# Preparing for the Quantitative Component of Case Interviews

This is one 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.

If you want to work in Management Consulting you should read this series. If you know somebody who wants to work in Management Consulting, you should share this series with them.

First article: **Math Skills Required in Case Interviews**Third article:

**Why Management Consulting Firms Assess Math Skills in Case Interviews.**

### Overview

The best way to prepare for the quantitative portion of Case Interviews is to practice the types of calculations you will need to perform with the types of numbers you are likely to encounter in Case Interviews and to be familiar with the types of problems you are most likely to encounter and how to solve them. Furthermore, candidates must understand how to analyze and interpret the quantitative results in order to make actionable recommendations in the Case.

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 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!

For example, Case Numbers are usually Round numbers with only a a few significant digits, such as 300,000 or 4,500,000. While some people may be intimidated by having to do calculations with large numbers like this without a calculator, once you learn how to efficiently manage the zeroes, calculations with numbers like these become straightforward.

When Case Numbers have multiple significant digits, they usually have additional properties that can make certain calculations like multiplication and division much easier if you are aware of the appropriate methods. For example, let’s say you are told a company sells a Product for $32 per unit and sells 25,000 units per year, and you are asked to calculate the Revenue. You would then need to do the calculation: ($32 × 25,000). Most people would use pen and paper for this calculation. You can make the pen-and-paper calculation more efficient by not writing out the three zeroes from “25,000” and then to add them to the end of the result of (32 × 25).

This example is representative of a typical Case Calculations, and here is an efficient method for this calculation (32 × 25). First decompose 25 as: (25 = ¼ × 100). Then we perform the calculation using this decomposition of 25.

32 × ¼ = 32 ÷ 4 = 8

8 × 100 = 800

Therefore (32 × 25 = 800) — you can check this with a calculator or pen-and-paper calculation. You calculation method is analogous to the following way of thinking: let’s say you are given 32 quarters (U.S. coins); how much money would you have? Most people would think to themselves: “there are four quarters in one dollar, and 32 ÷ 4 = 8, so I would have $8,” which is correct! Now think about how many cents would be in $8. The answer is clearly 800: (8 × 100 = 800), as there are 100 cents in 1 dollar. So 32 quarters is $8, which is also 800 cents. Now, since quarter is also worth 25 cents, and because we have 32 quarters, then 32 × 25 must also equal 800. So to multiply any number by 25, you divide the number by 4, and then multiply by 100. This method works because (4 × 25 = 100); so, when you divide 32 by 4, the result is the “number of groups of 100.” It follows the answer to our original problem of ($32 × 25,000) is therefore $800,000.

This method relies on the fact that 25 = ¼ × 100. These numbers are representative of actual Case Numbers, and most actual multiplication calculations in Case Interviews involve numbers like 25,000 or 25 Million, for which there exists a way to dramatically simplify the calculation.

Most other quantitative Case Interview preparation resources I have reviewed have example calculations that you are extremely **unlikely** to encounter in an actual Case interview, such as multiplying by a number like 23,487, which has five significant digits and no clear way to simplify the calculation. I believe that practicing with quantitative problems like this, that are not representative of actual Case Numbers or Case Calculations, is not an effective way to prepare, as you won’t be practicing for what you will actually face in an interview, and you therefore won’t be practicing skills you can actually use in Case Interviews. Practicing with problems that involve numbers with many significant digits can actually be counterproductive, as you will focus on the brute-force mechanical calculations, and won’t practice of think about the logical problem-solving aspects and developing recommendations from result of quantitative analysis, and you won’t get experience looking for and finding efficient calculation methods. In short, you wont’ be training for the types of problems you will actually face in Case Interviews.

### Effective Quantitative Preparation Resources

I created an online course, called *FastMath Ace the Case**, *that* *provides a comprehensive set of tools to prepare for the quantitative portion of Case Interviews. This online course uses video-based instruction to teach fast and efficient mental calculation methods for the most common types of Case Calculations (multiplication, division, percentages and compound growth calculations). The methods in this course are specialized for typical Case Numbers, and all the examples in this online course are based on real Case Calculations. The video lessons in this course demonstrate how to apply these efficient calculation methods to the most common types of quantitative Case Interview problems, such as Market Sizing problems and Break-even Analysis problems, and show how to solve these types of problems most efficiently. This course also has extensive practice problems that include **detailed** solution methods that reinforce the course material.

