# Math Skills Required in Management Consulting Case Interviews

This is the first of three articles in a series that discusses math skills in Management Consulting Case Interviews, and how you can effectively prepare for the quantitative component of Case Interviews.

### What is a Case Interview?

A “Case Interview” is a specific type of interview originally given to candidates applying to Consulting positions at Management Consulting firms like McKinsey, Bain, and BCG. These firms are some of the most prestigious companies in the world, and many future business leaders have worked at these and other Management Consulting firms. Anybody interviewing at a Management Consulting firm for a consulting position can expect Case Interviews with a quantitative component as part of their interview process. Former Management Consultants have subsequently taken jobs at other companies in Strategy, Marketing, Finance, Product Management or Business Development, and have take the Management Consulting-style Case Interview with them. Case Interviews have therefore started to be used to evaluate candidates applying for jobs outside of Management Consulting. In this series of articles the term “Case Interview” refers to the Management Consulting style of Case Interview, regardless of whether they are given by a Management Consulting firm, or another firm using the same style of interview.

The Case Interviews require the interview candidate to discuss, analyze, diagnose, and ideally solve a business problem encountered by a company. These “Cases” are usually based on real work that the Management Consulting firms has done for a client. Case Interviews frequently have a quantitative component that requires candidates to perform quantitative analysis without calculators, spreadsheets, or other electronic aids. A candidate’s performance in the quantitative component of the Case Interview often plays a significant factor in their overall evaluation and decision of whether or not to extend an offer.

### Math Problems within the Context of Case Interviews

A candidate’s quant skills are assessed within the context of a consulting “Case Interview,” which typically represents a scenario the consulting firm encountered when doing work for a real client. Some problems or questions frequently encountered in Case Interviews are: how a client can increase profits, increase revenue, or increase revenue growth rates; how a client can cut costs; whether the client should buy another company; or what the client’s international expansion strategy should be. Candidates will frequently need to quantitatively evaluate different potential actions to determine which one will have the most value to the client, which is often measured by the resulting increase in profit or valuation for a for-profit company, and other metrics for non-profit or government clients.

As an example, the overall Case could focus on identifying how a large multinational manufacturing firm could increase profits after having seen several years of declining profits (or increasing losses). As a candidate, you should begin by providing a Structure for the areas you would like to investigate further, and then ask more detailed Questions about each of these areas to specify what information you would like to gather. This phase of the Case Interview is meant to mimic the actual research you would conduct as a Consultant if you faced this scenario. For example, in your Structure, you might say that you would look at the different Products the manufacturing firm sells, the Competitors, overall Market Dynamics, and Cost information. For your more detailed questions, you might find it useful to know the Revenue and Profit for each Product the firm offers. You might also want to know Revenue and Market share for the Competitors. You would ask relevant and more specific questions for each topic in your Structure. If you identify the “Right” areas to investigate and ask the “Right” Questions, then the interviewer will usually provide additional quantitative Data and ask you to Analyze it.

The specific type of quantitative analysis required of candidates in Case Interviews varies substantially from interview to interview. You could be given numerical data and be asked to calculate metrics, where the specific mathematical operations you need to perform are relatively clear. An example of this type of calculation would be if you are given the Price and Quantity (number of units sold) for a series of Products and asked to calculate the Revenue for each Product, which would be the Price multiplied by the Quantity for each Product.

You could also be asked to perform more complex analysis where you need to identify the approach, and the specific calculations required may not be clear. For example, you may be asked how the client should prioritize their limited production capacity among several different products. You would then need to determine the criteria you would use to prioritize the products and perform the required analysis. You might be given data or shown graphs and asked what conclusions or insights you can draw from that data. Or, you may not be given any data until you ask specifically for data relevant to the Case. Once you ask the “proper questions” the interviewer will usually provide some guidance on the analysis they would like you to perform. Candidates are also frequently given charts, graphs or data tables and asked to interpret them.

After performing your analysis, you will usually be asked to provide Recommendations for the client, and you may be asked to Brain-Stormcreative ways to solve the problem. Finally, you will usually be asked to Summarize the overall problem, your research, and your “findings.”

