**Case Interview Prep**

# Preparing for the Quantitative Component of Management Consulting-style Case Interviews

### Many candidates aren’t sure what to expect in, don’t prepare adequately for, and/or don’t know how to prepare for, the quantitative component of Case Interviews. This article explains it all…

## About the Author

### Matthew A. Tambiah

- Former McKinsey Consultant
- Bachelors from Harvard
- MBA from MIT
- Creator of FastMath™

Matthew is a former McKinsey consultant, and graduated with highest honors from Harvard with a Bachelor’s in Electrical and Computer Engineering, and also holds an MBA from MIT. He has conducted workshops on quantitative skills for job interviews at Harvard Business School, MIT, Wharton, Columbia, London Business School, INSEAD, and many other leading schools.

**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 from companies like McKinsey, Bain, and BCG have subsequently taken jobs at other companies in Strategy, Marketing, Finance, Product Management or Business Development in many industries like software and technology, pharmaceuticals, financial services, and consumer products — they’re everywhere — and have taken the Case Interview with them. Case Interviews are therefore frequently used to evaluate candidates applying for jobs in many industries and roles outside of Management Consulting.

The term “Case Interview” in this article refers to the style of Case Interviews used in Management Consulting, 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 a real project the Management Consulting firm has done for a client — internally, they call these client projects ‘Cases.’ These 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 — often measured by the resulting increase in profit, revenue or valuation for a for-profit company, and/or other non-financial 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 you would look at the different Products the manufacturing firm sells, the Competitors, overall Market Dynamics, and Cost information. For your more detailed problems, 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 percentages 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 are then asked to calculate the *Revenue* for each Product, which would be the *Price* multiplied by the *Quantity* for each Product. You then be given Cost or Gross Margin information, and asked to calculate overall Profits. Next, you might be given a target overall level of Profit or Gross Margin, and asked what Price or Quantity for a single Product (while keeping all other variables constant), is required to hit the target — several examples appear later in the article.

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 **Brainstorm **creative 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 the 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
- Develop recommendations based on the result within the context of the business case

There may multiple iterations in the flow, where the results of one phase of Analysis will prompt more questions, which will lead to more Data, and further Analysis. Two phases of analysis is common, and some Cases may require three or more phases of Analysis to complete.

**Example Quant Case Interview Problems**

To illustrate the quant skills needed in Case Interviews, below are a few examples of quant problems and calculations given in real Management Consulting Interviews.

**Example 1: Beverages & Gross Margins**

A beverage manufacturer produces and sells three types of drinks: Vitamin Water, Energy Drinks, and Advanced Muscle. *Revenue*, *Quantity*, and *Cost per Unit* are shown in the table below.

**Question 1:
** What is their current

*Gross Profit Margin*?

**Question 2**:

What would the *Price* of Energy Drinks need to be to achieve an overall *Gross* *Profit Margin* of 40%? Assume the *Quantity* of Energy Drinks and all other products remains the same.

**Comments**

This is a fairly analytical quant Case Interview problem. The context for this Case was that the client’s profits had been declining and they were investigating options for increasing their profit margins. One option is raising prices. Before doing so, the company would want to do a thorough analysis to determine what the impact on gross margin is of raising prices, how much they would need to raise prices by (the question), and be sure that doing so would not cause them to lose market share. For example, if they needed to increase their *Price* substantially relative to competitors, that would likely cause the company to lose market share.

**So What?**

After doing a calculation in a Case Interview, you need to ask yourself: “What should we (i.e., the client) do based on that information?” That is — *“So What?!” *

Usually in Case Interviews there is a take-away from the calculation that is related to the Case. That is, the consultant is expected to make a recommendation based on the result, and the recommendation would be different depending on what the result is or was. The Cases in interviews are usually based on real client engagements and the companies usually pick Cases to use in interviews that have clear recommendations from the analysis. Also, think about how the recommendation would vary depending on what the numerical result was? What would you recommend if the answer is 10% compared to an answer of 80%?

