case study of math

Case interview maths (formulas, practice problems, and tips)

Today we’re going to give you everything you need in order to breeze through maths calculations during your case interviews. 

Becoming confident with maths skills is THE first step that we recommend to candidates like Karthik , who got an offer from McKinsey. 

And one of the first things you’ll need to know are the 6 core maths formulas that are used extensively in case interviews. 

Let’s dive in!

• Case interview maths formulas
• Must-know formulas
• Optional formulas
• Cheat sheet
• Practice questions
• Case maths apps and tools
• Tips and tricks
• Practice with experts

Click here to practise 1-on-1 with MBB ex-interviewers

1. case interview maths formulas, 1.1. must-know maths formulas.

Here’s a summarised list of the most important maths formulas that you should really master for your case interviews:

If you want to take a moment to learn more about these topics, you can read our in-depth article about  finance concepts for case interviews .

1.2. Optional maths formulas

In addition to the above, you may also want to learn the formulas below. 

Having an in-depth understanding of the business terms below and their corresponding formulas is NOT required to get offers at McKinsey, BCG, Bain and other firms. But having a rough idea of what they are can be handy.

EBITDA = Earnings Before Interest Tax Depreciation and Amortisation

EBIDTA is essentially profits with interest, taxes, depreciation and amortisation added back to it.

It's useful for comparing companies across industries as it takes out the accounting effects of debt and taxes which vary widely between, say, Meta (little to no debt) and ExxonMobil (tons of debt to finance infrastructure projects). More  here .

NPV = Net Present Value

NPV tells you the current value of one or more future cashflows. 

For example, if you have the option to receive one of the two following options, then you could use NPV to choose the more profitable option:

• Option 1 : receive $100 in 1 year and $100 in 2 years
• Option 2 : receive $175 in 1 year

If we assume that the interest rate is 5% then option 1 turns out to be slightly better. You can learn more about the formula and how it works  here .

Return on equity = Profits / Shareholder equity

Return on equity (ROE) is a measure of financial performance similar to ROI. ROI is usually used for standalone projects while ROE is used for companies. More  here .

Return on assets = Profits / Total assets

Return on assets (ROA) is an alternative measure to ROE and a good indicator of how profitable a company is compared to its total assets. More  here .

1.3 Case interview maths cheat sheet

If you’d like to get a free PDF cheat sheet that summarises the most important formulas and tips from this case interview maths guide, just click on the link below.

Download free pdf case interview maths cheat sheet

2. Case interview maths practice questions

If you’d like some examples of case interview maths questions, then this is the section for you!

Doing maths calculations is typically just one step in a broader case, and so the most realistic practice is to solve problems within the context of a full case.

So, below we’ve compiled a set of maths questions that come directly from  case interview examples  published by McKinsey and Bain. 

We recommend that you try solving each problem yourself before looking at the solution. 

Now here’s the first question!

2.1 Payback period - McKinsey case example

This is a paraphrased version of question 3 on  McKinsey’s Beautify practice case :

How long will it take for your client to make back its original investment, given the following data?

• After the investment, you’ll get 10% incremental revenue
• You’ll have to invest €50m in IT, €25m in training, €50m in remodeling, and €25m in inventory
• Annual costs after the initial investment will be €10m 
• The client’s annual revenues are €1.3b

Note: take a moment to try solving this problem yourself, then you can get the answer under  question 3 on McKinsey’s website . 

2.2 Cost reduction - McKinsey case example

This is a paraphrased version of question 2 on  McKinsey’s Diconsa practice case :

How much money in total would families in rural Mexico save per year if they could pick up benefits payments from Diconsa stores?

• Pick up currently costs 50 pesos per month for each family
• If pick up were available at Diconsa stores, the cost would be reduced by 30%
• Assume that the population of Mexico is 100m 
• 20% of Mexico’s population is in rural areas, and half of these people receive benefits
• Assume that all families in Mexico have 4 members

Note: take a moment to try solving this problem yourself, then you can get the answer under  question 2 on McKinsey’s website . 

2.3 Product launch - McKinsey case maths example

This is a paraphrased version of question 2 on  McKinsey’s Electro-Light practice case :

What share of the total electrolyte drink market would the client need in order to break even on their new Electro-Light drink product?

• The target price for Electro-Light is $2 for each 16 oz (1/8th gallon) bottle
• Electro-Light would require $40m in fixed costs
• Each bottle of Electro-Light costs $1.90 to produce and deliver
• The electrolyte drink market makes up 5% of the US sports-drink market
• The US sports-drink market sells 8b gallons of beverages per year

2.4 Pricing strategy - McKinsey case maths example

This is a paraphrased version of question 3 on  McKinsey’s Talbot Trucks practice case :

What is the highest price Talbot Trucks can charge for their new electric truck, such that the total cost of ownership is equal to diesel trucks? 

• Assume the total cost of ownership for all trucks consists of these 5 components: driver, depreciation, fuel, maintenance, other. 
• A driver costs €3k/month for diesel and electric trucks
• Diesel trucks and electric trucks have a lifetime of 4 years, and a €0 residual value
• Diesel trucks use 30 liters of diesel per 100km, and diesel fuel costs €1/liter
• Electric trucks use 100kWh of energy per 100km, and energy costs €0.15/kWh
• Annual maintenance is €5k for diesel trucks and €3k for electric trucks
• Other costs (e.g. insurance, taxes, and tolls) is €10k for diesel trucks and €5k for electric trucks
• Diesel trucks cost €100k

2.5 Inclusive hiring - McKinsey case maths example

This is a paraphrased version of question 3 on  McKinsey’s  Shops Corporation practice case :

How many female managers should be hired next year to reach the goal of 40% women executives in 10 years? 

• There are 300 executives now, and that number will be the same in 10 years
• 25% of the executives are currently women
• The career levels at the company (from junior to senior) are as follows: professional, manager, director, executive
• In the next 5 years, ⅔ of the managers that are hired will become directors. And in years 6-10, ⅓ of those directors will become executives. 
• Assume 50% of the hired managers will leave the company
• Assume that everything else in the company’s pipeline stays the same after hiring the new managers

2.6 Breakeven point - Bain case maths example

This is a paraphrased version of the calculation portion of  Bain’s Coffee Shop Co. practice case : 

How many cups of coffee does a newly opened coffee shop need to sell in the first year in order to break even?

• The price of coffee will be £3/cup
• Each cup of coffee costs £1/cup to produce 
• It will cost £245,610 to open the coffee shop
• It will cost £163,740/year to run the coffee shop

Note: take a moment to try solving this problem yourself, then you can get the answer  on Bain’s website .

2.7 Driving revenue - Bain case maths example

This is a paraphrased version of the calculation part of  Bain’s FashionCo practice case : 

Which option (A or B) will drive the most revenue this year?

Option A: Rewards program

• There are 10m total customers
• The avg. annual spend per person is $100 before any sale (assume sales are evenly distributed throughout the year)
• Customers will pay a $50 one-time activation fee to join the program
• 25% of customers will join the rewards program this year
• Customers who join the rewards program always get 20% off

Option B: Intermittent sales

• For 3 months of the year, all products are discounted by 20%
• During the 3 months of discounts, purchases will increase by 100%

3. Case maths apps and tools

In the case maths problems in the previous section, there were essentially 2 broad steps: 

• Set up the equation
• Perform the calculations

After learning the formulas earlier in this guide, you should be able to manage the first step. But performing the mental maths calculations will probably take some more practice. 

Mental maths is a muscle. But for most of us, it’s a muscle you haven’t exercised since high school. As a result, your  case interview preparation  should include some maths training.

If you don't remember how to calculate basic additions, substractions, divisions and multiplications without a calculator, that's what you should focus on first.

In addition, Khan Academy has also put together some helpful resources. Here are the ones we recommend if you need an in-depth arithmetic refresher:

• Additions and subtractions
• Multiplications and divisions
• Percentages

Scientific notation

Once you're feeling comfortable with the basics you'll need to regularly exercise your mental maths muscle in order to become as fast and accurate as possible.

• Preplounge's maths tool . This web tool is very helpful to practice additions, subtractions, multiplications, divisions and percentages. You can both sharpen your precise and estimation maths with it.
• Victor Cheng's maths tool . This tool is similar to the Preplounge one, but the user experience is less smooth in our opinion.
• Mental math cards challenge app  (iOS). This mobile app lets you work on your mental maths easily on your phone. Don't let the old school graphics deter you from using it. The app itself is actually very good.
• Mental math games  (Android). If you're an Android user this one is a good substitute to the mental math cards challenge one on iOS.

4. Case interview maths tips and tricks

4.1. calculators are not allowed in case interviews.

If you weren’t aware of this rule already, then you’ll need to know this: 

Calculators are not allowed in case interviews. This applies to both in-person and virtual case interviews. And that’s why it’s crucial for candidates to practice doing mental maths quickly and accurately before attending a case interview. 

And unfortunately, doing calculations without a calculator can be really slow if you use standard long divisions and multiplications. 

But there are some tricks and techniques that you can use to simplify calculations and make them easier and faster to solve in your head. That’s what we’re going to cover in the rest of this section. 

Let’s begin with rounding numbers.

4.2. Round numbers for speed and accuracy

The next 5 subsections all cover tips that will help you do mental calculations faster. Here’s an overview of each of these tips: 

And the first one that we’ll cover here is rounding numbers. 

The tricky thing about rounding numbers is that if you round them too much you risk:

• Distorting the final result
• Or your interviewer telling you to round the numbers less

Rounding numbers is more of an art than a science, but in our experience, the following two tips tend to work well:

• We usually recommend that you avoid rounding numbers by more than +/- 10%. This is a rough rule of thumb but gives good results based on conversations with past candidates.
• You also need to alternate between rounding up and rounding down so the effects cancel out. For instance, if you're calculating A x B, we would recommend rounding A UP, and rounding B DOWN so the rounding balances out.

Note that you won't always be able to round numbers. In addition, even after you round numbers the calculations could still be difficult. So let's go through a few other tips that can help in these situations.

4.3. Abbreviate large numbers

Large numbers are difficult to deal with because of all the 0s. To be faster you need to use notations that enable you to get rid of these annoying 0s. We recommend you use labels and the scientific notation if you aren't already doing so.

Labels (k, m, b)

Use labels for thousand (k), million (m), and billion (b). You'll write numbers faster and it will force you to simplify calculations. Let's use 20,000 x 6,000,000 as an example.

• No labels: 20,000 x 6,000,000 = ... ???
• Labels: 20k x 6m = 120k x m = 120b

This approach also works for divisions. Let's try 480,000,000,000 divided by 240,000,000.

• No labels: 480,000,000,000 / 240,000,000 = ... ???
• Labels: 480b / 240m = 480k / 240 = 2k

When you can't use labels, the scientific notation is a good alternative. If you're not sure what this is, you're really missing out. But fortunately, Khan Academy has put together a good primer on that topic  here .