### Case Flow

The diagram below shows overall flow of a typical Case Interview:

As a candidate, you will be expected to do the following in the quantitative portion of case interviews:

• Understand the quant question
• Determine what information or data is relevant
• Identify if additional data (beyond what has currently been given to you) is required and ask for that data
• Identify how to calculate the answer
• Perform the calculation
• Interpret the result and develop recommendations based on the result within the context of the business case

### Weight of Quant Skills in Interviews

Quant skills are only one of the skills assessed in Case Interviews. As a candidate, you are evaluated on your overall combination of skills and your perceived potential. Generally, a candidate’s quantitative assessment consists of 20–25% of the overall interview evaluation weight. The goal of Management Consulting firms is to find the set of candidates with what they perceive are the best set of skills overall.

Most firms have a “minimum bar” for quant skills for their candidates, which means you are unlikely to receive an offer if you do not exceed this bar — unless you are otherwise a highly exceptional candidate (such as being a Rhodes Scholar). At some firms, dramatically exceeding this “bar” does notstrengthen your candidacy — that is, the quantitative evaluation is binary: you either pass the quant assessment or you don’t. At other firms, such as McKinsey, if you can demonstrate very strong quantitative skills it will differentiate you from other candidates. The quant problems from these firms can be more complex, which allows candidates with strong quant skills to fully demonstrate their ability. The corollary is that, for some very complex quant problems, you don’t need to correctly solve all the components to receive an offer.

The relative importance of quantitative skills in the Case Interview varies by firm, office and the individual interviewer. As an example, if you are interviewing with a Consulting firm partner who has a PhD in a very quantitative field like Applied Mathematics or Physics, they may ask more challenging quant questions and place a greater emphasis on quant skills in their interviews. McKinsey, in particular, places more emphasis on quant skills than other firms, and, as mentioned previously, demonstrating exceptional quant skills can greatly strengthen your candidacy at McKinsey.

Given the large weighting of 20–25% that quant skills have in Consulting interviews — and the quant “minimum bar” at most Consulting firms — you can certainly sink your chances of receiving an offer if you perform poorly in the quant portion of the interview.

### No Calculators Allowed in Case Interviews

Be aware that, in Case Interviews, you cannot use calculators or spreadsheets. All your calculations must be done with pen and paper, or mentally.

There are a number of reasons calculations must be done without calculators or spreadsheets. First, the Management Consulting firms want their Consultants to be comfortable with numbers and don’t want employees to rely on calculators for simple calculations. Furthermore, clients pay a significant amount of money to hire Management Consultants, and the Consulting firms would lose credibility if their consultants make mistakes when doing mental or pen-and-paper calculations in meetings.

### Example Quantitative Case Interview Problems

#### Question Type 1: Break-even Analysis

A very common type of quantitative Case Interview problem is a Break-even Analysis, where you are given some financial information on a proposed project and asked to calculate how many units must be sold in order to “Break Even,” or recoup the initial investment. Below is a sample Break-even Analysis problem:

Question
Your client is evaluating whether to open a number of MRI scanning clinics. Each proposed MRI clinic would have two MRI machines, two Technicians and two Radiologists. The table below shows the annual cost for a single unit of each line item required to operate a clinic. Each MRI scan would generate \$420 in revenue and will have \$70 of costs for consumables, which are items that are consumed in the scanning process.

How many MRI scans would a clinic need to perform annually in order to Break Even on the costs of operating a clinic? Try to solve this without a calculator or spreadsheet.

If you are having trouble with this calculation, read below for a Hint

If you find this calculation difficult, keep in mind that there are two MRI Machines, two Technicians and two Radiologists per center, in addition to the cost of running the Facility itself. Using these values will make the math work out easier.

End of Hint.

Each MRI clinic would need to perform 3,000 scans annually to Break Even.

Watch this video to learn how to efficiently calculate this answer: Breakeven Analysis Video.

The next article has detailed explanation of how to solve all the problems in this article without a calculator.

Quantitative Conclusions & Recommendations

After finishing any quantitative analysis or calculation in a Case Interview, you should ask yourself what Conclusions you can draw and/or what Recommendations you should make based on the quantitative result you calculated. In short, you should be asking yourself, “So What?” That is: “What specific action(s) should the client take based on this result?”