The FastMath Ace the Case Online Course has a detailed solution method for this problem and discusses in further detail how to take the quantitative result of the analysis and make case-specific recommendations — i.e. how to determine the *“So What?!”*

**Example 2: Competitive Labor Costs**

Your client (Consumer Co.) is a U.S.-based manufacturer of consumer appliances like microwaves, refrigerators, and air conditioners. They distribute primarily through retailers, and conduct all their manufacturing in the United States. Their profitability has been reduced over the past few years because a foreign competitor (Foreign Co.) has been selling comparable products in the U.S. at lower prices.

Your client would like help analyzing the cost structure and P&L of Foreign Co. Your team has gathered cost data for both firms, which is shown below.

**Question:** What is the relative cost structure for Foreign Co. compared to Consumer Co.?

In addition to the table above, your team has gathered the following information:

U.S. Labor Rate: |
$25 per hour |

Foreign Co. Overseas Labor: |
$2.50 per hour |

Overhead: |
40% of costs related to Labor |

Other Overhead costs are 1:1 between the two companies | |

Material: |
30% of costs related to Labor |

Other Material costs are 1:1 between the two companies | |

Transportation: |
Foreign Co. has an 8% increase in costs to transport products to the United States |

**So What?**

How is this information relevant? If there was a 10% difference, what would you recommend? If there was a 50% difference, what would you recommend?

**Example 3: Analog to Digital**

Your client manufacturers and distributes analog medical image equipment (e.g., x-ray machines, CT scan machines), and film for medical imaging. They have long-term contracts with many hospitals. They have seen prices and margins decline as more and more imaging moves to digital imaging, away from consumption of analog imaging.

Financial data for the most recent fiscal year is shown below:

**Question:**

In the past 10 years, the client has seen *Revenue* from Film increase by 25% and number of *Units Sold* of Film increase by 66%. What was the net percentage change in the *Price* of Film over the same time period? What is the approximate *annual* percentage change in the *Price* of Film?

**So What?**

The rate of *Price* decline has a significant impact on the strategy this company should pursue, and how they should pursue it.

**Comments**

Most MBA students I have given this problem to have trouble calculating an accurate answer without a calculator. FastMath Ace the Case f will show you how to calculate the answer to this problem quickly and easily, without even calculating the *Revenue* from Film, or the *Price *of Film, 10 years ago. *How* is this possible? You’ll have to take the course…

**Example 4: Breakeven & Profitability Analysis**

A very common question is to calculate the number of units of a certain product that need to be sold in order to “Break Even.” Related problems are finding the number of units a company needs to sell to achieve a certain amount of profit, or finding the Profit given a production quantity.

Your client is considering opening a number of MRI scanning clinics. Each MRI clinic will have two MRI machines, two Technicians and two Radiologists. The table below shows annual costs for a single unit of these and other line items required to run a clinic. Each MRI scan will generate $420 in revenue and will have $45 of costs for disposable medical products, and $25 of cost for other consumables (i.e., items consumed in the scanning process).

**Question 1:**

How many MRI Scans will a clinic need to perform annually in order to break even?

**Question 2:**

How many scans would be required to achieve $1.4 *M* annual profit per clinic?

**So What?**

The entire purpose of a Breakeven Analysis is to quickly estimate whether the Breakeven Quantity is realistically achievable. You might need some additional information in order to determine this. An outstanding interview candidate would proactively identify the additional information you would need in order to answer this question and ask the interviewer for this information. If you don’t proactively do this, the Interviewer may ask you whether you think the Breakeven Quantity is achievable. Proactively doing this type of analysis shows you can anticipate questions and interpret results, and is usually viewed positively by interviewers.

So, what information do you think you’d need to answer this question?

**Question 3:**

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* available 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)?

**Example 5: 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 the Market Size. The following are example Market Sizing & Estimation questions candidates were asked in real Case Interviews:

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

This is only a small sample of potential Market 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., UK, Germany, France), or in the largest developing economies in the world (e.g., China and India).

Try estimating these values before reading the **Answers** below.

## Market Sizing Answers

**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”).

**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.

**How many weddings occur each year in the United States?**

Approximately 2.4 Million weddings occur annually in the United States.**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, and which sources you refer to. Most official figures state there are between 100,000 and 200,000 gas stations in the United States.

