or do the multiplication.

Where the case interview formulas meet our weights

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

Drilling the case interview formulas

1. Write the eight equations from memory, then check them. Profit, revenue, unit contribution, breakeven, margin, markup, compound growth, payback.
2. Work every profit two ways. Contribution route and full-cost route. Disagreement means an error, and you find it in twenty seconds.
3. Say the base with every percentage. Of revenue, of profit, of the gap, of the margin.
4. Do ten one-cent sensitivities. Take any thin margin, shave a cent, and compute both the margin change and the volume multiple. The two differ every time.
5. Convert units once per problem on purpose. Bottles to gallons, units to cases, kilowatt hours to sessions.
6. Check the rule of 72 against the exact power above 20 percent. Once is enough to remember the drift.

Bain publishes the reason all of this has to work unaided, at bain.com: calculators are allowed on its digital assessment and ruled out in the interviews that follow.

Our math drill times a chain and reports Answer, Working, Units and Interpretation separately, and the breakeven calculator will check a threshold while you learn to do it by hand. Both are free and neither needs an account. Neither one will tell you which convention a client applies to fixed cost, which is the question that changes the answer.

For the finance relations that sit above these, see consulting math formulas. For the speed techniques, see case interview mental math. For the habits that keep a chain honest, see case interview math tips.