• Multiplication example: 600 x 500 = 6 x 5 x 102 X 102 = 30 x 104 = 300,000 = 300k
• Division example: (720,000 / 1,200) / 30 = (72 / (12 x 3)) x (104 / (102 x 10)) = (72 / 36) x (10) = 20

When you're comfortable with labels and the scientific notation you can even start mixing them:

• Mixed notation example: 200k x 600k = 2 x 6 x 104 x m = 2 x 6 x 10 x b = 120b

4.4. Use factoring to make calculations simpler

To be fast at maths, you need to avoid writing down long divisions and multiplications because they take a LOT of time. In our experience, doing multiple easy calculations is faster and leads to less errors than doing one big long calculation.

A great way to achieve this is to factor and expand expressions to create simpler calculations. If you're not sure what the basics of factoring and expanding are, you can use Khan Academy again  here  and  here . Let's start with factoring.

Simple numbers: 5, 15, 25, 50, 75, etc.

In case interviews some numbers come up very frequently, and it's useful to know shortcuts to handle them. Here are some of these numbers: 5, 15, 25, 50, 75, etc. 

These numbers are common, but not particularly easy to handle.

For instance, consider 36 x 25. It's not obvious what the result is. And a lot of people would need to write down the multiplication on paper to find the answer. However there's a MUCH faster way based on the fact that 25 = 100 / 4. Here's the fast way to get to the answer:

• 36 x 25 = (36 / 4) x 100 = 9 x 100 = 900

Here's another example: 68 x 25. Again, the answer is not immediately obvious. Unless you use the shortcut we just talked about; divide by 4 first and then multiply by 100:

• 68 x 25 = (68 / 4) x 100 = 17 x 100 = 1,700

Factoring works both for multiplications and divisions. When dividing by 25, you just need to divide by 100 first, and then multiply by 4. In many situations this will save you wasting time on a long division. Here are a couple of examples:

• 2,600 / 25 = (2,600 / 100) x 4 = 26 x 4 = 104
• 1,625 / 25 = (1,625 / 100) x 4 = 16.25 x 4 = 65

The great thing about this factoring approach is that you can actually use it for other numbers than 25. Here is a list to get you started:

• 2.5 = 10 / 4
• 7.5 = 10 x 3 / 4
• 15 = 10 x 3 / 2
• 25 = 100 / 4
• 50 = 100 / 2
• 75 = 100 x 3 / 4

Once you're comfortable using this approach you can also mix it with the scientific notation on numbers such as 0.75, 0.5, 0.25, etc.

Factoring the numerator / denominator

For divisions, if there are no simple numbers (e.g. 5, 25, 50, etc.), the next best thing you can do is to try to factor the numerator and / or denominator to simplify the calculations. Here are a few examples:

• Factoring the numerator: 300 / 4 = 3 x 100 / 4 = 3 x 25 = 75
• Factoring the denominator: 432 / 12 = (432 / 4) / 3 = 108 / 3 = 36
• Looking for common factors: 90 / 42 = 6 x 15 / 6 x 7 = 15 / 7

4.5. Expand numbers to make calculations easier

Another easy way to avoid writing down long divisions and multiplications is to expand calculations into simple expressions.

Expanding with additions

Expanding with additions is intuitive to most people. The idea is to break down one of the terms into two simpler numbers (e.g. 5; 10; 25; etc.) so the calculations become easier. Here are a couple of examples:

• Multiplication: 68 x 35 = 68 x (10 + 25) = 680 + 68 x 100 / 4 = 680 + 1,700 = 2,380
• Division: 705 / 15 = (600 + 105) / 15 = (15 x 40) / 15 + 105 / 15 = 40 + 7 = 47

Notice that when expanding 35 we've carefully chosen to expand to 25 so that we could use the helpful tip we learned in the factoring section. You should keep that in mind when expanding expressions.

Expanding with subtractions

Expanding with subtractions is less intuitive to most people. But it's actually extremely effective, especially if one of the terms you are dealing with ends with a high digit like 7, 8 or 9. Here are a couple of examples:

• Multiplication: 68 x 35 = (70 - 2) x 35 = 70 x 35 - 70 = 70 x 100 / 4 + 700 - 70 = 1,750 + 630 = 2,380
• Division: 570 / 30 = (600 - 30) / 30 = 20 - 1= 19

4.6. Simplify growth rate calculations

You will also often have to deal with growth rates in case interviews. These can lead to extremely time-consuming calculations, so it's important that you learn how to deal with them efficiently.

Multiply growth rates together

Let's imagine your client's revenue is $100m. You estimate it will grow by 20% next year and 10% the year after that. In that situation, the revenues in two years will be equal to:

• Revenue in two years = $100m x (1 + 20%) x (1 + 10%) = $100m x 1.2 x 1.1 = $100m x (1.2 + 0.12) = $100m x 1.32 = $132m

Growing at 20% for one year followed by 10% for another year therefore corresponds to growing by 32% overall.

To find the compound growth you simply need to multiply them together and subtract one: (1.1 x 1.2) - 1= 1.32 - 1 = 0.32 = 32%. This is the quickest way to calculate compound growth rates precisely.

Note that this approach also works perfectly with negative growth rates. Let's imagine for instance that sales grow by 20% next year, and then decrease by 20% the following year. Here's the corresponding compound growth rate:

• Compound growth rate = (1.2 x 0.8) - 1 = 0.96 - 1 = -0.04 = -4%

See how growing by 20% and then shrinking by 20% is not equal to flat growth (0%). This is an important result to keep in mind.

Estimate compound growth rates

Multiplying growth rates is a really efficient approach when calculating compound growth over a short period of time (e.g. 2 or 3 years).

But let's imagine you want to calculate the effect of 7% growth over five years. The precise calculation you would need to do is:

• Precise growth rate: 1.07 x 1.07 x 1.07 x 1.07 x 1.07 - 1 = ... ???

Doing this calculation would take a lot of time. Fortunately, there's a useful estimation method you can use. You can approximate the compound growth using the following formula:

• Estimate growth rate = Growth rate x Number of years

In our example:

• Estimate growth rate: 7% x 5 years = 35%

In reality if you do the precise calculation (1.075 - 1) you will find that the actual growth rate is 40%. The estimation method therefore gives a result that's actually quite close. In case interviews your interviewer will always be happy with you taking that shortcut as doing the precise calculation takes too much time.

4.7. Memorise key statistics

In addition to the tricks and shortcuts we’ve just covered, it can also help to memorise some common statistics. 

For example, it would be good to know the population of the city and country where your target office is located. 

In general, this type of data is useful to know, but it's particularly important when you face  market sizing questions . 

So, to help you learn (or refresh on) some important numbers, here is a short summary:

Of course this is not a comprehensive set of numbers, so you may need to tailor it to your own location or situation.   

5. Practice with experts

Sitting down and working through the maths formulas we've gone through in this article is a key part of your case interview preparation. But it isn’t enough.

At some point you’ll want to practise making calculations under interview conditions.

You can try to do this with friends or family. However, if you really want the best possible preparation for your case interview, you'll also want to work with ex-consultants who have experience running interviews at McKinsey, Bain, BCG, etc.

If you know anyone who fits that description, fantastic! But for most of us, it's tough to find the right connections to make this happen. And it might also be difficult to practice multiple hours with that person unless you know them really well.

Here's the good news. We've already made the connections for you. We’ve created a coaching service where you can do mock interviews 1-on-1 with ex-interviewers from MBB firms. Learn more and start scheduling sessions today.

Case Interview Math: a comprehensive guide

• Case Interview: A comprehensive guide
• Pyramid Principle
• Hypothesis driven structure
• Fit Interview

Consulting math

• Consulting Math in Principle
• Interview Math in Practice
• Fundamentals: A Checklist
• Writing Equations
• Mental Math

There's no way around it - math is a key part of the management consulting selection process. You are going to need to prep your math if you want to have a chance of landing any consulting job, let alone at a top-flight role at an MBB or similar firm.

Even before you make it to case interviews, the latest aptitude tests and online cases being rolled out by the major consulting firms are featuring more and more mathematical questions. Particularly prominent examples are BCG's Casey chatbot case study and the new versions of McKinsey's Solve assessment - both have substantial, demanding math components.

If and when you make it through to case interviews, these will almost certainly feature another wall of math for you to make it through - and you will have to work hard to impress your interviewer here, as expectations will be high. This is somewhere where far too many candidates fail to prepare properly and really let themselves down.

Now, this might seem contradictory, but, whilst your math needs to be very sharp to land a consulting job, you simultaneously won't need a huge depth of mathematical knowledge to do well. You certainly don't need to come from a quantitative background at university - indeed, the math you were doing by age 16 in high school will be more than sufficient.

However,the key thing to note is that consulting math is very different to academic math . Even if you do have strong mathematcial training, you won't get far approaching problems the same way you did at university.

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 the areas in which you need to be proficient, whilst giving some tips on "hacks" that you can use to excel in tests and interviews.

Math is one of the most important elements of preparing for a consulting interview, and this article is a great point of entry into the subject. However, it is impossible to be fully comprehensive in any reasonable amount of space - for a start, we're not going to reproduce your high school math textbook here!

Where appropriate, we'll point you towards useful public resources, including other articles on this site. Generally, though, if you want a more comprehensive source, you should check out the full Consulting math content within our comprehensive Case Academy course:

Case Academy Course

If you want to start off with just that math content, you can find this in isolation in our Consulting Math package:

Consulting Math

Finally, if consulting prep rightly seems daunting to do alone, you can investigate getting some coaching from real consultants here:

Explore Coaching

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, being good at math in an academic context is a solid 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 don't approach problems in the way their interviewer expects.

Prep the right way

In academic math, the overriding concern is accuracy. It might take a lot of complex work and a great deal 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 heavy 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, precise answer is simply not required. Instead, consultants will simplify their analyses to be more time efficient .

In case interviews, special importance will be ascribed to mental math . Of course, being able to do mental math quickly demonstrates mental agility. However, consultants also frequently use quick mental math to impress clients (and thus help justify their fees). The sharper your mental math, the more impressed your interviewer will be. We include a brief section on mental math skills below, with much more detailed treatment in the MCC Academy or our specific math package .

Consulting math in practice

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 of how things will actually be in the consulting selection process. Let's get some of the most straightforward matters out of the way before we look at consulting math in more depth.

In this article, we'll primarily focus on math for case interview, just as that's the tougher nut to crack. Math for tests or online cases will generally be at the same conceptual level, but with calculators and/or Excel allowed and without having to constantly explain your reasoning to a harsh audience.

That said, the math for tests and online cases is certainly not easy, and we include some specific notes on prepping for these throughout this article.

Case Interview Specifics

Perform calculations on paper.

