Case Interview Math: a comprehensive guide

There is no way around it - your case interview is going to feature math and you are going to need to be good at it to get any consulting job, let alone an MBB one. This is not to say you need a math degree to be a consultant. In a standard case interview, the math will be no more complicated than what you learnt in high school. However, consulting math is very different to academic math and you won't get far approaching problems the same way you would have done in school.

In this article, we'll first look at what makes math so important for aspiring consultants and what makes consulting math different. We'll then run through a list of the areas in which you need to be proficient, whilst giving some tips on "hacks" which you can use to excel in interview.

Math is one of the most important elements of preparing for a consulting interview. This article is a great point of entry into the subject, but it is impossible to be fully comprehensive in any reasonable amount of space - for one thing, we're not going to reproduce your high school math textbook here! Where appropriate, we'll point you towards resources including other articles on this site. Generally, though, the if you want a more comprehensive source, you should check out our Consulting Math video lesson within the MCC Academy. Our Consulting Math packages are also great for practice, along with our free mental math tool. This article will get you started, but the these additional sources will give everything you need to have a real chance at landing your dream consulting job.

Consulting Math in Principle

You might think that math is math and that being good at academic math - perhaps at a university level - will mean you have nothing to fear from a consulting interview conducted at a high school level. Certainly, it is true that being good at math in an academic context is a great advantage going into a consulting interview. However, the style of math used in consulting is very different from that used in academia and takes practice to pick up. Even a very accomplished mathematician will struggle to impress if they do not approach problems in the way their interviewer is expecting.

In academic math, the overriding concern is accuracy. It might take a lot of complex working and a lot of time to get there, but what matters is that the answer is absolutely watertight. Consulting math is a very different beast. Working consultants - and consulting interview candidates - are always under time pressure. Results are what matter and answers are required simply to be good enough to guide business decisions, rather than being absolutely correct. A 90% accurate answer now is a lot more useful than a 100% accurate one after a week of in-depth analysis. The additional mathematical complexity required to reach such a totally accurate answer is thus not required. Rather, consultants will simplify their analyses in order to be more time efficient.

Your interviewer will also attach a special importance to mental math. Of course, being able to do mental math quickly shows general mental agility. However, consultants also specifically use quick mental math in order to impress clients (and thus help justify their fees). The sharper your mental math, the more impressed your interviewer will be. We have a brief section on mental math skills below, with much more detailed treatment in the MCC Academy.

Interview Math in Practice

So, now we know a little about how academic and consulting math differ. This is good knowledge to have, but we should keep an eye on practicalities and the nuts and bolts of how things will actually be in your interview. Let's get some of the most straightforward matters out of the way before we look at consulting math in more depth:

Working on Paper

You will be given a piece of paper and should feel free to use it when doing calculations.

Don't Rush

The time pressure in case interviews is severe and you cannot afford to waste time. By the same token, though, taking a few extra seconds to get to a correct answer is always preferable to producing an incorrect answer a few seconds more quickly. Don't be afraid to take the time you need. "Slow is smooth and smooth is fast" as Navy officers are taught!

Be Assertive

Candidates which are not really comfortable with math tend to state their answers as questions - with a rise in vocal pitch towards the end of the sentence. Interviewers will notice this and take note. Successful candidates will be confident and state their answers with an air of certainty.

Ask About Rounding

Ask your interviewer if it is okay to round numbers in your calculations. Generally, they will be fine with this and you can do so.

Fundamentals: A Checklist

So, which skills will you need? Here, we'll go through the main areas which you should cover to prep for a standard MBB interview.

We go into much more depth on each issue - along with worked examples and "hacks" for quicker calculations - in our video lesson in the MCC Academy. Of course, though, if you really weren't paying any attention in school and are totally in the dark as to what a fraction is - there is a point where you will simply need to simply pick up a basic math textbook from your local library.

