Why is math an important component of case interviews?
When you relay your desire to break into management consulting, a typical quip that follows is the “How many golf balls can fit in an airplane?”. While most management consulting firms have moved away from these types of questions, they still expect you to confidently, and reliably, navigate mathematical problems on the fly in your case interviews. Regardless of your degree, or past experience, you’ll need to quickly and accurately calculate while still demonstrating that you understand the overall structure of the problem and the implications of your solution.
You might be wondering why, in an age where a calculator or Microsoft Excel can be accessed in seconds on your phone or laptop, interviewers still judge applicants on the strength of their mental math. The rise of Excel and other mathematical modeling software has definitely saved a lot of time and effort, but in interactions with senior executives, where the perception of competence is important, mental math will probably never go out of style.
The typically limited amount of time available to successfully deliver a client project, combined with the endless potential areas of focus, makes solid mental math skills essential to prioritizing key analyses before you even open up a spreadsheet. Strong mental math skills are also pivotal to obtaining and retaining credibility with clients, for example in meetings where you need to calculate values at a moment’s notice — e.g., the impact of a cost-saving measure, the revenue of a potential new product, etc.
The good news is that you don’t need to be a budding Fibonacci to ace math problems in a case study interview. Advanced calculus and other fields of complex math are not a feature of case study interviewers. You are most likely to face basic math and business concepts. Here, accuracy is more important than speed, and you won’t be penalized for working through problems with a pen and paper — in fact, it is strongly encouraged.
This article will walk you through all you need to know to ace math questions in a case interview, including an overview of the style of questions to expect, the skills necessary to ace case math, and how to prepare for case math. This site also has a comprehensive Case Math Course, provided as part of the Interview Prep Course, and other resources to help you ace your case interviews.
- Despite the proliferation of digital calculation solutions, strong mental math skills remain relevant in consulting for efficient analytics as well as obtaining and retaining credibility with clients.
How is math tested in a case interview?
In this section, we’re going to address the types of math problems you’re likely to encounter in a case interview. Generally speaking, there are four types of problems: straight calculations, exhibits that require calculations, word problems, and estimation questions.
Let’s start with straight calculations. It’s possible that an interviewer will give you a (relatively) straightforward math problem and ask you to solve it. Usually, however, these will take the form of follow-up questions to a larger case study that you’ve been working on. For example, in a product launch case study, the interviewer might ask you to calculate revenue based on a number of units and a price provided by the interviewer.
Exhibits requiring calculations
Next are exhibits that require calculations. Numerous calculations could be requested or proposed based on charts and exhibits. In the chart below, a candidate could be required to calculate the total population across all five towns, the number of small, medium, or large companies, and so on. Further external information could even be provided, or requested, to calculate additional values needed to crack the case.
Word problems are another staple of case interviews. You’ll almost certainly have encountered word problems during your schooling at some point or another. In the context of a case interview, word problems usually imply that algebraic calculations are in order. Candidates are expected to extract the important facts and figures from the word problem, present them in the form of an equation, and solve them. One common type of word problem which often crops up in case interviews is the “breakeven analysis.” Breakeven analysis requires you to calculate the amount of sales needed in order for a company to recover their costs, i.e., make no profit or loss.
For example, let’s say your client is a mid-sized Peruvian copper-mining firm, CopperCollect. They’ve invested $50 million in land purchases, development costs, equipment procurement, and hiring staff, and now they’ve approached your firm to find out how much copper they need to sell in order to recoup that investment. In the current market, CopperCollect expects to gross about $10,000 per tonne of copper sold. It costs the firm about $3,000 per tonne in order to extract, ship, and store the copper. In this scenario, you would divide the firm’s total investment ($50 million) by the net profit per unit ($7,000) in order to solve for X, the number of units that need to be sold in order for CopperCollect to recover their investment.