**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/**

FastMath Ace the Case has detailed video explanations of how to approach Market Sizing and estimation problems, and how to efficiently perform the required calculations. You can see a video solution of estimating the revenue of McDonald’s restaurants here: **McDonald’s Market Sizing Example**.

**Desired Accuracy**

Note that, for Market Sizing and Estimation problems, getting an answer within 2*x* or 3*x* of the actual number is usually considered a “good” estimate. In some cases, within an “Order of Magnitude (i.e., within 10*x*) is acceptable. Some interview preparation resources state that you need to be within 10%–20% of the actual value for your answer to be considered “correct” — this is generally *false!*

Given the level of uncertainty on most of these questions, accuracy to within 10%–20% is often not possible or would take a considerable amount of time, and usually isn’t what the interviewer is looking for. Instead, I recommend that you try to get within 2*x* to 3*x*as *fast* as possible. Of course, the closer you are, the better, but there is a tradeoff between speed and accuracy. Even answers within 10*x* are often useful for clients.

**Example 6: Compound 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, or some point in the past. The growth rates can even vary over time.

**Example 1:**

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?

**Example 2:**

A PC manufacturing firm had the number of PCs sold grow by 4% per year for eight years, and then shrink by 3% per year for two years. What was the net change (as a percentage) in PCs sold over that 10 year period?

**Example 7: Margin Math**

A manufacturing company wants to have a 20% profit margin on all goods sold. Product A costs $50 to manufacture — what is the selling price?

What should the *Price* be of Product B, which costs $140 to manufacture?

How about a product that costs $30 to manufacture?

How about a product that costs $45 to manufacture and would need a 30% *Profit Margin*?

**Why Management Consulting Firms Assess Quant Skills**

Before going further, I think it’s worth explaining why Management Consulting firms, and other companies, assess a candidate’s quantitative (quant) skills. The short answer is that strong math and quantitative skills are essential skills for Management Consultants.

Clients frequently engage Management Consultants to develop recommendations that will maximize profitability or valuation for the client. To do this Management Consultants perform comprehensive quantitative analysis and model the financial results of a range of options the client could pursue. Therefore, the ability to perform quantitative analysis on data and develop insights and recommendations based on that quantitative analysis is a critical skill for Management Consultants.

In addition to being able to perform quantitative analysis, effective Management Consultants need several additional quantitative skills. They need the ability to determine what kinds of quantitative analysis are relevant to accomplishing the client’s specific goals. Furthermore, Management Consultants need to be able to identify what specific data and information are relevant to the client’s specific problem, and are required to perform the necessary analysis. Having these skills allows Management Consultants to collect only the data relevant to the problem, and to determine what additional data is needed beyond what is currently available.

Clients hire Management Consultants primarily for fact-based analytical research and quantitative analysis, and not for “creative” or “communications” work, such as developing advertising campaigns or for PR/communications. Companies will usually work with a specialized firm like an advertising agency, PR or communications firm for these needs. Historically, Management Consultants have not been very involved in the creative aspects of Product Design. Frequently, the quantitative and analytical work Management Consultants do will influence Product Design by showing that certain consumers’ needs are unmet . Recently, Management Consulting firms have begun to start doing more Product Design (often called “Design Thinking”), and McKinsey recently acquired several Product Design consulting firms (Lunar and Veryday). Management Consulting firms are generally not involved with the creative aspects of advertising.

Since quantitative skills are required to be an effective Management Consultant, Management Consulting firms naturally assess these skills in their interviews. Leading Management Consulting firms also have a surplus of qualified candidates, which would be magnified if the consulting firms did not eliminate people for having sub-par quant skills or did not include quant skills in their overall assessment of a candidate. For these reasons Management Consulting firms use a candidate’s quantitative skill level as an important factor in selecting candidates.

Management Consultants are often hired to provide advice on the best course of action, which naturally requires modeling the expected financial impact of each of these actions. Many of the quant problems in Case Interviews therefore ask candidates to calculate the financial impact of various actions and select the best option considering all information available.

## 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 purely 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.

Also, consider the client perspective — do you know how much clients pay per consultant?

