Case interview math: the insider guide

the image is the cover for an article on case interview math tips and tricks and preparation advice

Aspiring consultants at McKinsey, BCG, and Bain or any of the Big 4 and other consulting firms will have to go through some numerical problems during their case interviews.

In fact, in roughly 70% of consulting case interviews, candidates need to demonstrate the ability to handle quantitative problems confidently. For top-tier firms such as McKinsey, BCG, and Bain this probability is close to 100%.

The more prestigious, the more likely that you will have to handle consulting math during the cases.

For example, you first have to analyze a problem mathematically before qualitatively investigating the particular reason for the numerical result and then providing a recommendation.

In fact, many candidates fear this part of the interview and our experience as interviewers and case coaches with McKinsey shows that most errors during the case interview happen in the math part.

If you want to go deeper to brush up on your math and chart interpretation skills, we have created a program with detailed insider learning materials, 25 videos, and a guidebook as well as 2,000 case interview math drills (including full case interview examples and case interview practice) that mimick the McKinsey, BCG, and Bain case interview math as well as the aptitude and analytics test math for you here: the Case Interview Math Mastery.

This article is part of our consulting case interview series. For the other articles, please click below:

Why candidates struggle with consulting math

There is no need to fear quantitative problems in case interviews. The level of math required is not more complex than what you have already learned in school and you do not need a specific degree to pass the case interviews.

That being said, as with every other element in a case interview (structuring and analytics, exhibit and data interpretation), there is a very specific way of approaching case interview math, which candidates are not used to from their previous academic or professional experience.

They need to first plot a course of action, i.e. figure out what to calculate and then, second, perform the calculations. Adding to that, there is the stress of the interview.

Read our tips below to learn:

  • How to come up with the correct calculations in case interviews (the logic)
  • How to perform swift calculations both mentally as well as with pen-and-paper (the calculation)
  • How to stay cool under pressure (consultant mindset)

The purpose of case interview math

As discussed above, fact-based decision-making and recommendations in case interviews are often based on numerical results. Coming up with the correct logical approach to quantitative problems and following through with the correct calculations is one of the key skills needed for the case interview but also for every working consultant.

Broadly, quantitative analyses are conducted for two reasons.

Finding the problem and quantifying its impact

Before solving a problem, consultants need to figure out what is wrong. They will run quantitative analyses to get to the root of the problem(s), a process that will also give them insights into potential solutions.

During the case interview, you will have to do the same in an abbreviated task.

Supporting recommendations

Numerical answers are needed to make and support business decisions for the client, and in reality, every measure or recommendation that is proposed by consultants will have a quantitative backing and rationale.

The answers you will receive by conducting this analysis during the case interview will be crucial for your recommendation in the end

Consulting firms check your quantitative skills

You are expected to perform well in such an environment. Basically, consulting firms check whether or not you have what it takes for the daily consulting life before candidates start their McKinsey careers, BCG careers, or Bain careers.

During the case interview, the math you have to deal with to arrive at a certain conclusion will never be too difficult in nature.

What makes it more challenging is the fact that both planning the steps you need to take and then calculating happens in a stressful environment with the interviewer watching your every step along the way.

Of course, calculators cannot be used.

Being able to solve easy numerical questions, do fast mental math, basic calculus (addition, subtraction, multiply, division, percentages, and fractions) and give reasonable estimates is usually enough to survive a typical interview problem.

Advanced mathematical knowledge is not required. Some of the problems you have to solve may be tricky, include multiple steps, and therefore require a certain degree of logic but the calculations itself should not be too difficult.

How to approach a case interview math question

Different skill levels, same problem

Some candidates might need to (re)-acquire skills not used since high school, and others need to simply dumb down their approach and get used to basic math again. The latter part is especially true for engineers and other people with a quantitative background.

Both types of background need to adapt to the specific case interview math principles and process.

Other than math problems in school, the focus of case interview math is always on the context of the case. Getting to results is only the first step. You need to use your results to make sense of them in the context of the case. In turn, 100% accuracy is not needed.

It’s better to get directionally correct results swiftly and interpret them correctly than getting 100% accurate results and not providing any insights into the case problem.

Approach case interview math with this mantra

In general, have a quantitative angle in every case, even if the interviewer does not explicitly ask you for it. For example, try to relate numbers to each other, think about the potential impact of your recommendation, etc.