As an example, let’s say you have calculated the net profits generated for each product in a group of products a client sells. Based on these calculations, you might recommend the company invest further in the products that generate the most profits and/or eliminate products that contribute the least profit. If you calculated annual growth rates for different product lines, you might recommend that the client invest in the products with the highest growth rates or highest future net profit. Your Recommendations should be based on the specific context of the Case and your analytical results.

Returning to the MRI Clinic example, the purpose of a Break-even analysis is to quickly estimate whether the Break-even Quantity is realistically achievable. An outstanding interview candidate would proactively identify the additional information you would need in order to determine whether the Break-even Quantity is achievable and then ask the interviewer for this information. If you don’t proactively do this, the Interviewer may ask you whether you think the Break-even Quantity is achievable. Proactively doing this type of analysis shows you can anticipate questions and interpret results, and is usually viewed favorably.

Information Required

To determine whether the Break-even Quantity is achievable, you would need to know the average duration of an MRI scan and the number of hours an MRI clinic is open per week.

Supplemental Information

The interviewer states that an MRI scan takes 30 minutes. Based on this information, do you think the Break-even Quantity is achievable?

To answer this, we’ll use a value of 50 weeks per year (rounding from 52).

So we will need to perform 60 scans per week. Remember there are two MRI machines per clinic, so we would need to perform 30 scans per machine per week. Since it is reasonable to assume that an MRI clinic would be open at least 40 hours per week, and each MRI scan takes a half hour, the Break-even Quantity is logistically achievable as long as sufficient customers can be attracted.

Follow-up Questions

a) In order to have profits of \$1.4 Million per year, how many scans per year would an MRI clinic need to perform annually? Is this Quantity achievable?

b) How much profit would an MRI clinic generate annually if it were open from 8:00 AM – 10:00 PM, seven days per week, and operated at full utilization (i.e. all time slots are used to perform MRIs on customers)? How much profit would it generate if it operated at 80% utilization or 50% utilization (i.e. only 80% or 50% of available time slots are used to perform MRIs)?

c) How many MRI clinics would the company need to operate at 50% utilization to generate total annual profits of \$100 Million from the MRI clinics?

a) You would need to perform 7,000 scans per clinic per year. There is sufficient capacity to perform this many scans, as long as the MRI clinic is open at least 40 hours per week.

b) Each MRI clinic is open for 14 hours per day, which is (14 × 7 = 98 ≈ 100) hours per week. Rounding the number of weeks in a year to 50 (from 52), and the annual costs of operating a facility to ~\$1 Million, gives an annual profitability of \$6 Million at 100% utilization.

80% ⇒ \$4.6 Million annual profit.
50% ⇒ \$2.5 Million annual profit.

c) To generate \$100 Million in total annual profits from MRI clinics, the company would need to operate 40 clinics at 50% utilization. The Conclusion is that this quantitative result tells the client how many clinics they need to open to achieve their target overall profits of \$100 Million. This analysis also provides information on required levels of utilization to achieve target profitability.

The next article has an explanation of how to calculate the answers to these problems.

The FastMath Ace the Case online course provides a comprehensive set of resources (with video-based instruction) for solving Case Interivew quant problems without calculators or spreadsheets.

#### Question Type 2: Revenue and Profit Calculations

It is a very common Case Interview scenario for the candidate to be given information on Price, Quantity and Profit Margin for a Product, or a group of Products, and then be asked to calculate specific metrics regarding Revenue and/or Profitability.

Example Question

A firm sells three products, with financial data for each product given in the table below.
a) What percentage of the firm’s overall Revenue does each Product contribute?
b) What percentage of the firm’s overall Profit does each Product contribute?

The next article has a detailed explanation of how to calculate the answers to this problem.

The FastMath Ace the Case online course provides a comprehensive set of resources (with video-based instruction) for solving Case Interivew quant problems without calculators or spreadsheets.

#### Question Type 3: Percentage Growth Calculations

A common Case Interview problem is to be given annual Revenue for a company (or a Product Line) and an annual percentage growth rate, and then be asked to estimate Revenue at some point in the future.

Example:

A company had annual Revenue of \$400 Million in its latest calendar year, and their Revenue is projected to grow at 4% per year. Approximately, what will the company’s Revenue be six years in the future?

In Case Interviews, it is appropriate to say the company’s Revenue will be approximately \$500 Million. Using a calculator, you can determine that the precise value is \$506.12 Million.