The last type of quant question you’re likely to encounter in a case interview are estimation and market sizing questions. Estimation questions are probably the most complex maths questions candidates are faced with during their interviews. In them, candidates have to come up with a value based on little, or no, upfront data. Famously, or perhaps infamously, these are the types of questions that usually trip candidates up in case interviews. How many petrol pumps are there in the United Kingdom? How many deep-water ports are there on the west coast of the United States? Estimation questions are a good way to demonstrate your ability to make common-sense assumptions and to extrapolate from a small amount of data using a solid structural framework.
Our tips to ace case interview math
As with so many other aspects of case interviews, there’s a formula you can use in order to structure your thoughts and analysis. When you’re confronted with a math problem, we recommend that you follow a four-step process.
Follow a four-step process
The first step before you begin to solve the problem is to verbally describe your approach. This will not only help structure the question in your mind but also it will give your interviewer the opportunity to correct any mistakes or incorrect assumptions you’ve made right off the bat. In a best-case scenario, the interviewer might even “sign off” on your approach, either verbally or nonverbally, by indicating that they agree with how you’ve decided to tackle the problem.
The second step is to work through the calculations, making sure to write everything down and describe what you’re doing verbally to the interviewer. This allows them to follow along more easily.
The third step is what is called a “sanity” or “sense” check. After you’ve completed your calculations, take a breath and try to adopt a bird’s-eye view of the problem. Does your solution make sense? For example, if you’re trying to assess the yearly revenue of a well-established multinational law firm and your final result is just a few thousand dollars, common sense should tell you that you need to return to your calculations and find the error.
Finally, step four is interpretation and presentation. You need to demonstrate to the interviewer that you can glean meaningful (and hopefully practical) insights as a result of your calculations. Ask yourself whether your solution is realistically achievable and whether it supports your initial hypothesis.
Keep workings tidy and simple
Throughout this entire process, you must ensure to write things down and to make sure that your calculations are tidy and simple. Many a candidate has been tripped up by an untidy workspace which can confuse the candidate and give off a bad impression to the interviewer, who will be expecting a professional and organized approach to the case study at hand. Therefore, you’ll want to use a fresh sheet of paper for each problem (and for each non-trivial calculation), write neatly and give yourself ample space on the page and align your equal signs on the page to make things visually tidier.
Pick friendly numbers when you can, compute one operation at a time, and avoid multiplying percentages together (this often leads to errors). Also, try to recognise when to use a table to synthesize different units or types of data you’re given. Finally, ask your interviewer if it’s okay to round numbers in order to make things easier — they are often comfortable with this and it demonstrates that you have your eye on the essence of the problem rather than trivial details.
Work with your interviewer
On that subject, it’s a good idea to work with your interviewer throughout the case study interview. As we mentioned earlier on, try and get your interviewer to sign off on your approach upfront. State your assumptions and ask them to fill in the gaps in your knowledge of certain data sets. For estimation questions, where this might not be appropriate, state your own assumptions so that the interviewer can follow along with your thought process. And, as you do your calculations, verbally describe everything you’re doing. Many candidates think it’s clever to stay silent and then impress the interviewer with the correct answer after a few minutes of quiet calculation, but the most successful candidates try to draw the interviewer into the process. This allows you to react in real-time to verbal or non-verbal feedback. Not only does this give the interviewer the impression that you are confident and work well in a team, but also it gives them the opportunity to correct any errors.
In the following video, you can see these principles being applied to a case interview math problem by a candidate in a case interview.
What kind of math do I need to know to ace my case interview?
There are three categories of math problems you need to know in order to conquer the math portion of most case interviews. They are basic math, key math concepts, and business math. Below, we’ll give you a detailed rundown on all three.
Addition, subtraction, multiplication, and division. You learned these four basic operations at school, and yes, they will be pivotal in case interview math. You might already feel confident that you’ve got them down, but it never hurts to brush up on the four operations, particularly without the use of a calculator, i.e., get out your pen and paper and practice! Feel free to skip this section if you’re confident you don’t require a refresher for the basic operations.