Many candidates are simply scared of digging into the mathematics of a case. Don’t be that person!

In the following section, we will show you what you need to know to keep you on track during the case interview mental and pen-and-paper math.

Our approach to every case math problem

The challenge of case interview math usually consists of two components. You need to learn an approach that works for every type of case math question, regardless of the case or context. Resources such as Case in Point cannot help you in this regard.

First, you need to derive and lay out a path to get to the desired result. Second, you need to do the calculations to get to the actual numbers.

So, how should you approach quantitative problems during a McKinsey, BCG, or Bain case interview?

the image shows an 8-step process of how to approach every case interview math question in McKinsey, BCG, and Bain interviews
How to approach every case interview math question
  1. Listen. Carefully and actively listen to what your interviewer tells you
  2. Clarify. Before you dig into the quantitative problem at hand, slow down: clarify the numbers you heard from the interviewer or extracted from charts or data tables or discuss what additional information you would need. Clarify the desired outcome of your analysis
  3. Draft your logic. Set up your planned approach to the calculation. For more difficult tasks, you could ask for some time to get your head around the problem with one to two minutes being the upper boundary. Draft your logic on the piece of paper you have been handed by the interviewer
  4. Communicate your approach. Lead the interviewer through your approach. This way you’ll make sure that mistakes are spotted early.
  5. Calculate. When the interviewer agrees with your approach, follow through with the calculations – alone and in peace. Again, ask for some time and use the paper
  6. Sanity check your results. Make sure there are no mistakes in your calculations. Do the numbers make sense?
  7. Communicate your results top-down. Summarize the result(s) you got in a confident and assertive manner. Do not phrase the answer as a question and really focus on the key results when communicating. There is no need to go through each intermediate step of your calculation. Remember, consulting interview communication should always be top-down and follow the Pyramid Principle
  8. Come up with a hypothesis. When you get a final result, DON’T STOP THERE. Quickly explain and interpret the numbers. Relate the numbers to the problem at hand. Remember why you set up the calculation in the first place. How does it tie in your planned analysis? How does it impact your hypotheses? Is it the final result or just some intermediate result? It is important to discuss the ‘so what’ of your quantitative analysis.

Be cautious with mental math

A word of caution for mental mathematicians. Write everything down, even when you are doing mental calculations. When you make a mistake while calculating mentally you will face a serious problem. You have no track record of your mistake and would have to re-do all calculations. It is also more difficult to spot your own mistakes in time, which could save you.

Additionally, the interviewer is not able to help you or pick you up from the mistake. At least keep a record of your intermediate results for the interviewer to be able to follow and intervene and for you to quickly go back to find and solve a mistake.

Now that we have discussed the process, let’s discuss the skills needed that help you ace case math.

Typical math problems and key formulas

3 types of case math problems

90% of case interview math problems fall into one of the following categories and should help you come up with a recommendation on

  1. Market or segment sizing (e.g., ”how many sports cars can be sold in China in the next 5 years”)
  2. Operational calculations and decisions (e.g., ”if we reduced the lead time for each production step by 15%, how much time would we save in total?”)
  3. Investment and other financial and strategic decisions (e.g., ”Investment A would give us 12% annual return and Investment B 5.5% every 6 months – what investment should the client go for?”)

Case math formulas

Market or segment sizing. There is no shame in asking the interviewer for certain numbers in a market sizing exercise (e.g. the population of a specific country), however, to make your life easier you should have some numbers memorized:

  • World population
  • Population of US, UK, Germany, China, India and other large countries
  • Population of countries within your geographical region
  • Life expectancy
  • Average household size
  • Income levels

Operational calculations. To solve operational problems, you would need to come up with the formulas on the spot based on the situation and context of the case. Sometimes, you would work with linear optimization problems to maximize or minimize a certain function (for more, see below).

These two formulas could be a good starting point for your efforts.