This article has examples of the calculation methods taught in the *FastMath Ace the Case* online course and has efficient solution methods to the example problems from the previous article. You can learn more about and register for *FastMath Ace the Case* online course through the URL below, which applies a 25% discount to $59.*FastMath Ace the Case* Course URL

If you want a free assessment of your quant skills, there is also a Quantitative Assessment Quiz and Solutions to the quiz.

### Mental Math Skills Are 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 a very helpful Case Interviews.

Consider an example similar to the one in the previous article, where you are given Price, Quantity and Profit Margin for three different products and asked to calculate the percentage of the firm’s overall Revenue and Profit contributed by each product. We’ll use different numerical values given in the table below:

Let’s say your approach to answering this question is to first calculate the Revenue and Profit for each Product. If you are able to do these operations 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 using longhand multiplication will be **much** **slower** than a person who can do the calculations mentally. It is much faster and more efficient to simply do each calculation mentally (without needing the original numbers out in longhand format) and then write down the result. The speed improvement of mental calculations is magnified if you need to multiply several numbers together when calculating a value.

For example: *Profit** *= (** Price**) × (

**) × (**

*Quantity***)**

*Profit Margin*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, the *FastMath Ace the Case* online course 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.

### Addition and Subtraction Methods

Given that the Case Numbers have few significant digits — and often have trailing significant digits of “5,” “25” or “75” — performing addition and subtraction with Case Numbers is relatively straightforward for most people. You may want to practice adding and subtracting numbers if you are rusty or don’t feel confident in being able to do this smoothly in an interview, which has added stress.

**Example:** Add the following numbers: 250 Million, 300 Million and 150 Million.

Read below for **Methods for adding large numbers**

A helpful method for adding large numbers is to calculate with **Units** like Thousand (** K**), Million (

**), Billion (**

*M***), and Trillion (**

*B***):**

*T***Units K:** Thousand

**Million**

*M:***Billion**

*B:***Trillion**

*T:*When adding large numbers, **don’t** write out (or mentally picture the number with) all the zeroes; simply write (or mentally picture) the leading digits and an abbreviation for the unit. For example, to represent “250 Million” — which, in long form, would be written as “250,000,000” — write/think “250* M*,” or mentally remember the Unit and do the calculations and, when you write down the result, include the appropriate Unit.

In Case Interviews, if you are given printed sheets with numerical values or tables of numbers, the prinout will usually not include all the zeroes for a number like 25 Million; it will use a Unit like “Million” or the table will indicate the values are in “Millions” without writing out all the zeroes.

Using these Units, the prior example becomes:

The answer is 700 Million.

End of **Methods** **for adding large numbers**

### Multiplication

Multiplication is the **most** **important** quantitative skill to practice for Case Interviews because you will usually have to perform numerous (if not many) multiplication operations in Case Interviews — sometimes involving three or more terms. You can improve your speed and efficiency with multiplication while reducing errors by learning and practicing just a few simple *FastMath* methods taught in the *FastMath Ace the Case*** **online course. Below are a few examples of how to apply *FastMath* methods to Case Interview multiplication calculations.

**Multiplication Example 1 **(from prior article): Calculate 120 M × 250

Read below for the* FastMath* Solution Method

Writing this calculation out in longhand with all the zeroes is a slow, error-prone process, and many people will make a mistake with the number of zeroes. It is also difficult to do this calculation mentally while keeping track of all the zeroes, and people executing this calculation mentally often make a similar error regarding the number of zeroes, which causes the answer to be off by a factor of 10 or more. This is clearly a very bad mistake to make in a job interview.

Here’s the *FastMath*** **solution method:

First, we represent 120 Million using our units, which becomes 120 *M*.

Second, we treat or “decompose” 250 as: (250 = ¼ × 1,000 = ¼ × *K*)

Then we calculate using this “decomposition” of 250:

¼ × 120 *M* = 30 *M*30

*M*×

*K*= 30

*B*(since

*M**×*

**=**

*K***)**

*B*The answer is 30 Billion (30

*B*).

