Case math drills: percentages, growth, CAGR and break-even

In this article

1. Before you start
2. Why interviewers ask this
3. Case math from first principles
4. Worked example
5. Trade-offs and edge cases
6. How to answer it in an interview
7. Common mistakes
8. Follow-up questions
9. Sources

Before you start

This page sits in the consulting cases guide, next to the page on market sizing.

Why interviewers ask this

A case interview hands you a business problem and, partway through, a table of numbers. The arithmetic is rarely hard; the cost is in doing it out loud, in front of someone, without losing the thread of the question. The firms are open about what they want. Bain says it will be "looking for problem-solving skills and aptitude in the areas relevant to your role", and that a case interview is something candidates may face for consultant roles, where they "work through a problem". Numbers are one place that skill becomes visible.

A weak candidate goes silent, types into a calculator, and announces a result. A strong candidate says what equation they are about to solve, rounds to friendly numbers, calculates in short chunks they can narrate, and ends by saying whether the answer is plausible and what it means for the client. The page below trains that habit with six drills whose answers are computed, not guessed.

Case math from first principles

In plain words

Case math

Case math is the arithmetic in a case interview, done mostly by hand and out loud, built from percentages, ratios, growth rates, break-even and multiplication of stated assumptions.

Compound annual growth rate

The compound annual growth rate (CAGR) is the single yearly growth rate that would take a starting value to an ending value over a period of years, assuming each year grows on the year before.

Break-even point

The break-even point is the sales volume at which total revenue equals total cost, so profit is zero; above it the business makes a profit and below it a loss.

Percentage point

A percentage point is the arithmetic difference between two percentages, so a margin that moves from 12 percent to 15 percent rises by 3 percentage points.

Think of case math as cooking. You do not weigh each grain of salt; you use pinches you trust and taste as you go. Rounding to friendly numbers is the pinch, and the sanity check is the taste. The analogy breaks at one point: a cook can fix a dish later, but an interviewer hears your error the moment you say it.

Precisely: case math turns a spoken question into one equation, solves it with rounded numbers, and tests the result.

Two ideas do most of the work. First, growth compounds, so two 25 percent rises make 56.25 percent, not 50. Second, the rule of 72 is an estimate of how long a value takes to double: divide 72 by the yearly growth rate in percent. The rule of 72 article, last edited in September 2026, says it works well for annual compounding at roughly 6 to 10 percent and is less accurate at higher rates.

Step 1: Restate the question as one equation

Start by turning the spoken question into a single line, with units. "How many units do we need to sell to break even?" becomes "fixed costs divided by contribution margin per unit". "How fast is the market growing?" becomes "the ratio of end to start, raised to one over the number of years, minus one". This step takes a few seconds and stops two failures: solving the wrong problem, and starting the arithmetic before the interviewer has confirmed the inputs. Say the equation aloud and ask whether the numbers you were given are the ones to use.

The input is the interviewer's words and the table; the output is one equation whose every term has a number or a question mark. You now know what you must compute, so the next step can simplify the numbers.

Step 2: Round to friendly numbers, and keep the units

Replace each awkward number with a nearby friendly one, and write down which direction you rounded. 812 becomes 800 (illustrative), 47.3 percent becomes 47 or 50 percent, 1.74 becomes 1.75. Friendly numbers are those you can halve, double, or multiply by ten without effort. Carry the unit with every number, because most slips in cases are unit slips: millions against thousands, a month against a year, a percentage against a percentage point.

The input is the equation from step 1; the output is the same equation with friendly numbers and units attached. Rounding costs accuracy, and the worked example below measures how much, so you can choose between rounding to 50 percent and to 47 percent knowingly.

Step 3: Calculate in chunks you can say out loud

Break the multiplication into pieces that stay in your head. Split 8 million subscriptions times 25 per month times 12 months into 8 million times 25, which is 200 million per month, then times 12, which is 2.4 billion a year.¹ For percentages, find 10 percent first and scale. For growth, divide first, then use the rule of 72 to bracket the answer before you root. Say each chunk as you finish it, because that is what lets the interviewer follow, and rescue you if you slip.

The input is the friendly equation; the output is a number with a unit. You have a result that has not been tested, and step 4 is the test.