In short, Management Consulting firms evaluate a candidate’s quant skills because quant skills are important — and because they can; top Management Consulting firms can still find enough qualified candidates if they include quantitative skills in their evaluation criteria. Consider it from the Consulting firm’s perspective: given two strong candidates who are otherwise equal — except that one has strong quant skills and the other has poor quant skills and could potentially damage the firm’s credibility with clients — which candidate would you choose?

**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 not strengthen 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 difficult 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.

**How to Prepare for the Quantitative Component of Case Interviews**

The best way to prepare for the quantitative portion of Case Interviews is to practice the **types** **of calculations** you will need to perform on the **types of numbers** you are likely to encounter in Case Interviews and then learn how to apply these methods to the **types of problems** you are most likely to encounter in order to solve them quickly and efficiently.

**Properties of Case Numbers**

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*.

As a reminder, calculators and spreadsheets are not permitted in Case Interviews, but candidates may use pen and paper. All the calculations required in Case Interviews (Case Calculations) take this into account, and the specific numbers involved in Case Interviews (what I will call “Case Numbers”) often have specific properties you can use to simplify the calculations — if you *know* the proper method!

**Math Skills Required in Case Interviews**

The specific math and calculation skills and knowledge required in Case Interviews is rather narrow. The specific calculation skills you need are:

- Addition
- Subtraction
- Multiplication
- Division

That’s it!

You’ll need to be able to perform these operations with large numbers, fractions, decimals and percentages.

In addition, you’ll need to understand the following mathematical concepts:

- Percentages and percentage change
- Compound percentage growth
- Averages and expected value
- Algebra: solving linear equations with one or two variables; if there is a second variable, it is often easily eliminated.
- Net Present Value (NPV)

People generally learn the basic calculation skills required for Case Interviews in elementary school, and cover the mathematical concepts needed in their first year of Algebra, which many people take by the age 13 or 14. The only exception is Net Present Value (NPV), which is relatively straightforward and for which there are many resources available to learn. The calculations methods for NPV problems required in a Case Interview are covered in the FastMath Ace the Case Online Course.

Most candidates have learned these math concepts but encounter several challenges in Case Interviews. The first significant challenge is that many people have trouble performing the calculations required in Case Interviews with large numbers quickly and without calculators or spreadsheets — the pressure is further compounded (pun intended) given that a six-figure salary and your future career are on the line.

Another significant challenge for many people is that they are given types of question they aren’t familiar with, and don’t know how to solve. That is, they aren’t even able to identify the specific set of calculations that will give the correct answer — this is even worse than performing the calculations incorrectly!

If any of the problems in this article fall into this category for you, then you should definitely take FastMath Ace the Case.

**Addition and Subtraction**

While the numbers usually have only a few significant digits, the values can be in the thousands, millions or billions, so some candidates find it challenging to manage the zeroes. Furthermore, the numbers sometimes involve decimals. If the numbers have two or more significant digits, the last significant digit will often be “5” — but not always.

**Example 1:** Add 250,000,000, 150,000,000, and 300,000,000, and subtract 125,000,000 from the result.

**Example 2:** A company has *Revenue* from two products of $15.2 Million and $3.1 Million, respectively, and combined costs of $5.7 Million — what is their overall *Profit*?

It is highly unlikely you will need to add or subtract a series of numbers with more than **three** 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**

Multiplication and division are two of the most common calculations (operations) required in Case Interviews — and ones where candidates frequently make mistakes. Multiplication is very common, and required for calculating *Revenue* and *Profit*, and in Market Sizing and Estimation Problems.

Candidates are often given *Price* and *Quantity* and asked to calculate *Revenue*, or given *Price*, *Quantity* and *Profit Margin* and are then asked to calculate *Profit*. These calculations require multiplication as:

Furthermore, you may be given other parameters and asked to solve for *Price*, or asked to calculate a *Gross Margin* or *Percentage Change*, which would require 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 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.

**Multiplication Example:** Your client sells a product for $160 per unit and sells 35,000,000 units per year.

Many people have trouble with handling the zeroes for a calculation like this, and doing the multiplication without writing out all the numbers involved.