In general, multiplying by 1,000 (** K**) will change the Unit to the next larger Unit (“increase the Unit”) because 1,000 (

**) of a given Unit is equal to 1 Unit of the next larger unit, as there is a ratio of 10³ between each Unit:**

*K*** K:** Thousand (10³)

**Million (10⁶ =**

*M:***×**

*K***)**

*K***Billion (10⁹ =**

*B:*

*M**×*

**)**

*K***Trillion (10¹² =**

*T:*

*B**×*

**)**

*K*Therefore, multiplying any number by 1,000 (** K**) — such as 30

**×**

*M***— keeps the same digits (e.g. “30”) and just changes the Unit from**

*K***to**

*M***This is a much faster way to multiply these numbers than counting all the zeroes in the calculation, either in longhand format or purely mentally.**

*B*.We could also reverse the order of operations and multiply by 1,000 first and then divide by 4. However, many people find it easier to divide by 4 first because that reduces the number of digits you need to mentally keep track of (30 is smaller than 120).

End of *FastMath* Solution Method

**Bullet Operator for Multiplication**

To improve readability of equations, we may use the “bullet” (also called the “dot”) symbol “⋅” to indicate multiplication. Therefore, the equation (20 × 4** **= 80)

**could be written as (20 ⋅ 4 = 80)**

**with the bullet/dot symbol. This can improve the readability of equations where you are multiplying a number of terms as the dot symbol is smaller than the cross symbol “×” and so it’s easier to read variable names and numbers.**

**Multiplication Example 2:** Calculate 125 ⋅ 2.5 ⋅ 4

Read below for the* FastMath* Solution Method

Another method useful for multiplying numbers is to **reorder** the multiplication terms (numbers)into a different sequence that is easier to calculate. We can multiply these numbers in this sequence:

125 ⋅ 4 = 500

500 ⋅ 2.5 = 1,250

The answer is 1,250

I’ll call this the “**Reordering**” method, and it uses the mathematical property that when multiplying a set of numbers, the order in which you multiply terms doesn’t matter; we’ll call this property the **Reordering** **Property** of multiplication. There are many potential ways in which you can Reorder the multiplication terms, and you can choose the order that is easiest for the particular numbers involved. There may be multiple efficient or effective ways to Reorder multiplication terms, and there is no single “correct” order. We could also calculate this as follows:

2.5 ⋅ 4 = 10

125 ⋅ 10 = 1,250

Most people find these new orders are much easier to execute than multiplying the first pair of numbers in the original calculation (125 ⋅ 2.5).

End of *FastMath *Solution Method

These are a few of the efficient multiplication methods covered in the *FastMath Ace the Case*** **online course. You can preview the video lesson on multiplication here: Multiplication Lesson.

### Division Methods

Some Case Calculations require division, and you frequently need to express the result as a percentage. There is usually a way to simplify the division Case Calculations, so you frequently don’t need to use Long Division. Candidates should still be familiar with and be prepared to use Long Division if they cannot find a simplification method, and candidates may encounter some Case Calculations where traditional Long Division is appropriate.

Below are a few examples of how to apply *FastMath* methods to division Case Calculations.

Example: Calculate 42 Billion ÷ 500

Read below for the* FastMath* Solution Method

Executing this calculation using Long Division and writing out all the zeroes is a slow, error-prone process. Many people find it difficult to perform this calculation purely mentally because of all the zeroes.

Here’s the ** FastMath** solution method:

First, we write 42 Billion using our units, which becomes 42 B.

Second, we treat 500 as: (500 = ½ ⋅ 1,000 = ½ ⋅ K)

Then we calculate using this “decomposition” of 500:

42

*B*÷

*K*= 42

*M*

**(since**

**÷**

*B***=**

*K***)**

*M*Next we need to divide by ½, which is the same as multiplying by 2.

42 *M* ⋅ 2 = 84 *M*

The answer is 84 Million.

To divide by 1,000 (** K**), we simply do the inverse of multiplying by 1,000 (

**), and decrease the Unit size. Hence: (**

*K***÷**

*B***=**

*K***). We could do these operations in any order, so we could multiply by 2 first, and then divide by**

*M***.**

*K*End of *FastMath* Solution Method

### Multiplying and Dividing with “Clean” Numbers

Many of the Case Numbers involved in multiplication and/or division calculations in Case Interviews will have special properties, and are what we will call “**Clean** **Numbers,**” defined as being a “**Clean** **Fraction**” times an Integer power of 10. In this context, a “Clean Fraction” means both the numerator and denominator (what we will call the “elements” of the fraction) are non-zero Integers that are each smaller than 10 — and often smaller than 5.