Step 4: Check the answer and say what it means

Test the result two ways. First, plausibility: is the market share you computed above 100 percent, is the break-even volume bigger than the market, is the growth rate wildly above the industry's? Second, a second route: compute the same quantity another way, such as using the rule of 72 to confirm that a rate near 12 percent fits a value that nearly doubles in five years. Then state the "so what" in one sentence tied to the client's decision: "break-even is 60,000 units, which on an illustrative market of 460,000 units is about 13 percent, so the plan is feasible but not trivial". That sentence is the answer to the case question; the number alone answers only the arithmetic.

The common error is to stop at the number. A candidate who announces "11.8 percent" has answered the arithmetic question; a candidate who adds "so the market doubles in a little over six years, which supports a growth-led entry" has answered the case question.¹

1 · Restate2 · Round3 · Calculate4 · Check and sayReplay

Step-by-step description

1. Write the question as one equation with units.
2. Use friendly numbers and note the direction.
3. Work in chunks you can say out loud.
4. Test it, then state what it means.

• Restate the question as one equation with units first.
• End with a plausibility check and a sentence about what the number means.

Worked example

Six drills, each solved with the four steps, with every figure computed by a short script rather than by hand.¹ The companies and numbers are invented, so treat them as illustrative.

Drill 1, percent change and percentage points. Revenue rises from 240 to 300 million dollars, a ratio of 300 over 240, which is 1.25, so revenue is up 25 percent.¹ Margin moves from 12 percent to 15 percent: that is 3 percentage points, which is a 25 percent relative increase, because 3 is a quarter of 12. The two compound: profit goes from 28.8 to 45.0 million dollars, which is up 56.25 percent, since 1.25 times 1.25 is 1.5625.¹ The percentage point article, last edited in October 2026, gives the same distinction: a rise from 40 to 44 percent is 4 percentage points.

Drill 2, CAGR. A market grows from 80 to 140 over 5 years. The ratio is 140 over 80, which is 1.75. The CAGR article, last edited in October 2026, gives the formula as the end value over the start value, raised to one over the number of years, minus one. The fifth root of 1.75 is about 1.1184, so the rate is 11.84 percent a year.¹

By hand, test a round rate by repeated squaring: 1.12 times 1.12 is 1.2544, squared again is about 1.57 for four years, and one more 1.12 gives about 1.76 for five years, so 80 grows to about 141, a hair too high, and the answer is just under 12 percent.¹ Then check it a second way with the rule of 72: at about 12 percent a value doubles in 72 divided by 12, or 6 years, so 5 years should multiply it by a little less than 2, and 1.75 fits. The rule is close at this rate: 72 divided by 11.84 is 6.08 years against an exact 6.19 years, 1.8 percent short.¹

Text description of the diagram

• The ratio of 140 to 80 is 1.75.
• The compound annual growth rate is 11.84 percent a year.

Drill 3, break-even. Fixed costs are 1,200,000 dollars, the price is 50 dollars and the variable cost is 30 dollars a unit. Contribution margin, the price of one unit minus its variable cost, is 50 minus 30, which is 20 per unit, and break-even units are fixed costs over contribution margin, so 1,200,000 over 20 is 60,000 units.¹ The break-even article, last edited in May 2026, states the same formula.

Revenue at that point is 3,000,000. To earn a profit target of 300,000 you need 75,000 units, because the target is added to fixed costs before dividing. If variable cost rises from 30 to 32, break-even jumps to 66,667 units, an 11.1 percent increase from a 6.7 percent cost increase, which is the kind of sensitivity a client cares about.¹

Drill 4, chunked market size (illustrative). 8,000,000 subscriptions at 25 dollars a month is 200,000,000 dollars a month, and times 12 is 2,400,000,000 dollars a year. A 15 percent share of that is 360,000,000 dollars.¹ Say the unit at every line.

Drill 5, what rounding costs. To find 47.3 percent of 812, rounding to 50 percent of 800 gives 400, against an exact 384.08, which is 4.15 percent too high. Rounding to 47 percent of 800 gives 376, which is 2.1 percent too low.¹ Both are fine for a direction, but only the second is fine if the next step amplifies the error. The rule is to round to the number that keeps the error below what the decision can tolerate.

Drill 6, how far to trust the rule of 72. Across annual growth rates from 2 to 20 percent, the rule's doubling time is too long below about 8 percent and too short above it: within 1 percent of the exact figure at 8 percent growth, 1.9 percent low at 12 percent and 5.3 percent low at 20 percent.¹ That matches the source's statement that it is most accurate near 6 to 10 percent.