In case interviews, you will be given a piece of paper and should feel free to use it when doing calculations.

The time pressure in case interviews is severe, and you cannot afford to waste a second. 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".

Be Assertive

Candidates who 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 sound confident and state their answers with an air of certainty.

Ask About Rounding

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

Math for Aptitude Tests and Online Cases

We'll include some specific notes on math for screening tests and/or online cases as we go. Online cases in particular are increasingly being deployed either before or alongside first round case interviews and have been featuring more and more math.

In general, though, the kind of math, and the conceptual level it's pitched at, will remain the same as for case interviews. Nothing will be more complicated than basic high school math, with the focus still very much falling on things like percentages, charts, multiplication etc.

As we note in the relevant section in this article, one very small change has been a move away from "average" simply being synonymous with "mean". Instead, newer tests are increasingly asking candidates to calculate median and modal values as well.

However, the really salient difference between case interview math and that found in aptitude tests, online cases and the like is that the latter allow you to use calculators and/or Excel. This doesn't necessarily make things easier so much as change the emphasis of math questions quite a bit.

In case interviews, big part of the challenge is simply performing the calculations sufficiently quickly - with this entailing clever use of estimation/approximation to deal with large numbers in timely fashion. By contrast, with access to electronic help in online tests and cases, arithmetic becomes trivially easy and approximations become unnecessary.

Now, the emphasis is on how you set up calculations and figure out how to get to the answer you need. For sure, this is very important in case interviews as well, but the presence of calculators etc allows this aspect of the math to be made more demanding. Thus, you can expect to have to use a little more algebra and/or conduct more multi-step calculations. You will also be given less time to complete questions, in light of the fact you have help.

In terms of preparation, things stay very largely the same, and all the case interview focused material from this article will carry over directly.

The main thing to add is to spend some time solving problems with a calculator and/or Excel - especially if you don't do this day-to-day. If you aren't proficient with Excel already and don't have long until your test/online case, don't worry and stick to calculator practice. However, if you have a little more time and/or a little more starting proficiency, getting up to speed can provide a small, but real, advantage in certain questions - particularly where you need to calculate averages.

Forget outdated, framework-based guides...

Fundamentals: a checklist of consulting math skills.

So, which math skills do you need?

Here, we'll go through the main areas you should cover to prep for a standard MBB interview or aptitude test/online case.

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 and our math package .

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 pick up a basic math textbook or fire up Google.

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 case interviews, you should be readily able to add/subtract and multiply/divide fractions. There are a couple of ways to manipulate fractions that will be particularly useful:

Approximating Divisions

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

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

We know that:

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!

Efficiently Navigating Math Problems

Fractions also help simplify your analysis of certain problems. Let's take a relatively simple example:

1/3 of a company's employees are software engineers. Due to new generative AI tools increasing productivity, 1/3 of the software engineers are to be laid off. What fraction of the remaining employees are software engineers? Software engineers laid off: Remaining software engineers: Employees remaining in the company: Therefore, the fraction of remaining employees who are software engineers is:

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, four pencils, and one eraser, then the ratio between them is 3:4:1.

Join thousands of other candidates cracking cases like pros

Fractions come up in all kinds of business problems. For solving case studies, 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 using a similar method to how we solved our example of software engineers exiting a workforce, above.

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 more than that for drinks, what is the total profit?

You should be able to arrive at an answer very quickly - certainly in under a minute. We show you how to do so in a MCC Academy , also available in our specific math package .

3. Percentages

Similar to fractions and ratios, we can think of percentages as ratios 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 and online tests and cases. Indeed, the most recent online cases - particularly newer versions of the McKinsey Solve assessment - have asked candidates to make a lot of percentage calculations, especially percentage changes in quantities.

In case studies, we might be dealing with profits that 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 addressing issues around pricing - such as applying mark-ups on products to generate profit or offering discounts to promote sales.

Note that percentages will sometimes be discussed in terms of "percentage points". As such, if you are told that revenues are down by 20 percentage points - or even just 20 points - this simply means that revenues have fallen by 20%.

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

Marta has a shop selling handbags for €30. She offers a 20% discount for one day. She then realises that the price is now too low, so she increases the price by 10%. What it is the current price of Marta's Handbags?

In the MCC Academy math video, also included in our specific math package we show how to answer this question in just a few seconds.

4. Probability

Nothing is certain in the business world. Thus, when consultants make decisions, they must constantly evaluate the probabilities of different future events.

The probability of such an event will always be a number between 0 (impossible) and 1 (certain) , calculated as the number of ways that an event can happen, divided by the total number of possible outcomes. Therefore, the probability of rolling a six on a fair die is 1/6, as there are a total of six possible outcomes, only one of which is the event in question.

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

You also need to know how to calculate the probability of multiple chance events all occurring. Luckily, in case interviews, tests etc, you will only have to deal with independent events, where individual outcomes do not influence subsequent ones.

The standard example here 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 dependent events 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:

Expected Returns

Probability is especially relevant to business where we need to calculate expected returns. Here, we weight the yield promised by an investment by the probability that it will pay off . 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 choose between two opportunities that 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 a probability of only 0.7.

The expected returns are:

As such, we should favour option A as yielding a greater expected return, despite option B's greater headline payout.

This is a very simple example. However, we take a look how to calculate a more complex expected return in the MCC Academy video lesson, also available in our consulting math package .

5. Averages

We can think of an average as a measure of the "typical" value of some series of numbers .

Unless you are told otherwise, any talk of averages in a case interview will refer specifically to the mean (very specifically the arithmetic if you want to be nerdy about it...). This 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:

Means 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 means. 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% work 8 hours and the rest 12 hours a day. What is the average time for which an employee works?

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

Now, whilst averages are typically synonymous with means in case interviews, there has been a little more variation in kinds of average coming up amongst the recent proliferation of online cases as part of the consulting selection process. Specifically, questions have frequently been asking candidates to calculate the mode and/or median of datasets.

These averages can be a little more tricky to manually compute than the mean - not more difficult so much as more time consuming and annoying. Luckily, these online cases allow for calculators and/or Excel to make things more straigtforward. Thus, it's definitely worth getting good at using these to find the mean, mode and median before you sit tests like McKinsey's Solve or BCG's Casey

Rates are ubiquitous across the business world in general and within consulting in particular. We can think about rates as a ratio or fraction where the denominator is always 1. Some rates you will encounter include the interest rate, the rate of inflation, various tax rates, the 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 or 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 single set of units to facilitate comparison. For example, 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 in 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

A lot of business problems will boil down to the optimisation of one or more salient variables. Optimisation in a mathematical context can be a mind-bendingly complex affair. Indeed, 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 and tests is a pretty straightforward affair. The business problems you are given will almost invariably be linear. That is, their form will resemble something like y = ax + b .

This means that the relevant variable 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 instance.

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

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

Your client is Ginetto’s 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 maximize their profit.

This will seem pretty difficult if you don't know what you're doing. However, in the MCC Academy and our math package , we show you how to optimise Ginetto's ice cream production in two different ways, demonstrating how to deal with these kinds of case questions in straightforward and - crucially - time efficient fashion.

The article up to here pretty much covers the fundamental math you will need for case interviews and/or aptitude tests. However, there are other, related skills that 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 work through your 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 the interviewer and the candidate. Generally, you will need to acquire more and 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 charts in order to get the information you need.

Looking for an all-inclusive, peace of mind program?

Chart basics.

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

Charts in Online Cases and Aptitude Tests

Most of the time, the role of charts in aptitude tests and/or online cases will be very much the same as that in traditional case interviews. That is, you will be presented with charts to interpret so as to provide information to answer questions.

However, recent versions of the McKinsey Solve assessment in particular have turned things on their head and asked candidates to create charts to best express information.

Rather than start from scratch with something like Excel, though, test-takers have been asked to decide a few variables, such as the particular data set/s to be represented and the kind of chart to use - pie, line, scatter, bar or other. The test's software then does the work of actually generating the final chart.

Candidates we have spoken to have often regarded the choice of chart type as the most difficult aspect of this question, which leads neatly into our next section...

Charts at Higher Level

Even these basic kinds of charts 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 are 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.

Adding Complexity to Charts

The charts you have to interpret in case interviews and online tests will often be rather more complex than a basic pie chart or bar graph. Charts become more complex as more and more information 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 here (though we discuss case study charts in more detail in MCC Academy and our math package ). Indeed, as charts become more complex, they are often merged with graphic design elements, and there is an increasing trend in the business world towards producing 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. Often, 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 accounting for a food retailer's overall annual revenues:

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

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

Example: Complex Tables

Case studies will very often contain complex tables, displaying information in multiple dimensions. You will need 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 in order to secure public donations.

We discuss these complex tables in more detail in a lesson in the MCC Academy , also included in our separate math package .

Writing equations

In simpler case studies, 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, this becomes exponentially more difficult. Soon, it becomes impossible to keep track of all the variables and all the relationships between them.

In such cases, you should be able to express the problem as an equation. This will allow you to engage in more complex reasoning and keep track of more items than you can hope to verbally.

Let's look at an example of how we have to adapt 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?

Being comfortable with equations has other benefits too. In the simple, linear optimisations we looked at above, having the relevant equation and knowing the boundary conditions is enough to be able to optimise the function.

In the straightforward 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. Thus, the function will be maximised at the upper bound of x - which will be x = 4 in this case. Thus, we have an answer without drawing a graph or doing any calculations!

Mental math "hacks", tricks and timesavers

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 be able to draw upon.

Our video lesson on consulting math in MCC Academy and our math package covers a full set of these skills. Here, though, we'll just take a look at a couple of these techniques to get an idea of the kind of methods consultants use day-to-day to make quick calculations - and that 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. Since we should already know that 28 ÷ 4 = 7, 75% of 28 is just 3 x 7 = 21. Easy!

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 that 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 almost instantly calculate that 4900 ÷ 50 = 98, or that 387 ÷ 9 is 43, then you should check out the math content in our MCC Academy or our consulting math package .

It's tempting to think of these kinds of "tricks" as "optional extras" in your case interview prep. However, you must remember what we said earlier about consulting math being an entirely different beast versus the academic math to which you are accustomed. In this context, these kinds of quick calculation methods are core skills. Indeed, you can expect to need these skills to impress your interviewer enough to land an MBB or any top-tier consulting 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 some work in during case practice (remember to check out our free case bank ), but you should also be practicing math separately.

Our free mental math tool is a great resource here, as is our specialist math package :

• Video Lecture on Foundations of consulting math
• Video Lecture on Applied consulting math
• Video Lecture on Advanced topics in consulting math
• Video lecture on Advanced methods in mental math
• Actionable advice on how to improve your calculation speed and accuracy 
• 60+ chart based questions with detailed solutions
• 100+ business problems with detailed solutions
• Mental math tool to improve mental calculations speed

This article gives you a great idea of the math you need to cover as you prep for your case interviews and/or any aptitude tests or online cases. It might be a relief for some of you to find out that the mathematical concepts required are not hugely complex. However, it's crucial not to become complacent as a result!