Let’s start with addition. Addition is probably the simplest of the four operations, but it can still be a cause of errors in a case interview when performed in your head. When performing addition operations, always start with ample room to do your calculations with a pen and paper, and arrange the numbers you want to add in clearly aligned columns. Start by adding the digits furthest to the right and carry excess numbers over to the next column to the left. The same principles apply whether you’re adding small numbers or big ones. You’ll rarely have to add numbers with decimal points together, and if you encounter a problem like that, it’s always okay to ask if you can round. If you do have to add numbers with decimals, the same principles apply: start on the right, move left across the columns, and make sure the decimal point doesn’t move!
It’s probably been a while since you’ve subtracted the kind of large numbers that you’re going to see in cases. Subtraction is quite similar to addition, in terms of how you’ll structure your calculations, with one major difference: you must always try to put the bigger number on top, and subtract the smaller number from it. If you encounter a scenario where you have to subtract from a smaller number, reverse the order and put a negative sign in front of the larger one. In some cases, you might have to “borrow” numbers from a column on the left to enable subtracting in your current column. For example, if you’re subtracting 49 from 72, you’ll need to borrow a 1 from the left column in order to transform the 2 in the right column into a 12, which will let you subtract 9 from it. Remember to remove 1 from the column from which you borrowed it.
Next up is multiplication, which is the most common of the four operations used in case interviews. Because of this, interviewers need to see that you can multiply figures quickly, confidently, and without errors — all in a high-pressure environment. When multiplying, line your numbers up, one figure atop the other, and start on the right, as with the other operations we’ve covered. If you’re dealing with multi-digit multiplication, simply multiply all the numbers on the bottom by all the numbers on the top (one at a time), and then add them together. When multiplying numbers with decimals, do the same as before and ignore the decimal till the end. Once you’ve arrived at your solution, add up the total number of digits to the right of decimal points at the start of the problem. For example, if you’re multiplying 9.31 by 4.2, you will have 3 numbers to the right of decimal points in the solution. Start to the right of your sum and count 3 times to the left to figure out where to place the decimal point.
Division is undoubtedly the hardest of the four operations to perform, whether in your head or on paper, especially in the high pressure environment of a case study interview. Because of this, always use a pen and paper and take extra care to review long division in case prep for your interviews. The video below walks you through everything you need to refresh your memory on division for case interviews. We provide similar videos explaining the other 3 operations in our Case Math Course, provided as part of the Interview Prep Course.
In essence, though you may remember the four operations from your high school days, it’s worth practicing so that you can walk into a case interview without hesitation.
Key math concepts
Once you’ve got the four operations down, you can move on to a few key concepts that usually crop up in case interviews. You’ll most likely have had some experience with some of these concepts, such as fractions, percentages, and simple growth. Beyond these, you must also have a good grasp of compounding, weighted averages, probability, and algebra.
Let’s start with the easier concepts — in this case, fractions. Put simply, a fraction is a division of whole numbers that have yet to be computed. The number on the top is the numerator, and it is divided by the number on the bottom, which is called the denominator. For example, in the fraction 26/4, 26 (the numerator) is being divided by 4 (the denominator). Computing that fraction gives you 6.5. Once you refresh your memory with these basic principles, you can move on to more complex operations with fractions, such as simplification of fractions and carrying out the four operations with fractions.
Most business metrics are expressed as percentages (e.g., gross/net margin, ROI). Calculating with percentages is, therefore, an essential key math concept to master. Fundamentally, a percentage is a proportion of a whole, represented as a number out of one hundred. There’s an easy way to calculate the percentage of a whole number. For example, if you need to find 40% of $500, change the percentage into a fraction — 40/100. Then multiply that fraction by $500, which gives you your solution: 40% of $500 = $200.
Multiplying percentages may sound easy, but it actually tends to trip up lots of applicants during their case interviews. A good rule of thumb is to always convert percentages into fractions first. For example, if you’re asked to find 40% of 70%, what you’re actually doing is multiplying 40% by 70%. Convert to fractions (40/100 x 70/100) and solve to get 28%. Here’s a good sanity check: your final answer should always be lower than either of the percentages you multiplied together, if these were lower than 100%.