For example, the number 25 can be represented as ¼ ⋅ 100; hence, 25 is a Clean Number, as the elements of the fraction are “1” and “4,” which are both smaller than 5. Below are some examples of how to express certain numbers as Clean Fractions times an Integer power of 10, showing that they are Clean Numbers:

A few things to note about Clean Numbers:

- The Numerator can be larger than 1, as shown with 75 = ¾ ⋅ 100.
- The Numerator can be larger than the Denominator, as shown with 150.
- The Integer exponent of 10 can be
**positive**or**negative,**as shown with 0.05 — which has a negative exponent of 10. - You can represent percentages as a Clean Fraction, as shown with 2.5%.
- 33% and 67% are approximately equal to ⅓ and ⅔, respectively (the symbol “≈” means “approximately equal to.” These fractions are actually repeating decimals, but, in Case Interviews, you can often round these percentage values to these Clean Fractions.

There are specific *FastMath* methods you can learn and apply to make multiplying and dividing with Clean Numbers much easier than traditional calculation methods. The key concept of these *FastMath* methods is to identify Clean Numbers and multiply/divide with them while expressing them as a Clean Fraction times a power of 10. Keep in mind that, when multiplying two numbers, only one number needs to be a Clean Number to use the *FastMath* methods. With the example of 250 ⋅ 120 M, only 250 is a Clean Number.

The *FastMath Ace the Case* online course provides a comprehensive set of methods for multiplying and dividing with Clean Numbers, which are the most common types of Case Numbers in Case Calculations that require multiplication or division, along with practice problems and solution methods to reinforce these methods. You can preview the multiplication lesson here: Multiplication Lesson.

### Percentage Growth Methods

**Example:**A company had annual Revenue of $400 Million in the past 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?

Read below for the* FastMath***Solution Method**

The most efficient way to approximate the answer is to ignore compounding — which we will call the “non-compounding approximation.” Since the Revenue grows 4% per year, for six years, if we ignore compounding, total revenue would grow by (4% ⋅ 6 = 24%). Since compounding will contribute slightly more growth, it is appropriate to round the total growth to 25%, which is ¼. Since the original Revenue is $400 *M,* the net change in revenue is 25% ⋅ $400 *M* = $100 *M*. Therefore, the Revenue in six years is $500 *M* = ($400 *M* + $100 *M*).

It is appropriate to ignore compounding when the percentage change and the number of years of growth are relatively small. The *FastMath Ace the Case* online course explains in further detail when you can use the non-compounding approximation, and provides additional methods for performing percentage-growth calculations when it’s not appropriate to use the this approximation.

End of** FastMath Solution Method**

Net Present Value (NPV) Calculation Methods

**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%?

Read below for the *FastMath***Solution Method**

There is a formula to determine the net NPV for an annuity, which is fixed payment that occurs annually (**$ C**), for a given required annual return or interest rate (

**), which is:**

*r*Given that *r *=10%,

Therefore, *NPV* = 10 ⋅ $*C* = 10 ⋅ $20 *M* = $200 *M*.

Your company would therefore be willing to pay up to $200 Million to acquire this firm.

End of *FastMath* Solution Method

**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?

Read below for the *FastMath* Solution Method

This is a trickier NPV calculation than the prior one, as this is not an annuity, and we have to determine the NPV of a payment of $250 *M* in six years. There is a concept called the **Rule of 72 ****[1]****,** which tells us that something growing at 12% annually will double in approximately six years, as: (6 ⋅ 12 = 72). In general, the doubling time for a quantity will be approximately 72 divided by the percentage growth rate (expressed as an integer) — in this example, 72 ÷ 12 = 6. This approximation is accurate for percentage growth in the range of 0%–20%.

Therefore, the NPV of a payment of $250 *M* in six years, with a required return of 12%, is exactly half the future payment value of $250 M, which is $125 M.

The** ***FastMath Ace the Case* online course has an in-depth lesson on the Rule of 72, which explains when and how to use it, and why the approximation is accurate.

[End of ** FastMath** Solution Method]

[1] This is more of an approximation rather than a “Rule,” but that is how this mathematical property is generally referred to.

### Solving Break-even Analysis Problems

Now we’ll cover efficient solution methods for the Break-even Analysis question type. Below is the example question from the previous article.