  • Capacity = total capacity / capacity need per one unit
  • Utilization rate = actual output / maximum output

Investment, financial, and strategic decisions. In order to evaluate the financial impact of decisions, these few formulas below are key:

  • Profit = Revenue – Cost
  • Revenue = Price x Volume
  • Cost = Fixed cost (the cost that can’t be changed in the short term) + Variable cost
  • Profitability (profit margin) = Profit / Revenue
  • Market share = revenue of one Product / Revenue of all products (in one market)
  • Growth rate = (New number – Old) / old
  • Break-even time = Investment / Profit per specific time frame (e.g., annual)
  • Break-even # of sales = Investment / Profit per product
  • Return on investment = (Revenue – Cost of investment) / Cost of investment
  • NPV (Net present value) = Net cash flow of a period / (1 + discount rate)^number of time periods. The NPV is the present value of the sum of future cash in and outflows over a period of time and used to analyze the profitability of an investment or project

Case interview math tips and tricks

Keep the following tips in mind to 3x your case interview math performance and speed, while reducing the potential for errors and mistakes.

the image is a list of tips and tricks that increase the performance in a consulting case interview for McKinsey, BCG, and Bain
Keept these 10 things in mind in each case interview

Tackle the problems aggressively

McKinsey, BCG, and Bain interviewers want to see highly driven candidates. Show self-initiative. If you hesitate or make mistakes the interviewer will test if this was just an anomaly. They will give you even more calculations in the progress of the case, whereas a candidate that proceeded flawlessly through a calculus process often gets short cuts for the next quantitative parts or whole results readily delivered by the interviewer.

In case you messed up one calculation, don’t mess up the next one! Don’t calculate everything in your head. Use pen and paper to structure the numbers. Choose the fastest and easiest approach to set up your calculations.

Keep calculations organized along the way. Ideally, you find what works for you in the first mock interviews and then apply it consistently (habit-forming).

Re-learn and practice basic calculus

(Re-) learn basic calculus operations and practice until you can do it in your sleep. Many candidates struggle with the concept of being watched while doing these basic operations.

Therefore, the better your skill to compute quickly in a stressful environment, the bigger your quantitative muscle in the interview. Practice these calculations both mentally and with pen-and-paper under time-pressure and the vigilant eyes of friends, family, and colleagues. It certainly helps to build resilience and stamina.

Become comfortable thinking quantitatively

Get a feeling for numbers, especially percentages (e.g. be able to instantly estimate percentages in your head, also % of %) and magnitudes. This helps you to interpret results and put them into context and spot more obvious mistakes you made in your calculation.

Be able to interpret your results and demonstrate good business judgment (e.g. if the numbers indicate that the goal of the company seems to be out of reach). Make approximations and estimations quickly and correctly. See the implications of your calculations and conclude correctly (ask the question ‘so what?’).

Relate numbers to each other

Quickly relate numbers and outcomes with each other. You can make a habit out of using equations to describe particular relationships. It both helps your thinking and shows that you are structured in your approach. A brief example:

To improve the over-utilization of train tracks draw up the equation: utilization = demand/ capacity. From this equation, you can instantly see that you need to decrease demand and increase the capacity to improve the utilization situation. Obvious but very effective, such an approach also demonstrates composure and logical thinking.

Sanity check everything

Try to spot your own mistakes before the interviewer does, be vigilant, and sanity-check your approach to the problem and outcome of each calculation. Use your judgment to spot calculation and estimation results that seem out of line in the first place (e.g. 18.3% vs. 183%). Maybe you have done a mistake in the calculations or your assumption was off. In this case, act quickly to re-think and give reasons why your numbers might be off.

Keep track of units

Don’t lose track of your units. Is it kg, tons or USD? Set up the calculation before actually doing them, and already prepare (either mentally or preferably on paper) a space for the end result including the correct unit. Keep the units for your intermediate results organized and labeled.

Be efficient and use shortcuts

Be efficient in your calculations. Most answers rely on multiple assumptions and reasonable estimates anyway, therefore not providing a 100% level of precision. Use shortcuts!

Most of the time, close-to-correct answers are expected. If you come up with a population number use 80mn instead of 82.5mn (state it beforehand that you will trim the fat a bit; if the interviewer agrees, proceed with your calculation). Similarly, if you get 42.65 as an intermediate result say that in the following calculations this will be rounded to 40. Round numbers! Of course, ask the interviewer before you do so. 99% of the time, they will agree. The numbers won’t be 100% accurate either way and are not expected to be. Make plausible shortcuts in your approach and calculation to reach plausible numbers.