Simple and Compounding Growth
If you’re ever asked to project the future value or growth of a business, you’ll most likely need to know about compounding, which denotes year-on-year growth. To understand this, you need to know the concepts of percentage change and simple growth.
Percentage change is an increase or decrease over a starting value. Percentage change can be calculated as: (V2-V1) / V1. Simple growth is the same as the previous calculation except that you calculate the second value given the first value and percentage growth. The formula for simple growth is P + (r x P) or P x (1 + r) where P is the principal and r is the growth rate, usually represented by a percentage.
The formula for compound growth, while similar, differs in a couple of key regards — it spans multiple periods and the principal changes every period. The formula for compounding interest is: P1 + (r1 x P1), plus P2 + (r2 x P2), continuing until the last compounding year “n” is addressed. It’s usually best practice to set up compounding growth problems in tabular form. An example of this can be seen below, where a 10% annual compound growth rate applied to $40 for 3 years is solved in a tabular format.
An organization’s performance is often summarized by one overall metric, whether it is sales or profit. This requires aggregating different segments, such as product lines or distribution channels. Because of this, candidates need to have a strong command of weighted averages, which is the overall average of a number of data groups with different weights. In this type of problem, a set of numbers (Q1, Q2, etc.) have corresponding weights (W1, W2, etc.). The sum of the products of each number divided by the sum of their weights is the weighted average.
A simple example is a scenario in which a grocer is selling 150 oranges at 20p each and 250 apples at 30p each. You are asked to calculate the average price of fruit being sold by the grocer. Q1 here is 20p and its weight (W2) is 150. The second number (Q2) is 30p and its weight (W2) is 250. Calculating this leads to (3,000 + 7,500) / 400. This gives 26.25p, the weighted average price of a fruit sold.
Organizations often make decisions based on uncertain outcomes. That’s why a basic understanding of probability is critical to making business decisions — and acing case math! The probability of an event is the number of ways an event can happen divided by the number of possible outcomes. A probability of 0, therefore, means an event will never happen and a probability of 1 means it will always happen.
Let’s say you’re calculating the probability of rolling an even number on a six-sided die. Because there are 6 possible outcomes and 3 even numbers, you would divide 3 (the number of ways the event — getting an even number — can occur) by six (the total number of possible outcomes) to get 0.5, or 50%.
To ace probability questions in case study interviews, you also need to learn or refresh your memory of how to identify the probability of multiple events, be they independent, dependent, or simultaneous events.
In most cases, laying out a clear and optimal approach to solving a math problem, especially in a case interview, requires writing an equation. While you’ve almost certainly done some algebra at school, we recommend taking a five-step approach to creating your own equations using algebra during case interviews.
First, write down all the known values. Secondly, assign a letter to the unknown value (the variable) that you need to calculate. Thirdly, create an equation in which both sides of the equals sign are equal. Next, isolate the variable on one side. Finally, solve the equation. Remember to sense-check to ensure that your units make sense (if using different units) and to make sure you do your operations in the correct order following the BEDMAS (Brackets, Exponents, Division, Multiplication, Addition, and Subtraction) acronym.
All key math concepts discussed here are covered in detail in our Case Math Course, provided as part of the Interview Prep Course.
Most roles that case interviews are used to select for are focused on improving business performance. It shouldn’t come as a surprise, then, that during a case interview you’ll probably be tested on business math. Too many candidates, particularly those without a business background, stumble on basic financial concepts. To ace business math questions in a case interview, you need to learn the essentials of financial statements (income statements, balance sheets, and cash flows), how to make an investment decision, value a business, and optimize operations.
The income statement shows a company’s income and expenditures for a defined amount of time — usually a year. It is also known as the “profit and loss statement.” While the income statement can contain numerous complex line items, a few concepts are common to most income statements and needed for case interviews.