I’ll show you the *FastMath* solution method to the above problem, which will show you a simple way to handle this calculation without longhand multiplication. The steps in the *FastMath* solution method are different from how most people would normally perform the calculation, but each step is very simple.

- Don’t write out all the zeroes in 35,000,000 — we’ll replace them with “Million” or just ‘
*M*’ for short. This leaves us with 160 × 35*M*

- What’s 160 ÷ 2?
- What’s 35 × 2?

- What’s 80 × 70
The original multiplication problem of 160 × 35 would require four single-digit multiplication operations and several additional operations. We simplified this to 80 × 70 that requires one non-zero single digit multiplication operation (8 × 7), which is much simpler.

- So we are left with 5,600 Million, which is 5.6 Billion, which is the final answer.

FastMath Ace the Case provides a comprehensive set of methods for multiplying and dividing Case Numbers. I’ll teach you how to utilize the specific properties of the numbers themselves to efficiently perform the required calculations for any set of numbers you are likely to encounter in a Case Interview.

I’ll teach you methods for handling all the zeroes, without writing out all the zeroes. The approach is incredibly simple and easy to learn, and the full online course provides numerous practice problems to reinforce the material.

**Compound Growth**

Candidates also need to understand compound percentage growth and how compound growth behaves in several situations. 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 or in the past. The growth rates can even vary over time.

**Example 1:**

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?

**Example 2:**

A PC manufacturing firm had the number of PCs sold grow by 4% per year for 8 years, and then shrink by 3% per year for two years. What was the net change (as a percentage) in PCs sold over that 10-year period?

Candidates therefore need to be comfortable making compound growth calculations in Case Interviews. FastMath Ace the Case provides a comprehensive set of methods for quickly and efficiently performing compound growth calculations for Case interviews.

**Net Present Value**

This is an application of Compound Growth, and is the only concept not usually covered in a high school Algebra class. 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%?

**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?

**Mental Math Is Extremely Valuable in Case Interviews**

Because you cannot use calculators in Case Interviews — and because you may need to perform a large number of math operations — being able to do calculations mentally (without relying on pen and paper for the mechanics of calculations) is very helpful during Case Interviews.

Consider the table below, which gives *Price*, *Quantity* and *Profit Margin* for three different products, and imagine you are asked to calculate the percentage of the firm’s overall *Revenue* and *Profit* contributed by each product.

If you are able to do these operations purely mentally (and write down the answers) without writing out all the multiplication operations in **Longhand,** you will have a significant advantage in calculation speed. Longhand multiplication refers to doing a multiplication calculation like 7 × 25 as shown below:

In order to do longhand multiplication for a particular calculation, you would first need to copy the multiplication terms from the table onto pen and paper, and then execute each calculation. Since several multiplication operations are required to calculate the *Revenue* and *Profit* for **all** the Products, a person dependent on Longhand multiplication will be **much** **slower** than a person who can do the calculations mentally, as somebody who can do these calculations mentally can simply look at the table, do the calculations, and then write down the answer(s) to refer to later. They won’t need to write out the original numbers from the table to perform Longhand multiplication, and so will be much *faster!*

The speed improvement of mental calculations is magnified if you need to multiply several numbers together when calculating a value.

For example, in Case Interviews you frequently need to calculate *Revenue* and *Profit* given *Price*, *Quantity* and *Profit Margin* for multiple products, which requires multiple multiplication operations (say that twenty times fast).

For mental calculations, it is very helpful to “mentally picture” the numbers with which you are calculating — which I may refer to as “thinking” of a certain number or numbers. After doing a calculation mentally, you should always write down the result so you don’t forget it.

Given the benefits of mental calculations in Case Interviews,* FastMath Ace the Case* teaches a comprehensive set of mental calculation methods specifically designed for Case Interviews. These methods are also helpful for pen-and-paper calculations for those not fully comfortable doing the calculations purely mentally.

**Quant Problem Types**

Once you have learned the basic calculation skills required for Case Interviews, you’ll need to learn how to apply them to the different Quantitative Problem Types you are likely to encounter in a Case Interview.