The challenge is not in the level of the math itself, but in being able to conduct the relevant calculations efficiently and very quickly . In interviews, this will be without the help of a calculator or computer and with your interviewer impatiently bearing down on you.

Now that you know which mathematical topics you need to get up to speed 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 is our math package .

Mental math in particular is a skill in itself, though, and there are specific techniques or "hacks" that you should learn if you want to impress in case interviews. The content in this article is a great start, though the most comprehensive resource remains our material on the subject in the MCC Academy also included in the aforementioned math package. This takes a detailed look at a whole range of techniques to massively speed up your calculations.

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• McKinsey Solve Game (Imbellus)
• BCG Online Case (+ Pymetrics, Spark Hire)
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• Kearney Recruitment Test
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Case Interview Math: The Insider Guide

Last Updated on March 27, 2024

Embarking on a career in consulting at leading firms demands mastery over consulting math problems. Statistically speaking, a staggering 85% of management consulting case interviews put candidates to the test with case interview math questions. At top consultancies such as McKinsey , BCG , and Bain , this expectation skyrockets to nearly 100%.

During case interviews, candidates are tasked with dissecting complex mathematical business puzzles, delving into the qualitative aspects that underpin these numbers, and ultimately crafting strategic recommendations.

These mathematical challenges often emerge as the largest hurdle for many aspiring consultants. Drawing from our extensive background in conducting interviews at McKinsey and coaching thousands of interviews on platforms such as PrepLounge and StrategyCase.com, we’ve observed that the lion’s share of mishaps during case interviews arises within this quantitative segment.

Developing math skills in consulting interviews is crucial for candidates aiming for top-tier firms. The ability to navigate these numerical problems not only sets the foundation for success in case interviews but also mirrors the analytical challenges consultants face in real-world scenarios.

This article is your ultimate guide to consulting interview preparation, with a focus on math challenges. Our insights in this expert article aim to demystify the numerical proficiency required by top-tier consulting firms, preparing you to tackle these challenges head-on with confidence and strategic insight. It includes all relevant tips for solving consulting math problems, making complex calculations manageable.

It is a critical installment in our comprehensive consulting case interview prep series:

• Overview of case interviews: what is a consulting case interview?
• How to create a case interview framework
• How to ace case interview exhibit and chart interpretation
• How to ace case interview math questions (this article)
• How to approach brainstorming questions in case interviews

Why Candidates Struggle with Case Interview Math

The conundrum of case interview math is not intrinsically tied to the difficulty of the mathematical problems themselves, which often do not surpass high school-level arithmetic. You have solved similar problems before, maybe not in a business or interview context, but in a classroom setting.

Let’s start with some positivity.

There is no need to fear quantitative problems in case interviews. The level of math required is not more complex than what you have already learned in school and you do not need a specific degree to pass the case interviews.

The true challenge emerges from the synthesis of multiple skills under the high-pressure environment of a case interview.

Logical thinking is paramount, as you must not only arrive at the correct approach but do so swiftly and efficiently. This is compounded by the need to execute calculations with potentially large numbers accurately and quickly, all while maintaining composure to manage the interviewer’s impression. Communication also plays a critical role; articulating your thought process and conclusions in a clear and concise manner is essential.

When faced with the task of juggling these aspects simultaneously, it’s common for candidates to experience panic, leading to a decrease in overall performance. However, by deconstructing these skills and mastering each individually – logical problem-solving, fast and accurate arithmetic, effective communication, and impression management – you can significantly bolster your confidence. This strategic preparation not only mitigates the fear associated with case interview math but equips you with the comprehensive skill set necessary to excel.

That being said, as with every other element in a case interview ( structuring , brainstorming , exhibit and data interpretation ), there is a very specific way of approaching case interview math, which candidates are not used to from their previous academic or professional experience. Learn how to apply business case math to real-world consulting scenarios.

Let’s get to it!

Case Math Mastery Course and Drills

Learn how to tackle case interview math questions with the insight and precision of an experienced consultant with the most comprehensive preparation program on the market. Learn from our McKinsey interviewer experience and benefit from the detailed curriculum of the guidebook and the video program as well as 40 hours of practice.

The Purpose of Case Interview Math

Numerical analysis forms the backbone of decision-making and strategic recommendations in case interviews, reflecting the real-world consulting emphasis on data-driven insights.

In the context of a business problem usually found in a case interview, quantitative analyses are conducted for two reasons.

Identifying problems and quantifying their impact

Initially, consultants are tasked with identifying underlying issues within a business context. Through quantitative analysis, they delve deep into the problem, quantifying its impact to uncover root causes and, subsequently, potential solutions.

In the condensed format of a case interview, you’re expected to mirror this investigative approach, albeit in a more abbreviated manner.

Supporting recommendations

Quantitative data underpins every business recommendation, providing a solid foundation for decision-making. In consulting practice, every suggestion or strategic plan presented to a client is supported by numerical evidence.

Similarly, during a case interview, the quantitative analyses you conduct will critically inform your final recommendations.

Test of your quantitative skills

Moreover, case interviews serve as a proving ground for your quantitative skills, simulating the analytical rigor required in consulting. I cannot remember a single day in my McKinsey career, where I was not running some form of quantitative analysis.

Therefore, honing your ability to devise strategic, logical approaches to quantitative challenges and execute precise calculations is crucial not only for acing case interviews but also for thriving as a consultant.

This skillset ensures you’re well-equipped to deliver insights that drive impactful business decisions, marking your capability to thrive in the consulting domain.

A simplified version of reality

In the case interview context, the mathematical problems presented are invariably a streamlined representation of real business challenges, often drawn from the interviewer’s direct experience with actual clients. This means that while the scenarios aim to mimic the complexities of business decision-making, the approach and calculations are deliberately simplified for the sake of brevity and clarity.

For instance, scenarios might feature fewer market segments or shorter time periods than those in actual business cases, and variables are designed to be more straightforward, allowing for easier manipulation and calculation. It’s also common practice for candidates to round numbers to simplify the process further. Unlike the exhaustive analyses that can span weeks on the job as new insights emerge, a typical math problem in a case interview is designed to be resolved within a succinct 5 to 8-minute window from start to finish. That should give you an idea of how complex it can really be.

This distilled version of reality, however, does not make the task at hand any less challenging. The dual demands of strategizing your steps and executing calculations unfold under the watchful eye of the interviewer, all within a high-pressure, calculator-free environment.

Yet, mastering the basics – quick mental arithmetic, fundamental operations (addition, subtraction, multiplication, division, percentages, and fractions), and the ability to make judicious estimates – proves invaluable. These skills equip you to tackle most interview problems effectively, without the need for advanced mathematical knowledge.

While some problems might feature a complexity that demands logical problem-solving and potentially multiple calculation steps, the essence of case interview math lies in its reduced complexity, designed to assess your analytical acumen rather than your prowess in advanced mathematics.

The myth of perfection

In the high-stakes environment of case interviews, there’s a prevalent myth that perfection is the key to success. This belief leads many to think that any mistake, particularly in math, spells automatic rejection. However, this couldn’t be further from the truth. Mistakes, whether in calculations or pacing, are not uncommon and do not necessarily jeopardize your chances of success.

It’s important to recognize that errors, to an extent, are expected. You might miscalculate, take a bit longer to arrive at an answer, or even find the interviewer stepping in to guide you. These instances, in isolation, aren’t deal-breakers. They’re often seen as part of the process, providing insights into your problem-solving approach and resilience.

The critical factor is how you handle mistakes. An isolated error or a moment of slowness doesn’t doom your interview outcome. However, repeated errors, especially if they’re indicative of a pattern within the same interview or across multiple interviews, can raise concerns. Moreover, a single mistake leading to a cascade of follow-up errors, triggered by loss of confidence or panic, can be detrimental. This reaction, rather than the initial mistake itself, can hinder your performance significantly. I have seen this hundreds of times in live settings.

One key strategy to mitigate the impact of mistakes is to excel in other aspects of the case interview. Demonstrating exceptional analytical skills, creative problem-solving, or outstanding communication can offset occasional mathematical errors. Interviewers are looking for a well-rounded skill set, so performance spikes in areas other than math can greatly enhance your overall evaluation.

Ultimately, how you respond to mistakes is crucial. Viewing them as learning opportunities rather than failures can transform your interview experience. Showing the interviewer your ability to quickly recover, correct errors, and proceed with confidence speaks volumes about your potential as a consultant. It demonstrates resilience, adaptability, and a growth mindset.

Effective Strategies for Tackling Case Interview Math Questions

Different skill levels, same problem.

Understanding the unique challenges and logic behind math questions in case interviews reveals an interesting observation:

Candidates from various academic backgrounds might find themselves revisiting basic mathematical concepts not engaged with since high school. Conversely, individuals with a strong quantitative foundation, such as engineers, may need to simplify their analytical approach to align with the straightforward nature of case interview math. This adjustment is crucial for all candidates, regardless of their initial competency levels, to adapt to the nuances of case interview calculations effectively.

Both types of backgrounds need to adapt to the specific case interview math principles and process.

Unlike traditional math problems, case interview questions prioritize the relevance and application of mathematical solutions to the business scenario at hand. The aim is not merely to arrive at precise numerical answers but to receive directionally correct results to leverage these findings and inform strategic decisions within the case’s context. Thus, achieving perfectly accurate results is less critical than developing a sound, strategic approach that yields directionally correct insights swiftly.

It’s better to get directionally correct results swiftly and interpret them correctly than getting 100% accurate results and not providing any insights into the case problem. Approach case interview math with this mantra

Adopting a mindset that embraces quantitative analysis as an integral part of every case scenario is essential. This involves not just solving the problem at hand but also considering the broader implications of your calculations on the strategic recommendations you propose. The ability to relate different numerical factors and assess their impact on the business challenge is key.

The apprehension some candidates feel towards case math can be mitigated by understanding that these calculations are designed to reflect real-world business problems in a simplified manner. Therefore, embracing the opportunity to demonstrate logical thinking and analytical prowess through these mathematical exercises is vital.

Even more so, have a quantitative angle in every case, even if the interviewer does not explicitly ask you for it. For example, try to relate numbers to each other, think about the potential quantitative impact of your recommendation, etc.

Many candidates are simply scared of digging into the mathematics of a case. Don’t be that person! rather go where no one else is going and highlight your numerical prowess at every opportunity.

As we delve further, I aim to equip you with the knowledge and strategies to confidently tackle both the structuring and calculation aspects of math questions in case interviews, ensuring you’re well-prepared to handle the quantitative analysis that underpins effective case interview performances.