Watch the below video for an overview of all you need to know about income statements for case interview math.
We provide similar videos about the other key business concepts described below in our Case Math Course, provided as part of the Interview Prep Course.
The aptly-named balance sheet compares a company’s assets (what it owns) against its liabilities (what it owes, i.e., its debts) and its equity — the amount invested into the company by shareholders.
There are three primary types of assets: current assets, which represent short-term economic resources like cash or inventory; fixed assets, such as buildings, real estate, and other resources not easily convertible into cash; and, finally, intangible assets, which include non-physical resources such as patents, trademarks, copyrights, and goodwill.
Types of liabilities include current liabilities, which generally must be paid within one year (an example of this would be accounts payable), and long-term liabilities, such as loans, which usually don’t have to be paid for over a year or more.
Finally, shareholder equity is usually divided into equity capital (money invested in the company by shareholders) and retained earnings (past profits not paid to shareholders as dividends).
All balance sheets follow this equation: Assets = Liabilities + Shareholder Equity. Balance sheets are essential because they allow you to see a snapshot of a company’s assets, liabilities, and equity at a given point in time. This is especially useful when you’re trying to calculate a company’s cash flow, which you can find by comparing its balance sheets from two different points in time.
There are three types of cash flow: cash from operating activities (generated from goods or services), investing activities (generated from a company’s investments), and financing activities (from investors, including banks).
Cash flow differs from profit in that a company could conceivably be reporting regular profits but simultaneously experience cash flow difficulties because reported profits may not be received immediately. Similarly, a company might report very little profit but still have excellent cash flow as investors pour money in.
Finally, it’s important to be aware of EBITDA (Earnings Before Interest, Taxes, Depreciation, and Amortization) which is sometimes used to estimate cash flows, and of working capital, which is the current assets available for a company to use as it wishes and a good proxy for solvability.
Many cases are focused on whether a company should make an investment. To handle these questions, it is important to understand the analyses typically performed ahead of investment decisions. Candidates will need to understand and be able to apply the concepts of breakeven analysis, return on investment (ROI) and payback period to tackle investment questions in a case study interview.
For a business, the breakeven point is the sales volume that needs to be reached to turn a profit. This is, therefore, the volume where no profit or loss is made. To find the breakeven point, you need to calculate the volume where revenue equals total costs.
ROI (return on investment) shows how well an investment has performed. ROI is usually represented as a percentage and is calculated by dividing the profit or loss from an investment by the amount initially invested.
Payback period indicates how long it will take to recover an investment or reach the breakeven point. To calculate this, simply divide the cost of the investment by the annual return of the investment. Overall, short payback periods are considered better than long ones.
These, as well as a calculation shortcut known as the rule of 72, are all covered in detail in our Case Math Course, provided as part of the Interview Prep Course.
Although you’ll very rarely be asked to value a business in a case, understanding valuations may help you structure your approach and understand exhibits in relevant cases. In short, valuation is the process of determining the value of an asset or a company. Generally, that means calculating either its market capitalisation (the cost of purchasing all the shares of a company, calculated simply by multiplying the cost of one share of the company by all its shares and debts) or enterprise value (more useful when comparing similar businesses and calculated by subtracting cash and equivalents from the sum of market capitalisation and total debt).
Two key valuation methods that might be used in case interviews are comparables analysis and discounted cash flow, or DCF. While you do not need CFA-level expertise in these valuation methods, understanding the concepts will help you make the right judgment call in a case interview when faced with questions containing elements of these methods.
Many businesses operate a process to produce a standardized good or service. Some case interviews are based on optimizing such processes. Unless you are applying to an operations specialist role, knowledge of capacity, utilization, and efficiency should be sufficient to tackle potential case math questions.
Capacity is the maximum amount of output that can be produced in a given time frame. Utilization is the amount of capacity being used over a set interval. Lastly, efficiency (specifically process-cycle efficiency) is the amount of value-added time (time spent actually making a product or delivering goods) in the lead time (total time till output) of a process.