Simplify and round numbers

Similar to the above point, try to simplify calculations as much as possible, thereby reducing the number of steps you have to calculate and minimizing the chances of mistakes. As the interviewer, if it is ok to simplify beforehand.

One common way to simplify calculations is to round numbers.

  • 83 million Germans become 80 million
  • 328 million Americans become 320 or even 300 million
  • 365 days in a year become 350 or even 300 days (making some clever assumptions about weekends, etc.)
  • USD 983 million in revenue becomes 1 billion

The key with rounding numbers is to know when it is a good time to do so. Sometimes you need precise results, other times you have more leeway.

Some case interview case math questions demand precise results. For example, if you are asked whether or not an investment has an ROI above 12% and you can already spot that it is around that number, it would be a wise decision to calculate with precision.

For instance, if you are asked to size a market, there are a lot of assumptions you will make and there is uncertainty. In such cases, rounding numbers would be the way to go, In general, the more assumptions you use to get to a result, the more you can use the rounding technique.

For the latter, as a rule of thumb, you should round only within a 10% margin; otherwise, you might skew the results and provide a false recommendation based on that.

Also, think about rounding consecutive numbers both up and down sequentially to get a more precise result due to the effects cancelling each other out.

For instance, if you want to calculate the Revenue, which is Volume times Price and the Volume = 9,500 units and the Price = USD 35, calculate with a Volume of 10,000 and a price of USD 30. That will roughly keep you in a 10% margin of the precise result. If you round both up or both down, you would already be around 20% off the precise result.

Take your time

Ask the interviewer for some time to prepare your logic and then again to conduct your calculations. 1 to 2 minutes is fine for the logic and up to 5 minutes are ok for the actual calculations. Do not feel pressured to talk to the interviewer while you are thinking or performing calculations.

Focus on one thing at a time. Then communicate your logic or your result.

Watch the 0s

You would not believe how many candidates fall into this trap. Candidates struggle with large numbers, try to simplify, and then end up losing some 0s along the way. Watch out for 0s that you have trimmed or left out to facilitate calculations.

One way to simplify your calculation and keeping track of the 0s is to use notations such as labels or scientific notations.

For labels, add k for thousand (000), m for million (000,000) and b for billion (000,000,000) when manipulating larger numbers. That way you can simplify AND keep track of your 0s.

One particular way that we find quite useful is to make use of the powers of 10, the scientific notation (e.g. 1.4bn / 70mn = 1.4 * 109 / 7 * 107). You can trim the power of tens and do the simple division. Once you reach a conclusion you are able to immediately see the magnitude of your number (in this case: 0.2 * 102 = 20).

How to prepare for case interview mental and pen-and-paper math

There are several things you could do to get up to speed with mental and pen-and-paper math that should suffice for McKinsey, BCG, or Bain interviews. The trick is to be confident in your ability to efficiently do simple math and resilient enough to external pressures in the process.

Get number affine by working with numbers you encounter in your daily life, be it the bar tap or the receipt at the grocery store or figures and data you find in the news (especially business news). Put them into context; create relative numbers and percentages. Try to calculate some simple business cases (e.g. waiting at the doctor: how much profit does he make a month, a year, etc.). The opportunities are unlimited.

Additionally, get some apps for Android or iOS that train your mental math abilities. There are some really fun and entertaining apps out there and we will name a few later in this article.

Do all this in a stressful environment. You want to build stamina and resilience to outside influence and stress. Use the apps in the crowded and noisy subway; calculate in mock interviews, in front of friends and family or simply with a time limit (some apps include this function).

Make sure to be able to recite the 1×1 in your sleep. From there you mostly just add some 0s in case interview calculations.

Basic arithmetic calculations

Learn simple calculus shortcuts and see the examples below as starting points:

Addition

Build groups of numbers that add up to ten or multiples of ten.

7+3+12+8+5+5 = 40
(10)+(20)+(10) = 40

Subtraction

Learn quick subtraction by finding out what makes it to ten.