1. Fractions

Fractions are a convenient way to represent numbers between 0 and 1 as parts of a whole. For instance, we might write 0.5 as 1 / 2 (or simply 1/2). For your interview, you should be readily able to add/subtract and multiply/divide fractions. There are a couple of specific ways in which being able to manipulate fractions in these ways will facilitate your mental math:

Approximating Divisions

Say you have to work out 107 ÷ 13. You only have a few seconds and no calculator. Certainly, you won't have time for ling division - so what will you do? The interviewer is waiting...

One specific use of fractions is in allowing us to tackle complex divisions quickly. For example, let's imagine we do indeed have to divide 107 by 13:

We know that:

Approximating Divisions

Thus:

Approximating Divisions

This method gives us a good-enough answer to proceed with our analysis with only a few seconds work and no need for a calculator. Success!

Navigating Problems in an Efficient Way

Fractions can also help to simplify your analysis of certain problems. Let's take the following relatively simple example:

In a company 1/3 of employees are engineers. Due to staff restructuring, 1/3 of the engineers is laid off. What fraction of the remaining employees are engineers?

Engineers laid off:

Approximating Divisions

Remaining engineers:

Approximating Divisions

Employees remaining in the company:

Approximating Divisions

Therefore the fraction of remaining employees which are engineers is:

Approximating Divisions

2. Ratios

Ratios are close cousins of fractions, and tell us how much of one thing we have in relation to another. For instance, if we have three pens, for pencils and one eraser, then the ratio between them is 3:4:1.

Many different business scenarios require an understanding of ratios. As regards solving cases, it is often very useful to express ratios as fractions of the whole. For example, we can re-express the ratio between our items of stationery above as 3 / 8 : 4 / 8 : 1 / 8 . This then allows you to address problems with a similar method to how we solved the example above as to what fraction of a company's workforce are engineers.

Think about how you might address the following question:

Restaurant Barbello’s profits are split among food drinks and tips in a 7:3:2 ratio. If the profit for food is 360 USD more than the one for drinks what is the total profit?

You should be able to get to an answer very quickly - certainly in under a minute. We show you how to do so in the MCC Academy video lesson.

3. Percentages

Similar to fractions and ratios, we can think of percentages as ratios expressed where one number is fixed at 100, or as fractions where the denominator is always 100.

Percentages are as ubiquitous in the business world as they are in interview case studies. We might be dealing with profits which are down 40%, targeting increases in sales or revenue by 20% or attempting to cut costs by 15%. We are especially likely to deal with percentages when working with issues around pricing, such as applying mark-ups on products to generate profit or offering discounts to promote sales.

One item to note here is that sometimes percentages will be discussed in terms of "percentage points". As such, if you are informed that revenues are down by 20 percentage points - or even just 20 points - this simply means that they have fallen by 20%.

You can test your ability to work with percentages by seeing how quickly you can work out an answer to the following:

Marta has a shop and she sells handbags for 30 EUR. She offers a 20% discount for a day. She then realises that the price is too low so she increases the price by 10%. What it is the current handbag price?

We run through a method to answer this question in a few seconds in the MCC Academy math video.

4. Probability

Nothing is certain in the business world and consultants will constantly have to evaluate the probability of different future events when they make decisions. The probability of such an event will always be a number between 0 (impossible) and 1 (certain) calculated as the number of ways that event can happen divided by the total number of possible outcomes. Therefore, the probability of rolling a six on a fair dice is 1/6, as there are a total of six possible outcomes, only one of which is a case of the event in question.

The probability of an event not happening is 1 minus the probability that it will. In proper notation, this is:

Probability

You will also need to know how to calculate the probability of multiple chance events all occurring in future. Luckily, in case interview, you will only have to deal with independent events, where individual outcomes do not influence subsequent ones. The standard example is coin tosses, where the probability of heads on each new toss remains 0.5 regardless of the results of previous tosses (despite any intuitions in line with the gambler's fallacy). This is as opposed to dependent events, where the outcomes of one event can influence subsequent ones. You might recall examples of these events from school problems about taking coloured balls out of vases without replacing them - in any case, we don't need to worry about them here.