**Question:**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 $70 of costs for consumables (i.e. items consumed in the scanning process).

How many scans would each clinic need to perform annually in order to **Break** **Even**? Try to solve this without a calculator or spreadsheet.

**Hint**If you’re having trouble with this calculation, keep in mind that there are two MRI Machines, two Technicians and two Radiologists per clinic, in addition to the cost of running the Facility itself. This should make the math work out easier.

This is an example of a McKinsey final round interview question. They gave the candidate a printed sheet with the table and text description of the question, and this is how the problem was worded. If you didn’t read the text carefully, you would miss that there were two units of these items at each MRI clinic.

Read below for the *FastMath***Solution Method**

Many people try to solve this problem with an algebraic formula saying that, at Breakeven: (** Total Costs** =

**). They then write out an algebraic formula for Total Costs and Total Revenue based on Quantity sold and solve for that Quantity. This will give the correct answer, but it is relatively slow — and many people make errors. Below is the**

*Total Revenue***solution method for this type of Break-even Analysis Problem.**

*FastMath***Variable Definitions FC:** Fixed Costs of Operating a Clinic

**Price of Scan**

*P:***Marginal Cost of Scan**

*MC:***Marginal Profit of a Scan**

*MP:***Break-even Quantity**

*Q:*A very efficient way to solve this type of problem is to define the **Marginal Profit** as: *MP** *= ** P**−

*MC*Then the **Break-even Quantity **is equal to the **Fixed Costs** divided by the **Marginal Profit:**

Think of the **Marginal Profit** as the amount of money you make, on each incremental Scan, that you can use to pay down your **Fixed** **Costs,** which are incurred no matter how many Scans you sell (i.e. regardless of **Quantity**). This is often the most efficient approach for this “standard” type of Break-even Analysis problem.

The total annual Fixed Costs are $1,050 K. The Marginal Profit = $420 − $70 = $350. Therefore:

A ** FastMath** method for this calculation is to double both the Numerator and Denominator, which gives:

Cancelling zeroes gives and the currency unit of dollars ($) in the equation above gives:

You can see a video explanation of this here: Breakeven Analysis Video.

End of *FastMath *Solution Method

#### Solving Revenue and Profit Calculations

**Example Question**A firm sells three products. Data on each product is 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?

Read below for the *FastMath***Solution Method**

Most people would approach these questions by first calculating the Revenue and Profit for each Product. You may also encounter Case Interviews where you are given similar data and asked to calculate the Revenue and Profit for each product and then determine the percentage of the total. There are efficient methods for calculating the percentage values directly **without** first calculating the Revenue or Profit of each individual product, which will be covered in the ** FastMath Ace the Case** online course.

For now, we’ll proceed by calculating the Revenue and Profit for each Product. To do this, you should use the ** FastMath** methods for multiplying

**Clean Numbers**(where needed).

Calculating the Revenue for each Product is simple if you know your 10×10 multiplication table and basic rules for multiplying by 10. The table below shows Revenue information for each Product.

Adding the Revenue for each Product gives Total Revenue of $200 *M. *To calculate the percentage of Revenue contributed for each Product, we divide the Revenue of each Product by the Total Revenue of $200 *M. *Since all units are in $** M,** we can ignore both units ($ and

**) and calculate with the numbers only. Since we need to divide by 200, we’ll first divide the Revenue for each Product by 2, and then divide the result by 100. Dividing by 2 gives:**

*M***A:**25 ÷ 2 = 12.5

**B:**50 ÷ 2 = 25

**C:**125 ÷ 2 = 62.5

Notice that, if we divide each of these results by 100, we can simply add the percentage symbol to the end without modifying any digits (given that we want to express the answers as a percentage. For example, 5 ÷ 100 = 5%.**A:** 12.5 ÷ 100 =12.5%**B:** 25 ÷ 100 = 25%**C:** 62.5 ÷ 100 = 62.5%

In its simplest form, this calculation requires just dividing the Revenue numerical value (ignoring Units) of each Product by 2, and then writing the percentage symbol (which is equivalent to dividing by 100). The results are shown in the table below.