**Solving Breakeven & Profitability Analysis Problems**

One of the simplest question types concerns what I will call Simple Breakeven Analysis and Profitability Analysis. In this problem type, you will be asked to calculate the number of units of a certain product that need to be sold in order to “Break Even,” or to calculate the number of units a company needs to sell to achieve a certain amount of profit, or to find the *Profit* given a production *Quantity*. This problem type assumes a constant *Marginal Cost*, a constant *Price*, and *Fixed Costs* for a single product. More complex problems do not fall into this category.

**Question 1: MRI Example**

The prior example of the MRI scanning clinic falls into this category:

Your client is considering opening a number of MRI scanning clinics. Each MRI clinic will have two MRI machines, two Technicians and two Radiologists. The table below shows annual costs for a single unit of these and other line items required to run a clinic. Each MRI scan will generate $420 in revenue and will have $45 of costs of disposable medical products, and $25 of costs for other consumables (i.e., items consumed in the scanning process).

**Question A:**

How many MRI scans will a clinic need to perform annually in order to break even?

**Question B:**

How many scans required to achieve $1.4 *M* annual profit per clinic?

**Question C:**

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)?

Here are a few other problems that fit this category

- A company manufactures computers, which they sell for $1,500 each. For each computer sold, the company incurs incremental costs of $900 for parts, $100 in labor, and $50 in transportation costs. Per year, they spend $2
*M*on factory rent and maintenance, and $250*K*in utility costs.- How many units do they need to sell annually to break even?
- How many units do they need to sell annually to have a profit of $9
*M*? - What would their annual profit be if they sold 65
*K*units per year?

- A company manufactures and sells motorized scooters. Each scooter sells for $4,750 on average, and costs $2,750 in parts and material, and $500 in labor to manufacture. They spend $45
*M*on factory rent, utilities, and maintenance per year, $7*M*on other administrative overhead, and $8*M*on advertising per year.- How many units does the company need to sell annually to break even?
- How many units would they need to sell to have annual profit of $300
*M*? - How many units would they need to sell to have annual profit of $1.2
*B*? - What would their profit be if they sold 6
*M*units per year? - What would their profit be if they sold 2.8
*M*units per year?

- A company manufacturers injectable vaccines. Each vaccine sells for $530 and costs $180 to manufacture. Their production facility costs $1.15
*M*to operate annually. They spend $400*K*per year on sales and marketing expenses, and $200*K*per year on regulatory compliance.- How many unites does the company need to sell annually to break even?
- How many units would they need to sell to have annual profit of $21
*M*? - How many units would they need to sell to have annual profit of $49
*M*? - What would their profit be if they sold 400
*K*units per year?

## FastMath Resources

FastMath Ace the Case explains how to quickly and efficiently solve this type of problem. In particular, the course gives very efficient methods for performing related calculations under multiple scenarios with the same basic information — i.e., changing the *Quantity* or *Profit* given the same *Cost* and *Price* information.

The course has more than 40 examples of this type with detailed solution methods for each of these problems, and includes video-based explanations for this problem, and many other problems. This is the simplest problem type you are likely to encounter in a Case Interview, so you want to make sure you are well prepared for this type of problem.

**Solving Market Sizing & Estimation Questions**

As a reminder, below are a few example Market Sizing & Estimation questions candidates have been asked in real Case Interviews:

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

You could be asked to estimate the market size for any product category in any country or geographic region around the world. It’s not an effective use of time to memorize lots of statistics regarding market sizes for different products and industries around the world.

You need an approach that will allow you to estimate the market size for any product or industry in any country!

**Global Demographics Data**

You should start by learning basic demographic and statistical data about the largest economies in the world. Things like population, GDP per capita, and persons per household are the most important to know.

Do you know what the top 10 largest economies are in the world, and their GDP?

Can you list the top 10 countries with the largest GDP in order?

What is their GDP per capita in relation to the U.S.?

Do you know the GDP per capita of China and India?

What’s the population and GDP per capita of South Korea?

You should learn this information before going into a Case Interview.

**Framework**

Next, you’ll need a framework for estimating the market size of a given product category or industry.

**Practice**

Last, you’ll need to practice your method with many examples, and fine tune your method based on the accuracy.