The probability of multiple independent events all happening is calculated simply as the product of their individual probabilities. To illustrate, the probability of heads (P(H)) on the toss of a fair coin is 0.5. Therefore, the probability of tossing heads three times in a row is:

Probability

Expected Returns

Probability is especially relevant to business in calculating expected returns. Here, we weight the yield promised by an investment by the probability that it will pay out. This then acts as a guide to decisions about which investment opportunities should be pursued.

Say we have $100 to invest and that we can chose between two opportunities which will pay out after one year. Option A will pay out $120 with a probability of 0.9, whereas option B promises to pay out $150, but with only a probability of 0.7.

The expected returns are:

Option A: 0.9 x 120 = $108
Option B: 0.7 x 150 = $105

As such, we should favour option A as yielding a greater expected return, despite option B's greater headline payout. We take a look how to calculate a more complex expected return in the MCC Academy video lesson.

5. Averages

When we consider the average here, we will technically be referring to the arithmetic mean. We can think of this as a measure of the "typical" value of some series of numbers. It is calculated as the sum of all the numbers in the series, divided by the number of those numbers. We can state this more formally as:

Averages

Averages are fairly straightforward. The only complexities you will need to worry about arise when the values you are averaging do not have the same weight as one another. In such cases, the calculations will start to look rather like those for expected returns, where appropriate weightings are applied.

Let's take an example to see how well you can manipulate averages. How long does it take you to solve this problem? Could you do so under time pressure in a case interview?

A company has 80 employees. 25% work on average 6 hours a day, 65% 8 hours and the rest 12 hours a day. What is the average time an employee work?

We show you two different ways to solve this problem in the MCC Academy math lesson.

6. Rates

Rates are ubiquitous throughout the business world in general and in consulting in particular. We can think about rates as a ratio or fraction where the denominator is always 1. Some rates we will typically encounter include the interest rate, the rate of inflation, various tax rates, rate of return on an investment and the exchange rates between currencies.

Rates are very common in case studies and will generally be expressed per year (per annum). Candidates can easily become confused, though, where information is not all provided in the same units. As such, it is best to convert all such quantities into one set of units to facilitate comparison. For instance, with a mix of monthly and annual rates, it might be best (depending on the details of the problem) to convert all the relevant figures into per annum rates.

In the MCC Academy math lesson, we work through a business case study advising a firm whether to invest on new equipment based on an analysis of different rates. This demonstrates how central rates can be to business problems, as well as how to work with them efficiently.

7. Optimisation

Optimisation is of central importance in various fields, and the optimisation of complex, non-linear problems is a substantial area of academic study with real world applications ranging from engineering to finance.

Mercifully, though, optimisation in consulting interviews is a pretty straightforward affair. Any business problems you will be given will be linear - that is, will be of the form y = ax + b. This means that they will be optimised at one of the function's boundaries. To establish which boundary value yields the optimum, we simply need to work out the gradient of the function - or, more simply, whether this gradient is positive or negative.

As such, if we are trying to maximise y for the function below, where y = 2x + 1, between x=0 and x=4, we can see that the positive gradient (upward slope) of the line means that y will be maximised for the maximum possible value of x - which is 4 in this case.

Line graph visualising function from optimisation problem

Note that, in the section on writing equations below, we also discuss a way to solve linear optimisation problems without doing any calculations or referring to a graph.

We work though examples of optimisation in case studies in the MCC Academy math lesson.

Now, let's try an example of the kind of optimisation which you might have to deal with in a case interview:

Your client is Ginetto gelato; a shop that sells ice cream in London. They make fresh ice cream on site every day using high quality, organic ingredients. If they have excess ice cream, they freeze it to make ice lollies that are then sold to another retailer. Making a kilo of ice cream costs Ginetto £15, and it is sold for 30. Ice lollies, however, can only be sold for 12. On any given day, the shop expects to sell 100kg of ice cream if it is sunny and only 30kg if it is rainy. In London, the probability of rain on any given day is 75%. Ginetto has asked you how much gelato they should make to maximise their profit.

This will seem like a knotty problem if you don't know what you're doing. However, in MCC Academy, we show you how to optimise Ginetto's ice cream production in two different ways, demonstrating how to deal with these kinds of interview questions in straightforward and - crucially - time efficient fashion.