My approach to every case math problem

Mastering the art of solving quantitative problems in case interviews involves a two-pronged approach: developing a universal strategy applicable across various case scenarios and executing calculations to arrive at concrete insights.

How, then, should one tackle the numerical aspects of case interviews with a structured strategy that you can always rely on? Proving essential math skills for case interviews is less daunting with my step-by-step guide.

• Listening : Engage fully, paying close attention to the information provided by your interviewer. Active listening forms the foundation of your analytical process.
• Clarification : Pause to ensure clarity around the data presented or derived from visual aids such as charts and tables. It’s crucial to confirm the accuracy of these figures and understand the objective of your analysis before proceeding.
• Strategizing : Outline a clear, logical plan for your calculations. For complex problems, don’t hesitate to request a brief moment – typically a minute or two – to organize your thoughts and structure your approach on paper.
• Articulating your strategy : Communicate your planned methodology to the interviewer. This step is vital for preemptively identifying any potential errors and ensuring alignment on the approach.
• Calculation execution : With the interviewer’s nod, carry out your calculations diligently. It’s advisable to work through this phase methodically, allowing yourself time to focus without interruption.
• Verification : Review your work to catch and correct any errors. Ensuring your numbers are reasonable and accurate is key to building a solid argument.
• Presentation of results : Share your findings in a clear, concise, and assertive manner, avoiding presenting your conclusion as a question. Highlight the most critical results, adhering to a top-down communication style as recommended by the Pyramid Principle .
• Interpretation and next steps : Beyond just presenting numbers, interpret what they mean in the context of the case. How do they influence your analysis and recommendations? Always connect your findings back to the larger case narrative, exploring their implications and forming hypotheses based on these insights. Propose next steps.

The benefit of adopting a structured approach to quantitative problems in case interviews is twofold. Firstly, it showcases to the interviewer your ability to navigate complex situations with a level-headed, systematic strategy, effectively demonstrating case leadership qualities. This organized methodology signals that you possess the poise and strategic foresight necessary to dissect and solve business challenges – a trait highly valued in consulting.

Secondly, this approach creates an optimal environment for you to perform at your peak. By delineating the processes of thinking, communicating, and calculating, you’re able to maintain a sharp focus at any given moment. This separation ensures that each step of the problem-solving process receives your undivided attention, significantly enhancing your efficiency and effectiveness.

Conversely, when candidates attempt to juggle multiple aspects simultaneously – such as solving the problem while overly concentrating on managing the interviewer’s impression – results tend to suffer. This scattered focus often leads to underperformance in case interviews, as it dilutes the clarity and precision necessary for success.

By adhering to a structured approach, you not only present yourself as a composed and capable candidate but also set the stage for demonstrating your best analytical and problem-solving skills.

Exercise caution with mental math

For those adept at mental arithmetic, a word of caution: always jot down your calculations. Relying solely on mental computations can lead to significant challenges if errors occur. Without a written record, pinpointing and rectifying mistakes becomes a daunting task, necessitating a complete reevaluation of your work. This not only hampers your ability to quickly identify where you went wrong but also prevents the interviewer from offering guidance or corrections.

Moreover, maintaining written documentation of your steps and intermediate results serves a dual purpose. It enables the interviewer to follow your thought process more effectively, providing an opportunity for intervention if necessary. Furthermore, it allows you to efficiently review your calculations, ensuring accuracy and clarity throughout the problem-solving phase.

Typical Case Interview Math Problems and Key Formulas

3 types of case math problems.

In case interviews, math problems predominantly fall into three main categories, each designed to test your analytical prowess and decision-making capabilities. Understanding these categories not only aids in your preparation but also equips you with the insight to tackle these challenges methodically.

Roughly 90% of case interview math problems can be categorized as follows, guiding you toward strategic recommendations:

• Market or segment sizing : This type of problem requires you to estimate the size of a market or a specific segment within a market. For instance, you might be asked to calculate the potential sales of sports cars in China over the next five years. Alternatively, you might be asked to estimate something, i.e. the impact of an initiative. This involves understanding key influencing variables and making reasonable assumptions to provide a well-reasoned estimate.
• Operational calculations and decisions : These problems focus on the operational aspects of a business and often involve making calculations to improve efficiency, reduce costs, or enhance productivity. A typical question might involve calculating the total time saved if the lead time for each production step is reduced by 15%. Such questions require an analysis of current operations and an understanding of how changes can impact overall performance.
• Investment and financial strategic decisions : This category involves assessing various investment options or financial strategies to determine the most beneficial course of action. For example, you might need to compare the returns of two investment options, where Investment A offers a 12% annual return and Investment B offers a 5.5% return every six months. These problems test your ability to apply financial concepts and formulas to real-world scenarios, evaluating options based on their potential returns, risks, and strategic fit with the client’s objectives.

Extending beyond these primary categories, case interview math problems may also touch upon areas such as cost-benefit analysis, pricing strategies, and financial forecasting. Each type of problem requires a blend of quantitative skills, logical reasoning, and strategic thinking, allowing you to demonstrate your comprehensive understanding of business fundamentals. As you prepare for your case interviews, focusing on these core categories will help you develop a robust footing for tackling mathematical challenges, enabling you to approach each problem.

Case math formulas

Market sizing. When it comes to market or segment sizing questions, it’s perfectly acceptable to seek clarification from your interviewer on specific figures, such as the population of a particular country. Nonetheless, arming yourself with a foundational knowledge of key statistics can streamline your analysis and enhance your efficiency during these exercises. Familiarizing yourself with essential data points, including:

• Global population
• Populations of major countries such as the US, UK, Germany, China, India
• Demographic specifics of regions pertinent to your geographic area
• Average life expectancy rates
• Typical household sizes
• General income brackets

Equipping yourself with these statistics not only speeds up your calculation process but also demonstrates your preparedness and broad understanding of global and regional demographics. For a deeper dive into tackling market sizing questions with confidence and accuracy, including common formulas and strategic approaches, be sure to explore our dedicated article on market sizing questions . This resource is crafted to further refine your skills in estimating market potential, a critical component of case interview success.

Operational calculations. Operational calculations in case interviews demand a tailored approach, requiring you to devise formulas that are directly applicable to the case’s specific context and challenges. Unlike predefined equations, these formulas need to be thoughtfully constructed on the fly, taking into account the unique aspects of the business scenario at hand. Whether it’s streamlining processes, optimizing resource allocation, or improving operational efficiency, your ability to craft and apply these custom formulas is key.

In many instances, you might find yourself tackling optimization problems. These are designed to identify the most efficient way to allocate resources or adjust processes to maximize or minimize a particular outcome, such as cost, time, or production output. Understanding the principles of optimization and how to apply them in various business contexts can significantly enhance your problem-solving toolkit.

To get started, familiarizing yourself with a couple of foundational operational formulas can prove invaluable:

• Utilization rate = Actual output / Maximum output
• Capacity = Total capacity / capacity need per unit​
• Resources needed = Demand / Supply (e.g., Employees needed per day = 80 hours of customer requests per day / 8 daily working hours per employee; 10 employees are needed per day)
• Output = Rate (per time) x Time (e.g., Rate = 5 pieces per hour, Time = 5 hours; Output for 5 hours = 25)

These formulas serve as a foundational base from which to approach operational challenges within case interviews.

To evaluate the financial impact of decisions, these few formulas are key.

• Profit = Revenue – Cost
• Revenue = Price x Quantity
• Cost = Fixed cost (the cost that cannot be changed in the short term, e.g., rent) + Variable cost (the cost that changes with the number of products produced or services rendered, e.g., material cost)
• Contribution margin = Price – Variable cost
• Profitability (Profit margin) = Profit / Revenue
• Market share = Revenue of one product / Revenue of all products (in one market)
• Total market share = Total company revenue in a market / Total market revenue
• Relative market share = Company market share / (largest) Competitor market share
• Growth rate = (New number – Old number) / Old number
• Payback period = Investment / Profit per specific time frame (e.g., annual)
• Breakeven number of sales = Investment / Profit per product
• Return on investment = (Revenue – Cost of investment) / Cost of investment = Profit / Cost of investment
• Depreciation refers to the reduction in the value of an asset over time

There are also more advanced concepts, which are common for more specialized financial case interviews, not for generalist roles:

• The NPV is the present value of the sum of future cash in and outflows over a period (t = number of time periods, e.g., years) and is used to analyze the profitability of an investment or project
• Rule of 72: To find out how long it takes for a market, company, or investment to double in size, simply divide 72 by the annual growth rate
• The CAGR shows the rate of return of an investment or a project over a certain period of years (t = the number of years), expressed as an annual percentage
• The perpetuity is an annuity that lasts forever
• The ROE measures how effectively equity is used to generate profit
• The ROA measures how effectively assets are used to generate profit
• It measures how a change in price affects the change in demand
• Gross profit = Revenue from sales – Cost of goods sold (COGS, e.g., materials)
• Operating profit = Gross profit – Operating expenses (e.g., rent) – Depreciation (the spread of an asset’s cost over its useful lifetime, e.g., of a machine) – Amortization (the spread of an intangible asset’s cost over its useful lifetime, e.g., of a patent)
• Gross profit margin = Gross profit / Revenue
• Operating profit margin = Operating profit / Revenue
• The EBITDA looks at the profitability of the core business

Case Interview Math Tips and Tricks

Keep the following tips in mind to 3x your case interview math performance and speed, while reducing the potential for errors and mistakes.

Tackle the problems aggressively

Tackle case study math questions with confidence. Consulting interviewers want to see highly driven candidates who show self-initiative and engagement. If you hesitate whenever a number pops up or make mistakes in the quantitative section of the case, interviewers will test if this is just an anomaly or happens repeatedly. Candidates who struggle with math get more quantitative challenges during the case, whereas candidates who proceed flawlessly through the initial math question(s) often get shortcuts for the remaining quantitative parts or even whole results readily delivered by the interviewer as they have collected enough positive data points about their candidate’s performance in that area.

Hence, it is important to tackle math problems aggressively and with confidence. In most of my client interviews, I notice a hesitancy once the case moves into a more quantitative direction. Many are simply scared of digging into the numerical parts of a case or of discussing things in a quantitative context. Do not be that person!

If you mess up one calculation, you should not let this have a negative impact on the next one.

Re-learn and practice basic calculus

(Re-)learn simple arithmetic operations and practice until you can perform them in your sleep. While case math is never difficult, many candidates struggle with the concept of being watched while doing these basic operations. Therefore, the better your skill to compute quickly in a stressful environment, the bigger your quantitative muscle in the interview.

Practice calculations both mentally and with pen and paper under time pressure and the vigilant eyes of friends and peers. Go through number generators and math drill exercises to work on large-number additions, subtractions, multiplications, and divisions. Work with averages, percentages, and fractions. This certainly helps to build resilience and stamina.