When given this problem, many people will try to accurately compound the annual growth, but this is very difficult to do without a calculator. In a Case Interview, you would want to calculate an approximate answer with much less effort than doing a detailed compound growth calculation.

The next article has an explanation of how to calculate the answers to this problem.

The FastMath Ace the Case online course provides in-depth explanations for howto perform Percentage Growth calculations in Case Interviews without calculators or spreadsheets.

#### Question Type 4: Market Sizing & Estimation Questions

Another common Case Interview question is to estimate the Market Size (i.e. annual sales) for a given Product Category in a country or region, or to estimate another figure related to a Market Size. The following are example Market Sizing & Estimation questions candidates were asked in real Case Interviews:

Examples:

1. What are the total annual sales for McDonald’s restaurants in the United States?
2. Approximately how many school buses are there in operation in the United States?
3. How many weddings occur each year in the United States?
4. How many gas stations are in the United States?
5. How many passenger automobiles are sold annually in Germany?

This is only a small sample of potential Marketing Sizing questions. The interviewer could ask you to estimate the market size for any product category in any country or geographic region. Interviewers are most likely to ask candidates to estimate the Market Size of products in the country in which the candidate is interviewing or in other local countries with large economies, in industrial countries (e.g. U.S., U.K., Germany, France), or in the largest developing economies in the world (e.g. China and India).

1. What are the annual sales for McDonald’s restaurants in the United States?
In 2016, McDonald’s restaurants in the United States had total sales of \$36.3 Billion. This includes sales at both franchise restaurants (which are owned by third parties) and sales at restaurants owned by McDonald’s Corporation (“Company owned”).
2. Approximately how many school buses are there in operation in the United States?
There are approximately 500,000 school buses in operation in the United States.
3. How many weddings occur each year in the United States
Approximately 2.4 Million weddings occur annually in the United States.
4. How many gas stations are in the United States?
There aren’t exact figures on the number of gas stations, and it depends on how a Gas station is defined. Most official figures state there are between 100,000 and 200,000 gas stations in the United States.
5. How many passenger automobiles are sold annually in Germany
From 2004–2015, between 2.9 Million and 3.8 Million cars were sold annually in Germany. In 2015, 3.2 Million passenger vehicles were sold in Germany.
Source: https://www.statista.com/statistics/416827/passenger-car-sales-in-germany/

The next article has an explanation of how to calculate the answers to this problem.

The FastMath Ace the Case online course has detailed video explanations of how to approach Marketing Sizing and estimation problems, and how to efficiently perform the required calcuations. You can see a video solution of estimating the revneu of McDonald’s restauratns here: McDonald’s Market Sizing Example.

Accuracy
Note that, for Market Sizing and Estimation problems, getting an answer within 2x or 3x of the actual number is usually considered a “good” estimate. In some cases, within an “Order of Magnitude (i.e. within 10x) is acceptable. You do not need to be within 10% or 20% of the actual answer to be considered successful in answering the Marketing Sizing or Estimation question. Of course, the closer you are, the better. Answers within 25% or 50% are probably considered accurate enough that closer estimates aren’t needed.

### Math Skills Required in Case Interviews

#### Overview

Nearly all the math required in Case Interviews 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.

#### Case Numbers are “Round” with Few Significant Digits

The good news is that most of the numbers you encounter in Case Interviews (what I will 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.

Most 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.

700 Million

In this addition example, the leading digits (prior to the Million) are: 250, 300 and 150. These numbers are either one significant digit (300) or two significant digits, where the last significant digit was 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.

800 Million

You 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”).

#### Multiplication and Division

As 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. In Example 2 with the table of Price, Quantity and Profit Margin, the number of units sold for each Product is 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.

\$30 Billion

The next article discusses methods to efficiently multiply and divide these types of Case Numbers.

The FastMath Ace the Case online course has video explanations of efficient methods for multiplying and dividing Case Numbers without calculators or spreadsheets.

#### Compound Growth

Candidates 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.

In an earlier example you were told that a firm had Revenue of \$400 Million in the past year, with Revenue growth of 4% per year, and you were asked to calculate their Revenue six years in the future. The next article discusses methods for performing Case Calculations that involve Compound Growth without calculators or spreadsheets.

### 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%?

\$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?

\$125 Million