Calculations: When is an estimate not good enough?

Nearly all the math required in a case interview is arithmetic: addition, subtraction, multiplication, and division. You will likely need to perform these calculations with percentages, decimals or fractions, and calculate a percentage value (which requires division). Most candidates understand these math concepts; the challenge for many people is performing multiple calculations with large numbers quickly and without calculators or spreadsheets, in a high-pressure environment with a six-figure salary and your future career on the line.If practiced, the maths part of the case interview can be completed competently and without much stress. Demonstrating good maths skills can be your differentiator in your interview so it is worth spending the time to practice. Being quick is not only a major plus for your performance but it also helps keep a natural flow to your interview, helping you build rapport with your interviewer.Good mental maths skills are particularly useful in market sizing/guesstimate questions and consulting case studies.​Case numbers are “round” with few significant digitsThe good news is that most of the numbers you encounter in case interviews (what we call “Case Numbers”) are “round” numbers and have only a few significant digits (i.e. only a few digits are non-zero), but may be in the Millions or Billions. For example, the number 200 Thousand, which is 200,000 in long-form, has a single-digit that is not zero (“2”), and so has one significant digit. The number 2.5 Million, which is 2,500,000 in long-form, has two non-zero digits (“2” and “5”) and therefore has two significant digits. Case numbers frequently have only one or two significant digits, less frequently three significant digits, and in rare cases four or more significant digits. The term "Case Calculation" refers to a calculation required in a case interview, and may provide specific Case Numbers or refer to a general calculation without specifying the numbers involved, such as calculating Revenue, given Price, and Quantity. Examples​Here are some examples of the types of questions you would get in a case interview:​Q. A hairbrush manufacturer has fixed costs of £3,000,000 and they sell 175,000 units a year, what is the fixed cost per unit?​A. £3,000,000 ÷ 175,000 = £17.1​ Q. A global software company sells products in Germany, France, and the US. The German market had revenues of £2.35bn last year, France had sales 23% higher than Germany, what were their total revenues?​A. £2.35bn x 1.23 = £2.89bn Methodologies​Maths skills take practice but before you begin your practice it is best to choose calculation methods that you are most comfortable with. The method that you were taught when you were younger may not always be the method you want to use now.​Below we have found some methods that we think are good for speed and accuracy in pressure situations along with any useful tips or shortcuts that we are aware of. Addition and SubtractionMost of the Case Numbers you need to add and subtract will have only a few significant digits. If they have more than one significant digit, the last or “trailing” significant digits will often be “5” or “25,” which makes them easier to add/subtract. Example 1: Add 250 Million, 300 Million, and 150 Million.Answer: 700 MillionIn this additional example, the leading digits (prior to the Million) are: 250, 300, and 150. These numbers have either one significant digit (300) or two significant digits, where the last significant digit is a “5,” which makes addition easier.When you need to add Case Numbers with three significant digits, the last two significant digits are often “25” or “75,” which are also easy to calculate with. Example 2: Add 225 Million, 375 Million, and 200 Million.Answer: 800 MillionYou may need to add some Case Numbers with three significant digits that don’t end in “25” or “75,” but they most likely end in “5.” For example, you might need to add numbers like 115 and 165. Since these numbers both end in “5”, they are relatively easy to add without a calculator, and the answer is 280.It is highly unlikely you will need to add or subtract a series of numbers with three or more significant digits, where all the digits are effectively random, such as 147, 368, and 434 (where the last significant digits are not “25” or “75”).However, if you have to do so, we recommend the regrouping method.The regrouping method aligns the two numbers on top of each other by their units and then you sum each unit column to find the total number. It reduces the complexity of the equation down to single-digit additions and can be used for decimal places too. The long subtraction method we prefer is built on the same principle as the long addition and so you don't need to learn a new method but rather apply it differently. The two numbers are again aligned on top of each other and instead of summing each column, you subtract the two numbers.*box-open*For extra practice, check out our Quizzes on Logical Reasoning and Math Assignments for Case Interviews!*box-close*Multiplication, Division, and PercentagesAs you can see from the prior examples, candidates need to perform multiplication and division calculations using numbers in the thousands, millions, or even billions (but with a limited number of significant digits). In case interviews, candidates often need to perform multiplication and/or division with percentages, decimals, and fractions. For example, you might need to multiply a number by a percentage (e.g., calculate 25% of $500 Million), or divide two numbers and express the result as a percentage (e.g. what percentage of $80 Million does $16 Million represent?). Similar operations using fractions instead of percentages are also frequently required.Most of the Case Numbers you need to multiply or divide will be round numbers with only a few significant digits. For instance, the number of units sold for each product could be 5 Million, 10 Million, and 12.5 Million. In a case interview, you are