Consider the numerical impact in your analysis

Get a feeling for numbers, percentages, and magnitudes. You should be able to accurately and approximately estimate percentages, percentages of percentages, as well as magnitudes on the spot. This helps you to interpret results and put them into context as well as to spot more obvious mistakes.

You should always have a critical eye on the quantitative aspects of a situation, even if the interviewer does not explicitly ask you about it. For example, relate numbers to each other (e.g., “The total is x, which represents a y% increase” ) or automatically think about the potential financial impact of your recommendation (e.g., “While these measures would definitely help improve our client’s customer satisfaction, I would be curious to understand how much the implementation would actually cost.” ). In addition, put numbers you hear into perspective (e.g., “I heard you say a 12% decrease is needed to achieve our planned cost reduction. I believe that in the current market environment with increasing commodities prices, this could be a difficult undertaking.” ). By interpreting numerical results in that way, you demonstrate strong business sense and judgment. You spot the implications of your outcomes and conclude correctly by discussing the so-what? of your analysis.

Putting numbers into perspective is also a valuable skill during a sanity check (e.g., “Is it really possible that we could increase our revenue by 200 million if we currently only make 50 million? Let me check my calculations again because that doesn’t seem right.” ).

On the other hand, if you are basing your recommendation solely on the outcome of a calculation, it makes sense to also discuss qualitative arguments to demonstrate your holistic big-picture thinking. Management consulting math goes beyond simple calculations, involving strategic thinking and analysis. For instance, if you recommend choosing a supplier solely because it is cheaper than the others, you could discuss that you would also like to look at the quality of their products, the supply chain, the availability, etc. Supplement a quantitative result with qualitative factors and vice versa.

Express problems quantitatively

Instead of approaching problems purely from a qualitative side, make a habit of using equations to describe relationships, ideas, and parts of the issue tree (if appropriate). It helps your thinking, shows that you are structured in your approach, and demonstrates that you are not afraid to get your quantitative hands dirty. A brief example: “Our client’s train tracks on Route A suffer from more than 100% utilization during the peak hours, leading to delays for many trains and passengers. What ways can you think of that could improve the capacity issue?”

To investigate and improve the over-utilization of the route, you could come up with the following equation: Utilization = demand / capacity. From this equation, you can instantly see that you need to either decrease demand or increase the capacity to improve the utilization situation. Demand and Capacity could be potential top-level buckets for your issue tree. You can now list investigative areas or ideas below each to structure your problem analysis. This approach would help you to quickly isolate quantitatively where the problem is coming from and how big it is, then quantify your remedies as you go along, indicating the best levers to pull and the best course of action.

Sanity check everything

Quantitative problems come with the most potential for errors and mistakes as they involve multiple challenging steps and actions you need to go through before reaching a sensible outcome. You want to avoid mistakes in the first place, but we all know that they do happen; even on the job later on. If you cannot avoid a mistake, at least try to catch your own mistakes before the interviewer does. How can you do that?

• Do not assume that the approach you came up with on the spot is correct without double-checking or thinking it through properly (the importance of taking time) .
• Remain vigilant and aware that mistakes are common in the math section. Never communicate the outcome of a calculation before double-checking that it is at least in the right ballpark and not the result of a careless mistake (the importance of sanity checking).

This also applies (or even more so) when you think that the math seems to be relatively easy. I have seen many interviewees getting caught off guard with simple math problems since they pay less attention to them compared to more difficult examples, then falling into a trap or making avoidable mistakes.

In sum, sanity-check your approach to the problem and outcome of each (intermediate) calculation. Use your judgment to spot calculation and estimation results that seem out of line (e.g., 18.3% vs. 183%). There are eight typical error sources:

• The logic is off or too complex.
• Your calculation is wrong (e.g., forgetting to carry the one, magnitude errors).
• You use the wrong numbers for the right approach. I see this often when candidates do not have organized notes and – in the heat of the moment – plug in the wrong numbers to calculate, even though their approach is correct.
• Your assumptions are off.
• You round numbers too generously or simplify the calculations too much (more on rounding later in this chapter).
• You fail to keep track of units and compare apples and oranges (more on that next).
• You forget one or several steps of your calculation. I see this often when candidates are glad to have made it through the math section yet forget to work on the final step of their approach (e.g., adding up two numbers).
• You interpret the results in the wrong way. I see this often when candidates are happy to have finished their calculations and then jump to a conclusion without thinking first. For instance, if we are comparing several scenarios and are interested in the alternative with the best net benefit, you would want to recommend the alternative with the highest result (highest net benefit). Some candidates do not think and select the alternative with the lowest number (lowest net benefit) as they somehow confuse lower with being better in this situation, by mixing it up with costs in their mind. Always make sure to interpret your results correctly and define what your outcome should be when drafting and communicating your approach.

If you spot a mistake and have not yet communicated the faulty result, ask for more time to sanity-check the calculation or the approach. If you have already blurted out a wrong number, state “This cannot be right.”  Then, go back to think about your approach or re-do the calculation. Provide reasons why your numbers might be off. Fix the problem quickly if the interviewer does not intervene. Most importantly, do not get thrown off by a mistake, and keep your composure.

Do not go faster after a mistake. Often, follow-up mistakes occur due to your newfound sense of urgency and disappointment in your performance. From my experience, more than 50% of candidates who make a math mistake make another one in the next two minutes. Rather, slow down and take some extra time to pick yourself up! It is not necessarily over yet unless you let it impact your performance going forward.

Keep track of units

Do not lose track of your units. Is it kg or tons, is it USD or EUR, etc.?

• When receiving the brief for a math question, write down every number including its unit.
• While setting up the calculation already prepare (either mentally or preferably on paper) a space for the end result including the correct unit.
• Keep the units for your intermediate results organized and label every number.

Interviewers might use different units for different numbers to check if you are paying close attention or simply just to confuse you. Stay vigilant, play back the units to make sure you have noted them down correctly. You must track the units of the input variables, and manipulate them correctly (i.e., convert all to the same unit), to then get to the right output. Do not compare apples and oranges.

Sometimes interviewers also use multiple units for one variable. For instance, “Our client would pay USD 500 per employee per year with option #1 and USD 1000 per three employees for 10 months with option #2.” Pay close attention in such cases and convert both options to the same units before comparing them, e.g., cost per employee per year.

Use shortcuts in your approach

Set up efficient and effective calculations. Most analyses in the business world rely on multiple assumptions and reasonable estimates, therefore not requiring a 100% level of precision. Hence, most of the time, close-to-correct answers are expected. Employ shortcuts in your approach to get accurate and directionally correct answers. Less is often more.

A couple of examples:

• When drafting formulas, always look for the simplest way to get to an accurate answer. For instance, if you are asked to decide between two potential suppliers by comparing the cost of both over a 40-week period, yet all information in the brief is on a weekly basis, for your decision it is enough to calculate and compare the weekly cost for each. If for some reason you want to calculate the difference over 40 weeks, first take the difference of the weekly cost, then multiply it by 40. Alternatively, you could calculate the cost for each supplier for 40 weeks, and then calculate the difference, but you would end up with more calculation steps and more difficult calculations since larger numbers are involved.
• Think critically about what outcome is needed to support your decision. For instance, if you must find out if the profit margin of a deal for 30 aircraft is more than 10% there is no need to calculate the profit margin for all 30 units but calculating the profit margin for one aircraft is sufficient to evaluate the deal. This leaves you with smaller numbers which are easier to handle and interpret.
• When evaluating which option out of several is the best, only look at metrics that differ for every option. For instance, if the fixed cost for every option is the same, yet the variable cost and revenue are different, you would only need to consider the latter two to provide a recommendation (given that you are not asked to evaluate the total value of each option but just to pick the best).

Always explain your logic, shortcuts, and simplifications to the interviewer. They need to understand why your approach is enough to answer the question. Ninety-nine percent of the time, they will agree. Your final results won’t be 100% accurate either way and are not expected to be for most cases. Use plausible shortcuts in your approach and calculation to reach plausible numbers. The same is true for rounding.

Simplify and round numbers

Like the point above, use rounding to make your calculations easier and minimize the risk of mistakes. Ask the interviewer if it is okay to round beforehand and explain exactly how you want to do it. For instance, if you come up with a revenue number of 82.5 million, ask to use 80 million instead. State beforehand that you will trim the fat a bit; if the interviewer agrees, proceed with your calculation. Similarly, if you get 42.65 as an intermediate result say that in the following calculations, this will be rounded down to 40. Other examples include:

• 83 million Germans become 80 million
• 331 million Americans become 320 or even 300 million (by making some clever assumptions explaining why not everyone in the population should be included in your approach, e.g., by excluding certain demographic segments or areas)
• 365 days in a year become 350 or even 300 days (by making some clever assumptions about bank holidays, opening hours, weekends, etc.)
• USD 983 million in revenue becomes one billion.

The tricky part about rounding numbers is to know when it is a good time to do so. Some case math questions demand precise results. For example, if you are asked whether an investment has an ROI above 12% and you can already spot that the final result is close to that number, it would be wise to calculate with precision. Similarly, if you are comparing two alternatives or outcomes, be careful. Outcomes could be very close to each other so extensive rounding might just flip their ranking and the direction of your answer. That is why you should always ask if you can round and provide details on how you would like to do it. That way, the interviewer could provide feedback on whether rounding is a good idea or not.

On the other hand, rounding is especially helpful when 100% precise answers are not needed. For instance, when you calculate a singular outcome, i.e., not comparing multiple numbers or outcomes. You might also round if your calculations yield only directionally correct results anyway, and precise answers are not expected, for instance, when you need to rely on (multiple) assumptions in your approach. Examples would be estimating the size of a market or the impact of a measure, which come with many assumptions and degrees of uncertainty.

What are the best practices related to rounding?

You should round only within a ten percent margin, ideally less, and within five percent. Otherwise, you might skew the results, over or understate the outcome, and provide false recommendations. Think about the impact of rounding consecutive numbers. You can either get more precise results because the effects cancel each other out or magnify the blur of rounding.

For instance, if you want to calculate the revenue, which is quantity times price and the quantity is 9,500 units and the price is USD 35, you could calculate with a quantity of 10,000 and a price of USD 30. That roughly keeps you in a 10% margin of the precise result. If you round both numbers in the same direction, up or down, you would already be off by around 20% from the precise result.

To create a general rule: When you sum two numbers or multiply them, make sure to round one number up and the other one down, essentially rounding in the opposite direction. If you want to subtract or divide, make sure to round both numbers either up or down, rounding in the same direction. Lastly, whenever you deal with indivisible items, round them up to a whole. For instance, if you calculate that you would need to purchase 533.4 new cars for a taxi company to meet their demand, round it up to 534. There are no half-cars.