Pens pencils and rulers, illustrating various consulting interview math skills, including reading charts

Charts

The above much covers the fundamental math you will need to reach a solution to your case. However, there are other, related skills which you will need beyond familiarity with these basic concepts.

Some mathematical skills will be required throughout the case, not just in computing final solutions. In particular, it is likely that you will have to interpret charts as you lead the analysis. Case interviews are not like exams, where you simply receive a question and solve it without further input. Rather, there is an ongoing dialogue between interviewer and candidate.

Generally, you will need to acquire more information in order to eventually answer the interviewer's main question. This will often be provided to you in the form of charts - meaning that you will have to be able to interpret these in order to get at the information you need.

Basics

As a starting point, you should be familiar with the kind of basic graphs and tables which you might be familiar with from Excel. As well as standard tables of values, you should be absolutely comfortable reading the following:

Pie Charts - used to show data in one dimension, using percentage area
Bar Charts - used to display data in two dimensions - the x and y axes
Line Graphs - used to display data in two dimensions - the time axis and the y axis
Scatter Plots - used to display data in up to three dimensions - the x and y axes and the size of the "bubble" for each point, where this varies.

Even these basic kinds of chart can take multiple forms, though, and it can be a more useful distinction to categorise charts by their function in conveying information rather than their specific form. As such we can think about these charts as records of the following:

Comparisons/Relationships - showing a correlation or pattern - generally with a bar or line graph. For example, demand for a product versus the age of buyers.

Distributions - showing how data is distributed to provide the viewer with a sense of the mean, standard deviation etc, generally with pie or bar charts. For example, the bodyweights of a group of individuals.

Trends - quantities shown over a period of time so as to identify seasonal variations, generally with a line graph. For example, weekly sales of a product over a three year period.

Composition - showing how a whole is divided into parts, generally with a pie chart or scatter plot. For example, the market share of different car producers in a geographic region.

Charts for Case Studies: Adding Complexity

The charts you receive in case studies will typically be rather more complex than a basic pie chart or bar graph. Charts become more complex as more and more data is added to them - generally by allowing data to be encoded in additional dimensions. Given there are an indefinite number of ways for this to be done, it is impossible to give an exhaustive treatment (though we discuss case study charts in more detail in MCC Academy). Indeed, as charts become more complex and are merged with graphic design elements, there is an increasing trend across business to the production of fully fledged infographics.

Example: Stacked bar charts

To take what is still a relatively simple example, we can significantly increase information content by generating "stacked" bar charts, where each bar is subdivided into constituent portions. Frequently, even more data will be added by recording additional values against each bar. Below, we can see how a stacked bar chart provides information about the specific product breakdown which accounts for overall annual revenues for a food retailer:

Example of a stacked bar chart being used in a case study about a food retailer

Stacked bar charts can be used to provide information about the relationships between elements. For example, the chart below shows the effect of government subsidies on the returns generated by different energy sources in Canada:

Stacked bar chart showing the relationships between subsidised and unsubsidised energy sources in a case study about energy in Canada

Alternatively, stacked bar charts can also be used to show the differences between elements, as shown below. Here we see some data showing rising demand for various types of building in a region of England. The chart allows us to appreciate rising demand for buildings as well as the extent to which this might be ameliorated by existing buildings re-entering the market.

Stacked bar chart showing the differences between demand for buildings at different times in East Anglia from a  case study about the building industry
Example: Complex Tables

Case studies will very often contain complex tables which display information in multiple dimensions. You will need to be able to quickly interpret these and pull out key values. An example is shown below. Here, we see the success of a large, multi-channel advertising campaign made by a new political party to secure public donations.

Complex table showing the effect of a political party's advertising campaign on donations

We discuss these complex tables in more detail in our Consulting Math lesson in MCC Academy. There, we also discuss and give examples of how tables can be used to illustrate information about processes carried out by a client company.

Writing Equations

In simpler cases, you will be able to analyse scenarios verbally and move straight to the relevant arithmetic without having to resort to equations. However, as cases become more complex, working in this way becomes exponentially more difficult - such that it soon becomes impossible to keep track of all the variables as well as all the relationships between them.