All the above-mentioned concepts needed to ace case math in your interviews are covered in detail in our Case Math Course, provided as part of the Interview Prep Course.
Once you have a good command of the above mathematical concepts and problem types, you can focus on brushing up on your skills by practicing what you already know with our calculation drills, provided as part of the Interview Prep Course.
What you don’t need to know
There’s no need to learn advanced math, such as calculus or number theory. You also do not need to start buying textbooks on advanced finance or know how to perform detailed DCF (discounted cash flow) valuations. And, unless you’re applying for very specific roles, you can leave aside industry-specific concepts that have little relevance to the general business community.
How to stand out in case math
As we said at the beginning of this article, you don’t need to be a mathematical prodigy to impress even the most competitive management consulting firms with your skills during case interviews. Nevertheless, demonstrating proficiency and confidence with math will give you more time to work on other aspects of case problems, and impress the people interviewing you. Here are three pro tips that can help you stand out in your case interviews.
1) Keep track of zeroes
The number one math mistake that we observe in case interviews is miscounting zeros. Case questions often deal with large numbers, sometimes in the millions or billions. Because of this, it’s easy to misplace or leave out a zero, making your entire calculation inaccurate. You can avoid this using one of three methods of keeping track of zeroes: a) counting zeroes, b) using scientific notation, or c) assigning letter units. All three methods are detailed in our Case Math Course, provided as part of the Interview Prep Course. Try out all three methods and decide which works best for you.
2) Simplify your calculations
Being fast and reliable does not necessarily mean being a human calculator. The secret to being fast and reliable, and impressing your interviewers, is to make life easier for yourself by simplifying calculations.
There are multiple ways to simplify calculations in case interview math. One of them is by rounding numbers. Candidates in a case study interview typically have two opportunities to simplify calculations with friendly numbers. The first opportunity is when making assumptions, for example with population. The second opportunity is within calculations, by rounding numbers as you go. For example, in a calculation involving the US population and life expectancy, rounding the population to 320 million (actual value is ~330million) and rounding life expectancy to 80 years (actual value is ~79 years) will make for much faster calculations.
Effective rounding should ideally not change the answer by more than 10% and will save you significant time when tackling a case math problem. Whatever the margin, however, it is super important to always communicate any rounding assumptions to the interviewer.
Our Case Math Course, provided as part of the Interview Prep Course, contains three lectures with details on ways you can simplify your case math calculations with distributive properties, factoring and rounding.
3) Memorize frequently used factions
Some fraction values are used so often in case math that knowing them, their percentage value and their decimal conversions can save you significant time. First, remember that divisions can be represented either as fractions, decimals, or percentages. For example, the fraction 50/100 = 0.5 = 50%. A good rule of thumb is to memorize the fraction and corresponding decimal and percentage values of 1/2, 1/3, all the way through to 1/10.
How to prepare for case interview math
While you may think it’s a waste of time to brush up on your math skills, you’re probably a little out of practice when it comes to doing calculations by hand after years of Excel and calculators. Plus, remember that you’ll be doing these calculations with a pen and paper in a high pressure and time-sensitive environment. So, take the time to go over all the concepts mentioned in this article.
Our Case Math Course, provided as part of the Interview Prep Course, covers all you need to ace case interview math. It includes comprehensive but concise video lectures on the four operations, key math concepts, CaseCoach’s pro tips, and business math. Dig in whether you want to refresh your math skills, fill any skills gap, or bring your speed and accuracy up to the next level.
Once you’ve familiarized yourself with the concepts, it’s time to practise, practise, practise. You can, of course, practise these calculations by yourself, but we strongly recommend that you do so in the context of a practice case question.
Our Calculation drills (included in our Interview Prep Course) offer timed question sets across various topics that can help you boost your calculation speed and accuracy. For case math questions typically accompanying charts and exhibits, try out our Chart Drills (also included in our Interview Prep Course) to learn how to stand out in your case interviews.