We could do this calculation slightly more efficiently if we notice that the Revenues for each Product are multiples of $25 *M*. In terms of multiples of $25 M, the Revenue for each product is shown below:**A:** 1× ($25 *M* = 1 × $25 *M*)**B:** 2× ($50 *M* = 2 × $25 *M*)**C:** 5× ($125 *M* = 5 × $25 *M*)

1 + 2 + 5 = 8 ($200 *M *= 8 ⋅ 25 *M*). Therefore, the Revenue that Products A, B and C contribute is ⅛, ¼, and ⅝ of the total, respectively. Most people know that ¼ = 25%, and might need to calculate ⅛ and ⅝ as percentage values. An efficient method is to start with ¼, and realize that ⅛ = ¼ ÷ 2. Therefore ⅛ = 25% ÷ 2 =12.5%. To calculate ⅝, we can use the fact that ⅝ = ⅝ = ½ + ⅛ = 50% + 12.5% = 62.5%.

Next we’ll calculate the Profit for each Product, which is (** Revenue** ⋅

**). Profit for Product A then becomes (16% ⋅ $25**

*Profit**Margin**M*). At first, this might seem like a cumbersome calculation, as 16% is not a Round or Clean Number. However, we can actually swap percentage and number as (16% ⋅ 25 = 25% ⋅ 16). In general, (X% ⋅ Y = Y% ⋅ X). 25% = ¼, and (¼ ⋅ 16 = 4), so Product A has a Profit of $4 M. We can use a similar method for Product B: (12% ⋅ 50 = 50% ⋅ 12 = 6), so Product B contributes $6

*M*of Profit. For Product C, note that ($125 = ⅛ ⋅ 1,000), so (8% ⋅ 125 = 8% ⋅ ⅛ ⋅ 1,000 = 1% ⋅ 1,000 = 10). Hence, Product C generates $10

*M*of profit.

Filling in the Profit values and their total gives:

Once we have the Profit values, calculating the percentage of the Total is straightforward, as the Total Profit is $20 *M, *which is easy number to divide by. The table below shows the percentage of Total Profit that each Product contributes.

The *FastMath Ace the Case* online course has a comprehensive set of methods for multiplying and dividing Case Numbers, and includes methods for dealing with percentage values, and includes a table of the most important fraction-to-decimal conversions.

As mentioned earlier, there are efficient methods to directly calculate the percentage of Revenue and Profit contributed by each Product, without calculating the Revenue or Profit of each individual product. These methods will be covered in the *FastMath Ace the Case* online course.

End of** FastMath Solution Method**

### Answering Market Sizing & Estimation Problems

#### Desired Accuracy

The first thing to be aware of is the desired level of accuracy and tradeoff of accuracy against speed. In general, your goal should be to get within 2× or 3× of the actual answer quickly. Even an answer that is within 10× too big or too small is still the right “Order of Magnitude” and would generally be considered in the right “ballpark.” You do **not** need to be within 10% or 20% of the correct value to answer these questions “correctly.”

#### Process

It is useful to use the following three-step process for answering Market Sizing and Estimation Problems:

**Parameters & Framework:**Identify**Parameters**you need and develop a**Framework**for how you would use them**Estimate:****Estimate**values for your**Parameters**(use Clean or Round numbers)**Calculate:****Calculate**a result using your**Framework**and**Estimated**values — use approximations and round results to simplify calculations

### Statistics to Learn

It is extremely helpful for Market Sizing and Estimation Problems to know some basic statistics about the United States and other countries, so you can minimize the number of Parameters of which you are estimating the value. For example, to accurately estimate many Market Sizes in the United States, it is useful to know the following statistics:

**U.S. National Data**

The ** FastMath Ace the Case** online course provides comprehensive data tables for the largest Industrial and Developing countries and information by geographic region.

You can preview a video lesson providing an overview of how to approach Marketing Sizing and Estimation Problems and relevant data to learn here: Market Sizing & Estimation Overview.

#### McDonald’s Annual Sales Example

Here’s how to apply this framework to the example problem of estimating the annual sales of McDonald’s restaurants in the United States:

**Parameters & Framework**Below is a visual representation of the

**Parameters**and

**Framework.**This is meant to be a very simple framework so we can quickly calculate an estimated Market Size.

The **Total U.S. Sales ( TS) **is the product of four different parameters labeled in the diagram: U.S. Population, the

**ratio**(or percentage/proportion) of people regularly purchasing at McDonald’s, the

**frequency**of their purchases, and the average

**Price**per meal.

**Estimate**

*Pop*= 300 M

*r*= ⅔

*f = 1 per week ≈ 50 per year*

P = $5

P = $5

Note: *“≈” *means approximately equal to. We round weeks per year to 50 per year to use a Clean Number.