In such cases, you should be able to express the problem you are working with as an equation. This will allow you to move towards a solution via more complex reasoning and keeping reliable track of more items than you can do verbally.

Let's look at an example of how we have to change how we work as problems become more complex:

Q1: I am 25 years old and my sister is 3 years older than me. What is my sister’s age?

This problem is easy to solve with basic arithmetic. Thus, the sister's age is simply 25+3=28yrs.

Q2: I am 25 years old today. 5 years before I was born, my father’s age was 19 years less than double my age 5 years ago. What is my father’s age today?

Now it is much harder to solve the problem directly as above. However, an equation makes matters easy. If we make our own age the variable a we can solve as follows:

Writing Equations

Being comfortable with equations has other benefits. In the simple, linear optimisations which we looked at above, having the relevant equation and knowing the boundary conditions is enough to be able to optimise the function. In the simple example we looked at, if we are trying to maximise y = 2x + 1 for x between 0 and 4, then the fact that the coefficient of x (that is, 2) here is positive is enough for us to know that the graph will have an upward slope the function will be maximised at the upper bound of x - which will be x = 4 in this case. This is, without drawing a graph or doing any calculations!

Mental Math

As we noted at the start of this article, consultants take mental math very seriously and you will need your calculations to be sharp in interview if you want to get a job. We have already noted a few "hacks" that will help you perform some operations more quickly. However, these are just a small subset of a whole host of such skills which you should ideally be able to draw upon.

Our video lesson on consulting math in MCC Academy explains a full set of such skills. Here, we'll just take a look at a couple of these techniques to get an idea of the kind of methods which consultants use day-to-day to make quick calculations and which are invaluable in case interviews.

X% of Y is Y% of X

What is 28% of 75? Difficult, isn't it? Well, not really. The answer will be the same as 75% of 28, which is much easier to calculate, as we should already know that 28 ÷ 4 = 7, so 75% of 28 is just 3 x 7 = 21. Easy!

Rule of 11

63 x 11 = what? If you have to think about this for more than two seconds, you are too slow. Luckily, there is a rule here which can help. Specifically, if you have to multiply a two digit number by 11, you simply add the two digits together and place whatever the result is between them. As such, for 63 x 11, we add 6 + 3 = 9 and put that 9 between 6 and 3 to get 693 - the correct answer! Similarly, if we wanted to multiply 26 by 11, we would add 2 + 6 = 8, giving an answer of 286.

If you want to learn similar techniques to be able to calculate almost instantly that 4900 ÷ 50 = 98, or that 387 ÷ 9 is 43, then you should check out the math content in MCC Academy.

Note that it is tempting to think of these kinds of "tricks" as somehow "optional extras" in your case interview prep. However, you need to keep in mind what we said earlier about consulting math being an entirely different beast to the academic math you will be accustomed to. In this context, these kinds of quick calculation methods are core skills, which you can expect to need to impress your interviewer and land an MBB job.

To make sure your mental math is as sharp as it possibly can be, you should be practicing constantly right up until your interview. You will get a certain amount of practice from case practice (see our free case bank), but you should also work on math separately. Our free mental math tool is a great resource here, as are our specialist math packages.

Takeaway

This article gives you a good idea of the math need to cover as you prep for your case interview. For some of you, it might be a relief to find out that the mathematical concepts required are not actually terribly complex. However, it is crucial not to become complacent about case interview math! The challenge is not in the level of the math, but in being able to conduct the relevant calculations efficiently and quickly, without the help of a calculator or computer and without making your interviewer wait for you to painstakingly solve long divisions or the like.

Now that you know which mathematical topics you need to read up on, you should get the basics firmly established in your mind and immediately move strait to practice. Our free mental math tool is a great resource here, as are our various math packages.

Mental math in particular is a skill in itself, though, and there are specific techniques or "hacks" which you should learn if you want to impress in interview. The content in this article is a great start, though the most comprehensive resource remains our math lesson on MCC Academy, which takes a detailed look at a whole range of techniques to greatly speed up your calculations.