Text description of the diagram

• The rule of 72 is within 1 percent of exact doubling time at 8 percent growth.
• At 20 percent growth the rule is 5.31 percent off.

Trade-offs and edge cases

Case math rewards accuracy you can defend, not accuracy to the last digit. These are the places where the textbook method misleads.

When rounding hides the answer

If two options differ by less than your rounding error, rounding decides the case. Drill 5 shows a 4.15 percent error from a harmless-looking rounding to 50 percent.¹ If a recommendation depends on a gap of 3 percent, carry more digits, or say that the answer sits inside the margin of error and name the extra data you would ask for.

When CAGR hides the path

CAGR reports a smooth rate even when the real path was bumpy. A market that was flat for four years and then jumped has the same CAGR as one that grew evenly. The CAGR article defines it as a mean annualized rate for compounding values, which is why it describes a period, not a forecast. When the interviewer gives yearly figures, look at them before you compress them into one number.

When the calculator question comes up

Bain's interview pages do not state a calculator rule, so ask your recruiter before the round and practise by hand either way. Case math is not about doing long division faster than a machine; it is about showing a clean method while talking.

How to answer it in an interview

The spoken structure is: restate the equation, round and calculate in chunks, check against a second route, then say the so-what and the next question.

Common mistakes

• "Revenue went from 240 to 300, so it grew 60 percent" (wrong). The 60 is the change in units, not a percentage (right: divide the change by the starting value, 60 over 240, which is 25 percent).
• "Margin rose from 12 to 15 percent, a 3 percent increase" (wrong). That mixes the two measures (right: say 3 percentage points, or 25 percent relative, and name which you mean).
• "Growth was 40 percent over 5 years, so 8 percent a year" (wrong). Dividing ignores compounding (right: take the fifth root of 1.4, which is about 6.96 percent a year).¹
• "Break-even is fixed costs divided by price" (wrong). Price ignores the cost of making each unit (right: divide by the contribution margin, price minus variable cost).
• "The answer is 11.8 percent" and nothing else (wrong). The number is not the answer to the case (right: say what the rate means for the client's decision, and check it first).

Follow-up questions

What happens to break-even if the variable cost rises from 30 to 32 dollars?

Break-even rises from 60,000 to 66,667 units, an 11.1 percent rise from a 6.7 percent cost increase, because the contribution margin falls from 20 to 18 dollars a unit¹. Say the direction first, then the size, then what it means: the client needs a tenth more volume to stand still.

Can you do this without a calculator?

Yes: restate, round and calculate in chunks out loud, as in the four steps above. Bain's interview pages do not state a calculator rule, so ask your recruiter. Prepare as if you will not have one, because the method, not the machine, is what the interviewer is watching.

What is the difference between percent and percentage points?

Percent describes a relative change and a percentage point describes the arithmetic gap between two percentages. A margin moving from 12 to 15 percent is up 3 percentage points and up 25 percent in relative terms; the percentage point article gives the same distinction.

How do you calculate CAGR without a calculator?

Divide the end value by the start value, then test a round rate by repeated squaring¹. For 80 to 140 over five years, 1.12 squared is about 1.25, squared again about 1.57, and one more 1.12 gives about 1.76, so a ratio of 1.75 means just under 12 percent a year. The rule of 72 cross-checks it: at 12 percent a value doubles in about 6 years, so a ratio a little under 2 after 5 years fits.

Does that number look right to you?

Say so, and recompute aloud from the equation. Interviewers watch how you react to a wrong-looking number, and "this implies share above 100 percent, so I have a unit slip; let me redo it in millions" is a strong signal. The interview loops guide shows where case rounds sit in a full process, and the probability and statistics guide covers the estimation habits behind the check.

For related practice, the finance and investment banking guide uses the same growth and margin arithmetic, and the product management guide uses it for metric questions.

Sources

1. Bain & Company: Our Hiring Process — Bain & Company (no page date shown; date = read date), 24 Oct 2026
2. Compound annual growth rate — Wikipedia (last edited date), 13 Oct 2026
3. Rule of 72 — Wikipedia (last edited date), 18 Sep 2026
4. Break-even point — Wikipedia (last edited date), 6 May 2026
5. Percentage point — Wikipedia (last edited date), 6 Oct 2026