**Calculate**

*TS*=

*Pop*⋅

*r*⋅

*f*⋅

*P*

Pop⋅ r = 300 M ⋅ ⅔ = 200 M (number of regular customers)

Pop

*f*⋅

*P*= 50 ⋅ $5 = $250 (annual expenditure per regular customer)

*TS*= (

*Pop*⋅ r) ⋅ (

*f*⋅

*P) =*200

*M*⋅ $250 = 200

*M*⋅ ¼ ⋅ $

*K*= $50

*M*⋅

*K*= $50

*B*

This very quick calculation gives us an estimate of $50 *B* for McDonald’s Total Sales in the United States. Looking at McDonald’s 2016 annual report, we see that the actual number was $36.3 *B* in 2016, so our estimate was within 50% of the actual answer, which is good considering a goal of being within 2× or 3×, and the speed at which we arrived at the estimate. In general, there will be a tradeoff between the speed of calculating a Market Size or Estimate, and the accuracy of that Estimate. Your goal in Case Interviews is usually to arrive at a “reasonable” estimate/value relatively quickly.

Note that you were asked for the Total Sales of McDonald’s restaurants in the United States, which is **different** from the Revenue for McDonald’s in the United States because many McDonald’s restaurants are owned by third-party franchisees. You therefore have to read McDonald’s annual report carefully. The figure for Total Sales in the United States is the sum of Sales at franchise-owned restaurants in the United States (page 19), and Sales at Company-operated restaurants in the United States (page 18).

You can view a video explanation of this example here: McDonald’s Market Sizing Example.

#### Additional Resources for Market Sizing & Estimation Problems

The *FastMath Ace the Case* online course covers Market Sizing and Estimation problems in great detail. This course has a comprehensive set of data tables with key statistics for countries around the world, outlines the **two** most effective types of Market Sizing & Estimation **Frameworks,** illustrates each of these Framework types, provides video explanations of several Market Sizing problems, and gives recommendations, best practices and other tips for Market Sizing and Estimation questions.

### FastMath Ace the Case Online Course Summary

The *FastMath Ace the Case* online course provides a comprehensive set of resources to prepare for the quantitative portion of Case Interviews. This online course uses video-based instruction to teach efficient calculation methods that have been proven to enhance performance in Case Interviews, and how to apply these methods to the most common types of Case Interview 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.

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**Course Features:**• 10 full quantitative Case Interview problems and solutions

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**Registration & Additional Information**You can learn more about, and register for, the

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The *FastMath Ace the Case*** **online course is the recommended quant training for consulting clubs at Wharton, Columbia Business School, LBS and numerous other MBA programs. More than 5,000 MBA students around the world have used this online course to prepare for consulting interviews. Learn for yourself why top MBA students love *FastMath***,** and maximize your own interview performance by registering for the *FastMath Ace the Case* online course.

The *FastMath Ace the Case* online course is different from other quantitative interview-preparation resources in several ways. First, it uses on-demand video instruction to teach fast and efficient calculation methods, whereas other quant interview-prep resources usually just provide practice problems and answers, without teaching practical methods for performing the calculations without a calculator. Second, the methods taught in the course are specifically designed for the types of calculations you are likely to encounter in a Case Interview, and all the examples are based on real Case Interview calculations.

The *FastMath Ace the Case* online course teaches calculation methods to improve speed and efficiency while reducing errors for multiplication, division, percentages and compound growth calculations. This online course provides examples of the most common types of quantitative Case Interview problems, such as Market Sizing problems and Break-even Analysis problems, and demonstrates how to solve these problems efficiently. This online course also provides practice problems and detailed solution methods that reinforce the course concepts and material.

**About Course the Author**Matthew Tambiah is the author of the

*FastMath Ace the Case*

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*Highest Honors*in Electrical and Computer Engineering from Harvard, and an MBA from the MIT Sloan School of Management. Matthew has hosted workshops on quant skills for Case Interviews at Harvard Business School, Wharton, MIT, London Business School, Georgetown and numerous other leading universities. Matthew’s students have joined McKinsey, Bain, BCG, PwC, Accenture, and other leading consulting firms.