Take your time

The single biggest lever to improve the outcome of your quantitative analysis is to take time and perform numerical tasks on your terms. What this means is that you should not get pressured to answer or calculate on the spot but rather ask the interviewer for some time to prepare your logic and then, again, to perform your calculations. One minute is usually fine for the logic and up to three minutes are okay for the actual calculations. Of course, faster is better but faster and wrong is worse than slow, steady, and accurate.

Remember our initial discussion. You do not need to have a spike in every area of the case, yet you should avoid mistakes at all costs. A slow but accurate math answer helps you get the offer if you demonstrate spikes in other areas. A wrong but fast answer might lead to a rejection, even if you spike in other areas.

Do not feel pressured to talk to the interviewer while you are thinking or calculating. Focus on one thing at a time. Only communicate your logic, your results, or if you want, your intermediate outcomes once you are done with each step.

Watch the 0s

You would not believe how many candidates fall into this trap. Many people struggle with large numbers, simplify them by cutting zeros, and then end up losing zeros along the way or even adding some to the result. Watch out for zeros that you have trimmed or left out to facilitate your calculations. There are two best practice solutions to deal with and keep track of zeros:

• scientific notation.

For labels, add k for thousand (000), m for million (000,000), and b for billion (000,000,000) when manipulating larger numbers. That way you can simplify and keep track of your zeros.

Alternatively, by applying the scientific notation, you can trim the power of 10s and then perform simple calculations. Once you reach a conclusion you can add your zeros back. Let’s look at one example: Calculate 96 x 1,300,000.

First, just calculate 96 x 13 x 10 5 , essentially getting rid of the five zeros of the second number: 96 x 13 = 96 x 10 + 96 x 3 = 1,248

Add the 5 zeros back, which makes it to 124,800,000.

Another example, a division: 1.4bn / 70mn = (1.4 x 10 9 ) / (7 x 10 7 ) = 0.2 x 10 2 = 20

When adding the zeros back, for a multiplication you would add the superscripted numbers, for a division you would subtract one from the other.

Adopt one of the two options discussed above when practicing so it becomes second nature to you. You will never struggle with zeros again.

Case Interview Math Practice Questions

Practice case math question #1.

It’s important to understand what to expect when preparing for your case interviews.

Let’s look at the following case interview math example:

Scenario : Imagine you are a consultant working for a beverage company, “RefreshCo,” which is considering launching a new line of herbal tea products. RefreshCo aims to understand the potential market size, profitability, and key financial metrics associated with this launch to make an informed decision. Your task is to help RefreshCo by analyzing if the breakeven will be achieved within 5 years. Data provided : RefreshCo estimates the initial investment for launching the new herbal tea line at $2 million. The expected lifetime of the product in the market is 5 years. The target market size for herbal tea in Year 1 is estimated at 2 million potential purchases initially, with a 5% annual growth rate. RefreshCo aims to capture a 10% market share in Year 1, with a 10% growth in market share each subsequent year. The selling price per unit is set at $4, with the cost of goods sold (COGS) at $2.5 per unit. Fixed costs (excluding the initial investment) are estimated at $500,000 per year. Prompt for a case interview math problem

Take some time to work on this question and then come back to the solutions below.

Let’s go through the calculations for each section in detail:

Market size calculation

The market size for each year is calculated using the compound growth formula: Market size=Initial market size×(1+Growth Rate)^Years

• Year 1 : 2,000,000 (Given)
• Year 2 : 2,000,000×(1+0.05)=2,100,000
• Year 3 : 2,000,000×(1+0.05)^2=2,205,000
• Year 4 : 2,000,000×(1+0.05)^3=2,315,250
• Year 5 : 2,000,000×(1+0.05)^4=2,431,013

You could also calculate each year based on the number of the previous year.

Revenue projections

Revenue is calculated as the product of potential customers and selling price, considering the annual growth in market share.

• Year 1 Revenue : 800,000 (Calculated based on market share, which is growing by 10% every year, and the selling price)
• Year 2 Revenue : 924,000
• Year 3 Revenue : 1,067,220
• Year 4 Revenue : 1,232,639
• Year 5 Revenue : 1,423,698

Profitability analysis

Profit for each year is calculated by subtracting total costs (COGS per unit multiplied by the number of units sold plus fixed costs) from total revenue.

• Year 1 Profit : −200,000 (Revenue minus costs)
• Year 2 Profit : −153,500
• Year 3 Profit : −99,792
• Year 4 Profit : −37,760
• Year 5 Profit : 33,887

Break-even analysis

The break-even point is not reached within the 5-year period as cumulative costs exceed cumulative revenues throughout the period. Based on the calculations, RefreshCo will not achieve breakeven within the first 5 years of launching the new line of herbal tea products.

By the end of the 5th year, the cumulative profit (including the initial investment as a negative profit) is still negative, amounting to approximately -$2,457,166 .

To facilitate and speed up your calculations you could also work with shortcuts such as generous rounding or estimating the impact of the growth rate in market size and market share. The result would still be directionally correct, indicating that this is not a good business idea.

Practice case math question #2

Let’s look at another example:

Scenario : AutoPartsCo is a manufacturer specializing in automotive parts. Due to increasing demand, the company is exploring ways to optimize its production process for one of its key products: brake pads. The company operates two production lines, Line A and Line B, each with different capacities, costs, and output levels. Your task as a consultant is to analyze the provided data and recommend which production line should be optimized to maximize efficiency and reduce costs, based on average cost per unit. Data provided : Line A : Capacity: 10,000 units/month Current monthly production: 8,000 units Fixed costs: $120,000/month Variable cost per unit: $15 Line B : Capacity: 15,000 units/month Current monthly production: 12,000 units Fixed costs: $150,000/month Variable cost per unit: $12 Based on the average cost per unit, recommend which production line AutoPartsCo should focus on optimizing. Consider factors like capacity utilization and potential for cost reduction. Prompt for a case interview math problem

• For Line A and Line B, calculate the total costs (fixed costs + total variable costs) and then divide by the number of units produced to find the average cost per unit.
• Total Variable Costs for each line are calculated as the product of the variable cost per unit and the number of units produced.
• Compare the average costs per unit between Line A and Line B to determine which line is currently more cost-efficient.
• Assess the capacity utilization for each line (current production divided by total capacity) to identify potential for optimization.
• Based on the cost efficiency and capacity utilization, recommend which production line offers the best opportunity for optimization and why.

Average cost per unit:

• Line A : The average cost per unit is $30.
• Line B : The average cost per unit is $24.5.

Capacity utilization

• Both Line A and Line B have a capacity utilization rate of 80%.

Recommendation

Based on the average cost per unit, Line B is currently more cost-efficient than Line A, with a lower average cost per unit of $24.5 compared to $30 for Line A. Additionally, both production lines are operating at the same capacity utilization rate of 80%, suggesting that neither line is currently overburdened.

Considering the lower average cost per unit and equal capacity utilization, AutoPartsCo should focus on optimizing Line B . Optimizing Line B could further reduce costs and enhance efficiency, given its already lower cost base and potential for increasing production closer to its full capacity without the immediate need for significant capital investment.

This recommendation is made with the assumption that demand can absorb the increased production and that similar quality standards can be maintained across both lines. Further analysis could involve exploring ways to reduce the variable and fixed costs of Line A or increasing its production volume to improve its cost efficiency.

Mental Math Concepts and Shortcuts

Mental math for consulting requires practice and strategy. Below are some tricks to become faster, more accurate, and more comfortable with case math as well as more advanced concepts that you might encounter during interviews. The more often you employ these tricks during practice and work with certain concepts, the more it becomes second nature to you. Sometimes you might be able to combine a couple of tricks to become even faster.

While there are many specific calculation shortcuts (e.g., when multiplying a number by eleven), you should focus on a couple of shortcuts that are replicable and can be used for most situations. Don’t try to memorize many different shortcuts that only have highly isolated use cases. Internalize and use a few shortcuts well. Like everything else in consulting interviews: Do not boil the ocean.

Basic arithmetic calculations

Master quick and effective arithmetic shortcuts essential for acing Bain, BCG, and McKinsey math case interviews:

Learn these simple shortcuts and use the examples below as pointers.

Build groups of 10

When adding up numbers, build groups of numbers that add up to 10 or multiples of 10.

7 + 3 + 12 + 8 + 5 + 5 = 40

(10) + (20) + (10) = 40

Go from left to right

356 + 678 = (356 + 600) + 70 + 8 = (956 + 70) + 8 = 1026 + 8 = 1034

This is a simple way to become faster and more accurate once you have internalized it.

Subtractions

Make it to 10

When performing quick subtraction, figure out what makes it to 10.

For instance: 4 2 – 2 5

• Reverse the subtraction for the unit digit (5 – 2 = 3)
• Add the number that would make it to 10 (3 + 7 = 10); this is the units digit of the result
• Add 1 to the digit on the left of the number you are subtracting (2 + 1 = 3)
• You end up with 7 on the unit digit and 4 – 3 = 1 on the 10s place, which is 17

Let’s use another example: 3853 – 148

• Reverse the unit digit (8 – 3 = 5)
• Add the number that would make it to 10 (5 + 5 = 10)
• Add 1 to the digit on the left of the number you are subtracting (4 + 1 = 5)
• You end up with 5 on the unit digit, 5 – 5 = 0 on the 10s place, and 8 – 1 = 7 on the hundreds place, which gives you a result of 3705

With a bit of practice, the what do you need to add to make it to 10 becomes an automated habit for your subtractions: 1 + 9, 2 + 8, 3 + 7, 4 + 6, 5 + 5, 6 + 4, 7+ 3, 8 + 2, 9 + 1

You can use the same approach we’ve discussed for additions for subtractions as well:

42 – 25 = (42 – 20) – 5 = 22 – 5 = 17

Multiplications

Get rid of 0s

To make your calculations simpler, get rid of the zeros at first, adding them again at the end. For instance, if asked to calculate 34 x 36,000,000: convert it into 34 x 36m, which is 1,224, then add six zeros to that number which is 1,224,000,000

Use the label method ( “m” ) or the scientific notation (x10 6 ). If you had to multiply 3,400 times 36,000,000: convert again to 3.4k x 36m, which is 122.4, then move the comma to the right side of the 4 and add eight zeros (the sum of the zeros you got rid of in the beginning: 3 + 6), which is 122,400,000,000. Using the scientific notation, we would end up with this: (3.4 x 10 3 ) x (36 x 10 6 ) = 122.4 x 10 9 = 122,400,000,000.

Break apart multiplications by expanding them and breaking one of the terms into simpler numbers. For instance: 18 x 5 = 10 x 5 + 8 x 5 OR (20 – 2) x 5 = 20 x 5 – 10 = 90

Factor with five

Factor common numbers to simplify your calculations when dealing with multiples of 5. For instance: 17 x 5 = 17 x 10 / 2 = 85.