## FastMath Resources

FastMath Ace the Case covers Market Sizing and Estimation problems in depth. First, the course has all the demographic and statistical data you need in easy-to-read tables and downloadable files. The course also has a universal framework and procedure for estimating the market size for any industry or product. Furthermore, the calculation methods covered in the course are directly applicable to Market Sizing and estimation problems as you will usually need to multiply many values together to get your estimate.

FastMath Ace the Case also covers topics like the optimal range for the number of parameters, the tradeoff between speed and accuracy, and how to recognize when you reach diminishing returns. The course also has numerous practice problems with detailed solution methods — not just the answer — that you can compare with your solution method so you can learn how to calculate the answer more efficiently.

**Other Quant Problem Types**

There are a wide variety of other quantitative problem types frequently asked in Case Interviews. Below is a sample of the types of problems you might encounter:

- Investment Decision Analysis
- Price Optimization
- Chart Reading
- Growth Rate Analysis
- Economies of Scale
- NPV Analysis
- Production Analysis
- Response Rate Analysis
- Pipeline Analysis

## FastMath Resources

FastMath Ace the Case provides examples and detailed solution methods for each of these types of problems. Furthermore FastMath Ace the Case explains what to do when you encounter a type of problem you haven’t seen before or aren’t familiar with. Hint — return to first principles.

Lastly, FastMath Ace the Case shows candidates how to take the results of the quantitative calculations and develop case-specific recommendations and insights.

## FastMath Ace the Case Online Course

When I was in business school at MIT, I saw that many of my classmates struggled with the quant component of Case Interviews. At the time, there weren’t good resources to teach people how to do calculations without calculators — as needed in Case Interviews. This experience inspired me to create FastMath Ace the Case. I designed this course to specifically teach the quantitative and mental calculation skills needed to succeed in Case Interviews. Math is always something that came naturally to me, and, in this course, I’ll teach you the calculation methods and approach to math I have used throughout my educational and professional career.

FastMath Ace the Case teaches fast and efficient calculation methods designed to solve the calculations most frequently encountered in Case Interviews. The course also demonstrates the most efficient solution methods to the most important quantitative Case Interview problem types, such as Market Sizing and Estimation problems and Breakeven Analysis problems. The methods in this course are specifically designed for typical Case Calculations, and all the examples are based on questions given in actual Case Interviews.

FastMath Ace the Case is different from other quant resources because it uses on-demand video lessons to teach efficient calculation and problem solving methods — instead of simply providing practice problems and answers — and is tailored to the problems and calculations actually encountered in Case Interviews. Thousands of MBA students from leading schools such as Wharton, Harvard Business School, Columbia, London Business School, MIT Sloan, and many other schools have used FastMath Ace the Case to prepare for Case Interviews and ultimately join elite Management Consulting firms such as McKinsey, Bain, BCG, PwC, Accenture, and many other leading consulting firms.

The video lessons in this course also show candidates how to develop relevant insights based on quantitative calculations. Discover simple and intuitive calculation methods that will save you time and effort while delivering results in your interviews.

**Course Features**

- 7+ hours of on-demand video lessons
- 50+ lessons/modules
- Hundreds of practice problems and solutions
- Detailed solution methods to all problems
- Market sizing data tables
- McKinsey PST methods and material
- Downloadable reference files
- Mobile accessible platform (iPad, iPhone, and Android)
- 30-day money-back guarantee
**All calculation methods and examples are based on real Case Interview calculations**

Learn more about **FastMath Ace the Case**

## About the Author

Matthew Tambiah is the creator of the FastMath Ace the Case Online Course. Matthew is a former McKinsey consultant and has a Bachelor’s degree from Harvard with *highest honors* in Electrical and Computer Engineering, and an MBA from MIT Sloan. Matthew is an internationally recognized expert on quantitative preparation for case interviews and has hosted workshops on quantitative interview skills for MBA, undergrad, and PhD students at Harvard Business School, Wharton, MIT, London Business School, Georgetown, and other leading universities. Furthermore, thousands of candidates from around the world have used the FastMath Ace the Case Online Course to prepare for case interviews. Matthew’s students have joined leading Management Consulting firms such as McKinsey, Bain, BCG, PwC, Accenture, and many other elite consulting firms.