  1. Reverse the subtraction (5-2 = 3)
  2. Find what makes it to 10 (3+7 = 10)
  3. Add 1 to the digit on the left of the number you are subtracting

Multiplication

  • Multiply any 2 digit numbers within 3 seconds
  • Cut the 0s, but be careful to add them again in the end, e.g. change 34x36mn to 34×36 = 1224, then add six 0s –> 1,244,000,000; use the label method we descirbe above
  • Break apart multiplications by expanding them. Break one of the terms into simpler numbers, e.g. 18×5 = 10×5+8×5 OR (20-2)x5 = 20×5-10=90
  • Exchange percentages and simplify the calculation, e.g. 60×13% = 0.6×13 or 6x 1.3=7.8
  • Factor common numbers to simplify your calculations when dealing with multiples of 5, e.g. 17×5 = 17×10 / 2 = 85. The most common numbers to keep in mind are: (5 = 10 / 2; 7.5 = 10×3 / 4; 15 = 10×3 / 2; 25 = 100 / 4; 50 = 100 / 2; 75 = 100×3 / 4

Division

  • Convert percentages into divisions, e.g. 33% of 500 = 500*1/3 = 500 / 3 = 167
  • Split numbers into tenths, e.g. 60% of 200 = 10% of 200*6 = 120
  • Apply factoring and expanding as described for multiplications
  • Use the table below to learn the division table and fractions by heart:
the image displays a division and fraction table to be learned for consulting case interview math
Division table and fractions for consulting case interviews

Learn the basic case interview math formulas and consulting math tricks.

Other common mathematical operations

Averages

For our purpose, we refer to the average as a number expressing the mean value in a set of data, which is calculated by dividing the sum of the values in the set by their number.

In case interviews, calculating the average or a number of averages is very popular, since it is simple, yet demands several calculations to arrive at a result. It is a good pressure test of the candidate.

For example, you might be presented a table with three products, each with different production cost. The produces the same amount per product and you might have to calculate the average production cost across all products.

A variation, which is common, would be weighted averages. Instead of each of the data points contributing equally to the final average, some data points contribute more than others and therefore, need to be weighted in your calculations.

To stick with the example above, Product A might be responsible for 20% of the sales, whereas Product B and C for 30% and 50% respectively.

Other common contexts, where you are asked to calculate an average could be

  • Growth rates
  • Population
  • Geographies and countries
  • Product categories and segments

Fractions, ratios, percentages, and rates

Fractions, ratios, percentages, and rates are all different sides of the same coin and can help you expedite your calculations.

For instance, fractions can be used to represent a number between 0 and 1, and calculate. Expressing numbers as fractions and using them for additions and subtractions as well as multiplication and divisions can help you solve problems faster and more conveniently through simplification.

For example, you can write 0.167 as 1/6, or 0.5 as 1/2. Take a look at the fraction table on top to learn these by heart. This will go a long way in your case interview performance.

Ratios are comparisons of two quantities, telling you the amount of one thing in relation to another. If you have 5 apples and 4 oranges, the ratio is 5:4 and you have 9 fruit in total.

In case interviews, one tip is to write ratios as fractions of the total.

Next, percentages are a specific form of ratios, with the denominator always being fixed at 100.

From experience, almost 80% of case interviews will include some reference to or use of percentages. Discussion points such as ”Revenue increased by 15% YTD” or ”Costs are down 4% over the last 6 months” are all too common.

Percentages are also very useful when you want to put things into perspective and state your hypotheses. ”Is a 15% increase realistic?” ”What would we need to do to achieve this?”

Be careful not to mix percentage points with percentages. A percentage point or percent point is the unit for the arithmetic difference of two percentages. For example, moving up from 40% to 44% is a 4 percentage point increase, but is a 10 percent increase in what is being measured.

Rates are ratios between two related quantities in different units, where the denominator is fixed at 1. If the denominator of the ratio is expressed as a single unit of one of these quantities, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable), then the numerator of the ratio expresses the corresponding rate of change in the other (dependent) variable.

One common type of rate is per unit of time, such as speed or heart rate. Ratios that have a non-time denominator include exchange rates, literacy rates, and many others.