unlikely to receive a similar problem where the number of units sold is something like 9,618,493, which has many significant digits that appear random. The percentage values you need to multiply/divide within case interviews will also usually have only a few significant digits. A typical Case Calculation would be calculating 20% or 25% of another number. It is unlikely you would need to calculate 23.7% of a value in a Case Calculation.Example 3: Calculate $120 Million times 250.Answer: $30 Billion*box-open*Don't know how to derive that answer? Check out our next article for practical tips & tricks*box-close*MultiplicationLong multiplication methods have the most variance of the calculations but most follow the simple principle of breaking the large numbers into their component parts e.g. 728 is 700, 20, and 8. To keep things as similar as possible across the different calculations we again choose to work vertically with the numbers aligned by units.DivisionDivision gets most difficult when dealing with decimals but the method we prefer allows you to continue into decimals seamlessly, this is particularly helpful when dealing with small amounts of money e.g. $3850.45, the 45 cents make the calculation more difficult.​This method moves away from the vertically aligned method but we find it easiest and most versatile for long division.PercentagesRelativity is an important concept in consulting case interviews as a number on its own does not demonstrate the value of the numbers in comparison to the other number(s) in the question. Being able to convert to percentages is an important skill and will often be required in case interviews.​There are a number of simple ways to find percentages but if you are stuck try finding 10%, 5%, and 1% first, using a combination of these amounts you will be able to find other percentages fairly quickly e.g. 37% = (10% x 3)+5%+(1% x 2). Compounded GrowthCandidates also need to understand compound percentage growth and how to make approximations with compound growth. It is a very common Case Calculation to be given a firm’s revenue (or another metric), and the associated Compound Annual Growth Rate (CAGR), and be asked to estimate that value at some point in the future. Net Present Value (NPV)Another common Case Calculation involves assigning a financial value to monetary payments that will occur in the future. Management Consultants usually determine the current value of future payments using a financial method called Net Present Value or NPV. Hence, candidates need to be able to calculate the NPV under a variety of scenarios. NPV Example 1: How much would your company be willing to pay for another company that generates $20 Million in profit annually, if your firm requires an annual Return on Investment of 10%?Answer: $200 Million NPV Example 2: A real estate development firm is evaluating a project that involves buying a parcel of land and building condominiums on that parcel. The company forecasts they can sell the condominiums for a total of $250 Million six years in the future. What is the maximum the real estate company would be willing to spend now to buy the land and develop the condominiums, if all the associated costs for the project would be incurred today, and they require a 12% annual return on invested capital? Answer: $125 Million*box-open*Don't know how to derive that answer? Check out our next article for practical tips & tricks*box-close* More Useful TipsTackling questions like this can be daunting. To learn how to approach questions like this, have a look at our 5 top tips:1. Don’t worry about getting the EXACT answerIt is important to remember that the interviewer is unlikely to be directly testing your maths skills, they just want to see how you approach the question so finding the answer to 5 decimal places is not important. They will usually be happy with an approximate answer, especially if it makes your calculations quicker. 2. Round the numberIf you have the choice, make the numbers easy for yourself. As an example, it is helpful to take the UK population as 60 million rather than its true value of 66 million. They are not always looking for perfect answers, just a good thought process and rough figure. 3. Shorten long numbersIf you have the opportunity to write down the numbers, shorten a thousand to ‘k’, million to ‘m’ and billion to ‘b’. This will help you write faster and keep the numbers smaller as too many 0’s can be confusing. 4. Verbalize your reasoning to the interviewerExplain out loud the calculation you are about to make. The interviewer may agree it is the right direction, they may give you part of the answer, or they could course correct and push you to a different part of the problem. It is important to give the interviewer the opportunity to help you as much as possible. 5. Make reasonable assumptionsOften you will be asked questions in which you are required to come up with the numbers yourself. These questions do not test accuracy but test your logic skills and common sense. As long as you are clear about the assumption you are making e.g. Every person living in the city gets their haircut once a month, then the interviewer can see your logic and challenge it if they think they need to. Helpful ResourcesMake sure to not only read through this article but to actually practice! There are a number of resources out there to help you with your maths skills and to improve your performance:PrepLounge Mental Math Tool - interactive and user-friendly tool to train your case interview math with respect to all basic operations (e.g. addition, multiplication, percentage). You can even compare your performance to the overall PrepLounge community!The Khan Academy - the website and app provides videos and practice for all your maths needs. They have exercises for all different levels and on an extensive range of topics.​BBC Skillswise - BBC has curated lessons to help adults gain skills for the workplace. Their maths skills section provides a good overview of the maths skills you would need.