Case interviews will often present you with the following rates:

  • Growth rate: the ratio of the change of one variable over a period of time versus the starting level
  • Exchange rate: worth of one currency in terms of the other
  • Inflation rate: the ratio of the change in the general price level in a given period to the starting price level
  • Interest rate: the price a borrower pays for the use of the money they do not own (ratio of payment to amount borrowed)
  • Price-earnings ratio: the market price per share of stock divided by annual earnings per share
  • Rate of return: the ratio of money gained or lost on an investment relative to the amount of money invested
  • Tax rate: the tax amount divided by the taxable income
  • Unemployment rate: the ratio of the number of people who are unemployed to the number in the labor force
  • Wage rate: the amount paid for working a given amount of time (or doing a standard amount of accomplished work) (ratio of payment to time)

Keep an eye on the timeframe rates are expressed in. This could be annually (p.a.), quarterly, per month, etc. Often, the information is provided for different timeframes or denominators. Convert to the same before conducting your analysis, calculations, or comparisons.

Growth rates

You should be comfortable with calculating growth rates. That is fine for one time period.

  • (Increase of 30% in year 1): 100m x 1.3 = 130m

It gets more tricky when you have to calculate growth over multiple periods. You need to get the compound growth rate first.

  • (Increase of 30% in year 1, and 25% in year two): 100 x 1.3 x 1.25 = 100 x 1.625 = 162.5

The latter you can do if you want to calculate growth over 2 to 3 time periods. Everything after that would become tedious. If you want to calculate growth of several periods, it is better to estimate. The common shortcut for this would be to use the growth rate and multiply it with the number of years.

  • (Increase of 4% p.a. over 8 years) = 4 x 8 = 32 (%) = 100 x 1.32 = 132

If you would use the exact CAGR, you would end up with roughly 137 in our example. The deviation of 5 or roughly 3.5% with your simplification is close enough.

However, be aware that the divergence increases with larger numbers, higher annual growth rates, and the number of years. In a case interview, you could account for that by saying that you would like to add 5-10% to your calculated value.

Optimization problems

In a mathematical optimization problem, you need to either maximize or minimize some function relative to a given set of alternatives. This function is called the objective function.

A typical linear optimization problem would look like this.

A local teddy bear factory wants to optimize its product mix in order to maximize its profit. They produce personalized large and small bears. The profit of a large bear is $25 while the profit of a small bear is $20. Each large bear requires 1kg of material to produce while small bears require 0.65kg each. The daily supply of material is limited to at most 50kg. About 8 bears of either product can be produced per hour. At the moment the family wants to limit their workday to 10 hours.

Your objective function would be to maximize the profit of the factory in this example.

Expected value and outcomes

Sometimes, you will have to compare the impact and success of different recommendations or the expected return of an investment. One way to do this is to work with probabilities and calculate the expected value of a course of action.

The expected value for each recommendation is calculated by multiplying the possible outcome by the likelihood of the outcome. You can then compare the expected value of each and make a decision that is most likely to get to the desired outcome.

For example, if you have to decide between two projects and your analysis shows that Project A will yield USD 50 million with a likelihood of 80% and Project B will yield an outcome of USD 100 million with a likelihood of 20%, you would decide for Project A, with an expected value of USD 40 million (Project B: USD 20 million).

If you want to compare the outcome of bundles of recommendations, the expected value is calculated by multiplying each of the possible outcomes by the likelihood of each outcome and then summing all of those values for each bundle.

Links to practice resources

Regardless of your skill level, devote some time in your case interview preparation to brush up your mental and pen-and-paper math skills. If you are struggling with math, there is no reason not to practice case interview math drills 2 hours per day for a few weeks. It is important to have a structured consulting mental practice (as found in our math mastery product).

The basics (free online learning)

If you are just starting out, check out the Khan Academy, which is an excellent source to (re)-learn basic calculus such as additions and subtractions, multiplications and divisions, averages, percentages, and fractions.

Once you have the basics down, you should start practicing using some of the following free iOS and Android apps

iOS

Android

How we can help you get into McKinsey, BCG, or Bain

If you want to improve your case interview and problem-solving skills and learn the key habits and tricks that make you succeed in any McKinsey, BCG, and Bain interview, try our case practice. Many students that started their BCG careers, Bain careers, or McKinsey careers, went through our training.

Together, we spent 9 years with McKinsey and coached 100s of candidates to receive their desired offers. Our mock interviews have a 100% satisfaction rate and 9 out of 10 candidates that go through our Ready-for-McKinsey coaching program receive the offer. If you want to get the same knowledge in an extensive 40-part video series, check out our Ready-for-McKinsey video academy.

 

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