When it comes to case interviews, data interpretation is a crucial skill that sets top candidates apart. Whether you're analyzing market trends, customer segments, or financial figures, your ability to quickly understand and communicate data-driven insights can make or break your performance. In this guide, we'll break down the process into three simple steps to help you excel in reading and interpreting charts and data during your interviews. Two Types of Data Interpretation in the Recruiting ProcessWhen preparing for your career in consulting, you'll encounter two primary scenarios where data interpretation comes into play:Solving specific tests (like the McKinsey Solve Game): This requires precise calculations and often involves mental math.Analyzing and communicating business insights: This is the core of case interviews, where you’ll need to interpret data, connect it to the case context, and use it to drive the discussion.This article will focus on the second aspect: how to analyze and communicate business insights in a case interview. If you're interested in improving your skills for specific tests, be sure to check out our resources on the McKinsey Solve Game and BCG Online Case.💡 Bonus Tip: The sooner you develop excellent data interpretation skills, the better. Check out your skill level now by taking our quiz on Graph and Chart Reading! A Three-Step Guide to Reading Charts and Data in Case InterviewsIn a case interview, you won’t be solving straightforward math problems; instead, you’ll be tasked with interpreting data to uncover business insights. Data in various forms is likely to make an appearance in many different types of cases. For example, when you ask for market, customer, or cost information while analyzing the case, the interviewer may provide you with a slide that has a few graphs detailing the various segments of the target market and several numbers (revenue, growth, etc.) pertaining to the segments. To work with this data effectively, you can follow a simple three-step approach: analyze, contextualize, and interpret. Step 1: Analyze – Understanding the DataWhen you're presented with data during a case interview, the first step is to analyze it thoroughly. There are three forms in which the data can be presented to you:Only numerical data is in a tabular form.Bar or pie charts with markings on the axes indicating units and reference data points.A mixture of tabular and graphic representations of data.Start by examining the exhibit. Reading the title will give you an initial understanding of what the data represents. Pay close attention to the axes, units, and labels, as these elements clarify what's being measured. For instance, if you're analyzing market size, quickly identify the most relevant numbers such as revenue and growth rates and use them to work through your analysis.Example: Suppose you're working on a case for a boot manufacturer that produces two types of boots: work boots and casual boots (👉 here you can find the whole case). You're tasked with planning a strategy in response to a competitor launching a new product. To understand the market, you ask the interviewer for market size data, and you're presented with the following exhibit: Take the following steps to analyze this data:Keep the question in mind: Always remember the question you asked the interviewer when they handed you the exhibit. Your primary job is to answer that question. In this case, you're focused on determining the market size – a figure in euros. Therefore, you must assimilate all the customer segmentation, product, and pricing data into the total market size.Study the exhibit: Begin by reading the title of the exhibit and then studying the axes and units of the data. This will tell you what the data is about. In this case, the exhibit is a mixed chart with both numerical and graphical representations. The first part shows buying propensity (the percentage of the population that bought boots last year). The data is presented as a percentage for different customer segments. Additionally, you have the average price paid and total population for each segment.Work the math: Figure out the most efficient approach to get to the final result, and do the calculations. If all the required data isn’t present, don’t hesitate to ask for it.For instance:Total revenue from all customers would equal the market size.Total revenue = Number of boots bought * Price per pairNumber of boots bought = Number of buyers * Boots bought per buyerNumber of buyers = (Total Population) * (% of the population buying boots)Now, do we have all the information required to calculate the revenue? Not quite yet. You need to know the number of boots bought per capita by each consumer segment. Quickly explain your approach to the interviewer and ask for the required information. Suppose the interviewer asks you to assume that every customer bought just one pair of boots in the last year.Now, you have all the information needed to calculate the market size for the total boots category. However, you cannot calculate the size of the work boots versus casual boots market separately since you only know the overall average price and not the category average for each. While these individual prices for the categories can technically be calculated using simultaneous equations, it’s overkill for a case interview, so you should ask the interviewer for this information to calculate the different segments.After completing the calculations, let's make some straightforward observations. Blue-collar workers buy more work boots than casual boots. Since they’re paying more than others on average, it could imply that work boots are more expensive than casual boots. Additionally, the work boot category is significantly larger than the casual boot category. Step 2: Contextualize – Connecting Data to the CaseAfter analyzing the data, the next step is to contextualize it within the broader case. This means linking the numbers back to the overall business problem you're trying to solve. For instance, relating the data to the case objective is crucial. If the case is about entering a new market, consider how the market size data influences your decision-making process. A large, growing market might suggest a strong opportunity, but you must also consider factors like competitive intensity and customer needs.By contextualizing the data, you ensure that your analysis supports a strategic recommendation. This means not just identifying what the data says, but also why it matters to the client’s goals. For instance, if the data reveals that a specific customer segment is both sizable and underserved, your strategy might focus on targeting this segment with a tailored product offering. This approach not only answers the immediate case question but also aligns your recommendation with the broader business objectives of growth and market penetration.In our boot manufacturer example, if you’ve determined that the work boot segment is larger and more profitable, and that it’s primarily driven by blue-collar workers, this information will guide your strategy. You might hypothesize that focusing on the work boot segment could be the most effective strategy for capturing market share from the new competitor, especially if you identify opportunities to differentiate your products or address unmet needs in this segment as a competitive response. Step 3: Interpret – Drawing Insights and Driving the DiscussionThe final and most important step is synthesizing all the analysis and business reasoning that you just completed and presenting it to the interviewer effectively. Communicate your insights clearly and concisely, ensuring that your conclusions are backed by the data. This approach not only demonstrates your analytical skills but also shows that you can use data to make strategic business decisions.In our boot manufacturing example, this could look as follows:"Our strategy should focus on further capturing the blue-collar worker segment, which is the most profitable for our work boots category. This segment accounts for [X]% of our total revenue, with an average price of [Y] euros per pair—higher than other segments. Given its significant contribution and [Z]% growth over the last [A] years, we should enhance product features such as durability and safety to meet their specific needs. Additionally, targeted marketing and partnerships with industrial employers or safety equipment suppliers could expand our reach and strengthen our position in this key market."Based on the Pyramid Principle of effective communication, the recommendation is communicated upfront, followed by the supporting data and rationale. Moreover, the hypothesis is clearly tied to specific data points, making the reasoning behind the strategy transparent and grounded in the analysis. Extra Tips to Work with Data and Charts in Case InterviewsMastering data interpretation requires more than just understanding the process during the interview; it’s also about thorough preparation. The following tips will help you sharpen your skills:1. Practice with real case studies: One of the most effective ways to prepare is by working through real case studies that include data-heavy exhibits. This will not only help you get familiar with the types of data presentations you might encounter, but it will also improve your ability to quickly extract relevant insights under time pressure. As you practice, focus on interpreting different types of charts, such as bar charts, line graphs, and pie charts, as well as numerical tables. The following cases can be a good starting point:👉 Company Case by Deloitte: Footloose 👉 Company Case by tkMC: Portfolio optimization of a holding company👉 Expert Case by Luca: Espresso, Whatelse? To really get comfortable interpreting data under pressure, simulate the interview environment. Set a timer, review a case with a data exhibit, and practice explaining your analysis and conclusions out loud. The best way to do this is by conducting mock interviews with peers. Check out our Meeting Board to find like-minded candidates to practice with. This will not only improve your speed and accuracy but also help you build the confidence needed to communicate your findings effectively in a real interview.2. Enhance your mental math skills: Although many cases don’t require complex calculations, being quick with mental math can significantly speed up your analysis. Work on improving your mental arithmetic, especially in areas like percentages, ratios, and basic multiplication/division. Our Mental Math Tool can get your math skills up to speed.3. Familiarize yourself with common data representations: Charts and graphs come in many forms, and each has its nuances. Spend time learning about different types of data representations – such as histograms, scatter plots, and stacked bar charts – and understand when and why they are used. This familiarity will allow you to quickly interpret the data during your interviews, as you’ll already know what to look for and how to analyze it. Final Thoughts About Charts and Data in Case InterviewBy following our three-step guide – analyzing the data, contextualizing it within the case, and interpreting it to drive the discussion – and integrating our preparation tips into your study routine, you’ll be well-equipped to tackle any data-heavy case interview with confidence. Remember, the key to success in these interviews isn’t just about crunching numbers; it’s about turning those numbers into actionable business insights. Keep practicing, stay curious, and approach each case with a strategic mindset.