Another example would be 20 x 15 = 20 x 10 x 3 / 2 = 300

The most common numbers to keep in mind are: 5 = 10 / 2; 7.5 = 10 x 3 / 4; 15 = 10 x 3 / 2; 25 = 100 / 4; 50 = 100 / 2; 75 = 100 x 3 / 4

Exchange percentages

Sometimes you can exchange percentages to simplify the calculation. For instance:

60 x 13% = 0.6 x 13 or 6 x 1.3 = 7.8

Convert to yearly data

If you want to convert daily to yearly data, instead of multiplying by 365, multiply by 30 and then by 12, which would add up to 360 days. For most cases, this is close enough and can be argued for well by using certain assumptions, e.g., bank holidays, and downtimes. Always notify interviewers about your assumptions and simplifications.

Convert percentages

Convert percentages into divisions. For instance: 20% of 500 = 500 x 1 / 5 = 500 / 5 = 100

Split into 10ths

Split numbers into 10ths. For instance: 60% of 200 = 10% of 200 x 6 = 120

Apply expansion in a similar manner as described already for multiplications:

• Simple example: 35 / 5 = 30 / 5 + 5 / 5 = 6 + 1 = 7
• More complex example: 265 / 5 = 200 / 5 + 60 / 5 + 5 / 5 = 40 + 12 + 1 = 53

This can be extremely useful when trying to estimate a number as you do not need to perform all calculations up to the last digit to get to a ballpark estimate, e.g., 200 / 5 + 60 / 5 = 52 ≈ 50

Advanced case math concepts

In case interviews, calculating the average is popular since it is simple, yet demands several calculations to arrive at a result. It is a good pressure test for candidates. For example, you might be presented with a table containing data on three products, each with different production costs and the same production quantity. You might have to calculate the average production cost for one unit. The average is the sum of terms divided by the number of terms. For instance, the production cost of Product A is 5, of Product B, 10, and of Product C, 15. The average production cost is (5 + 10 + 15) / 3 = 30 / 3 = 10 for one unit.

A common variation is weighted averages . Instead of each of the data points contributing equally to the final average, some data points contribute more than others and therefore, need to be weighted differently in your calculations. If the weights add up to one, multiply each number by its weight and sum the results. If the weights do not add up to one, multiply each variable by their weight, sum the results, and then divide by the sum of the weights.

To stick with the example above, Product A might be responsible for 20% of the sales, whereas Product B and C for 30% and 50% respectively. Alternatively, it could be written as the following: There are 40 units of Product A, 60 units of Product B, and 100 units of Product C. The weighted average is: 5 x 20% + 10 x 30% + 15 x 50% = 5 x 0.2 + 10 x 0.3 + 15 x 0.5 = 1 + 3 + 7.5 = 11.5. For the second set, you could calculate it as: (5 x 40 + 10 x 60 + 15 x 100) / 200 = 11.5. Other common contexts, where you are asked to calculate averages could be growth rates, demographics, economic data, geographies and countries, product categories, business segments and units, revenue streams, prices, cost data, etc.

Fractions, ratios, percentages, and rates

Fractions, ratios, percentages, and rates are all different sides of the same coin and can help expedite your calculations.

For instance, fractions can be used to represent a number between 0 and 1. Expressing numbers as fractions and using them for additions and subtractions as well as multiplication and divisions can help you solve problems faster and more conveniently through simplification. For example, you can write 0.167 as 1/6, or 0.5 as 1/2. You can also combine fractions with large number divisions. For instance, let’s assume you want to see how much percent 400k is of 1.7m.

Write it as a fraction: 4/17 = 1/17 x 4

Now look at the division table for 1/17, which is 0.059, essentially 0.06.

1/17 x 4 = 0.06 x 4 = 0.24 or 24%

If you had calculated it more accurately by taking three times as long, you would get to 0.235 or 23.5%, rounding it up again to 24%.

As you can see, using fractions for larger number divisions can be a huge time saver. I would recommend you learn all fractions up to a divisor of 20 (e.g., 1/20) by heart, using the fraction table I shared earlier. It increases your speed and accuracy in interviews.

Ratios are comparisons of two quantities, telling you the amount of one thing in relation to another. If you have five apples and four oranges, the ratio is 5:4 and you have nine fruits in total. In case interviews, one tip is to write ratios as fractions of the total, e.g., apples are five out of a total of nine fruits, which is 5/9.

Percentages are a specific form of ratios, with the denominator always being fixed at 100. From experience, almost 80% of case interviews include some reference to or use of percentages, pun intended. Discussion points such as “Revenue increased by 15%” or “Costs are down four percent over the last six months” are common. Percentages are also useful when you want to put things into perspective, state your hypotheses, or guide your next steps. For instance, “That would translate to a 15% increase compared to our current revenue. Now, is a 15% increase realistic? What would we need to do to achieve this?”

Be careful not to mix percentage points with percentages. A percentage point or percent point is the unit for the arithmetic difference of two percentages. For example, moving up from 40% to 44% is a four-percentage point increase, but it is a 10% increase in what is being measured. Interviewers might ask for one or the other.

Rates are ratios between two related quantities in different units, where the denominator is fixed at one. If the denominator of the ratio is expressed as a single unit of one of these quantities, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable), then the numerator of the ratio expresses the corresponding rate of change in the other (dependent) variable. To make this more practical, let’s look at common rates. One common type of rate is per unit of time , such as speed or heart rate. Ratios with a non-time denominator include exchange rates, literacy rates, and many others. Case interviews might include some of the following rates:

• Growth rate: the ratio of the change of one variable over a period versus the starting level
• Exchange rate: the worth of one currency in terms of another
• Inflation rate: the ratio of the change in the general price level in a period to the starting price level
• Interest rate: the price a borrower pays for the use of the money they do not own (ratio of payment to amount borrowed)
• Price-earnings ratio: the market price per share of stock divided by annual earnings per share
• Rate of return: the ratio of money gained or lost on an investment relative to the amount of money invested
• Tax rate: the tax amount divided by the taxable income
• Unemployment rate: the ratio of the number of people who are unemployed to the number of people in the labor force
• Wage rate: the amount paid for working a given amount of time, or doing a standard amount of accomplished work (ratio of payment to time)

If you are not familiar with these or others that might come up, it is always okay to ask the interviewer for a clarification of the definition. Keep an eye on the time frames rates are expressed in. This could be annually (per annum = p.a.), quarterly, per month, etc. Often, information is provided for different time frames, divisors, or units (e.g., “the top speed of vehicle A is two miles per minute, the top speed of vehicle B is 150 miles per hour” ). Interviewers often use different units for different figures to trick you. For instance, when dealing with two different currencies, always convert all numbers to the same currency by using the exchange rate first. Otherwise, you are comparing apples and oranges. Convert to the same before conducting your analysis, calculations, or comparisons.

Growth rates

You should be able to work with growth rates, which is easy for one time period.

• (Increase of 30% in year 1): 100m x 1.3 = 130m

It gets trickier when you must calculate growth over multiple periods. You need to get the compound growth rate first.

• (Increase of 30% in year 1, 30% in year two): 100 x 1.3 x 1.3 = 100 x 1.69 = 169

The latter can be done quickly if you want to calculate growth over two to three time periods. Everything beyond that becomes tedious and lengthy. If you want to calculate growth for several periods, it is better to estimate the outcome. A shortcut is to use the growth rate and multiply it by the number of years.

• (Increase of 4% p.a. over 8 years): 4% x 8 years ≈ 32%; 100 x 1.32 = 132

If you use the exact compound annual growth rate (CAGR), you end up with roughly 137, more accurately 100 x (1+0.04)^8= 136.85. The total deviation of five or roughly 3.5% (5 / 137) due to your simplified approach is close enough. However, be aware that the divergence (the underestimation) increases with larger numbers, higher annual growth rates, and the number of years. In a case interview, you can account for that by adding between 1% and 10% to your outcome value, depending on the numbers you are dealing with. Keep it simple. Adding 5% to the 132 brings us to 138.6, even closer to the exact number.

To use the same approach with varying growth rates, sum them up. For instance:

• (Increase of 4% in year 1, 10% in year 2, 5% in year 3, 7% in year 4): 4% + 10% + 5% + 7% = 26%; 100 x 1.26 = 126

If we calculate the exact number, it is 128.5; again, the shortcut is close enough and much faster. If you add 1% or 2% you are even closer.

You can apply the same tricks to negative growth rates, keeping in mind that you are overestimating the decrease. Lastly, you can use this trick for combinations of positive and negative growth rates as well.

Expected value and outcomes

You might have to compare the impact and success of different recommendations or the expected return on investment. One way to do this is to work with probabilities and calculate the expected value (EV) for a course of action. The expected value for each recommendation is calculated by multiplying the possible outcome by the likelihood of the outcome. You can then compare the expected value of each option and make a decision that is most likely to achieve the desired outcome.

For example, if you have to decide between two projects and your analysis shows that Project A yields USD 50 million with a likelihood of 80% and Project B yields an outcome of USD 100 million with a likelihood of 30%, you will decide for Project A, with an expected value of USD 40 million (Project B: USD 30 million).

• EV(A) = 50m x 0.8 = 40m
• EV(B) = 100m x 0.3 = 30m

If you want to compare the outcome of bundles of recommendations, the expected value is calculated by multiplying each of the possible outcomes by the likelihood of each outcome and then summing all values for each bundle. Sometimes, interviewers keep it simple and set the expected outcome for each alternative to 100%. In such cases, just take the alternative with the better outcome, i.e., the one with a lower cost or the one with a bigger (net) benefit, depending on the question or goal.

Avoid the Most Common Pitfalls and Mistakes

There are several potential pitfalls you need to avoid when approaching case interview math problems.

Avoiding these common pitfalls requires thorough preparation, including practicing mental math, familiarizing oneself with quick calculation techniques, and simulating the interview environment to improve performance under pressure.

What Should You Do When You Get Stuck?

Most candidates start to panic when they don’t know how to structure a math problem. Not all is lost at this point if you stay calm and collected and have a plan to deal with the situation. So, what can you do when you get stuck and don’t know how to proceed?

Before you ask for help, think through the following checklist:

• Do I know what the objective is? What do I need to solve? If you don’t have an answer for that, clarify with the interviewer.
• Do I understand the problem and the details of the question? If you are missing some context, clarify with the interviewer.
• Do I have all the data that I need? Am I missing something, or am I confused due to a large amount of (irrelevant) data?
• What would be the simplest way to approach this? Am I approaching this from a too complex perspective? This is a big sticking point for most candidates.
• Is this similar to something I have worked on during practice?

If you are still not able to move forward, ask for help in a targeted way by offering the interviewer something in exchange. Do not say: “I don’t know what to do. Can you help me?”

Do not stay silent either. Rather explain your current understanding of the situation to the interviewer (e.g., “I believe that in this case, I would need to look at the net benefit of the decision.”).

Discuss your thinking about how you would like to solve the problem on a higher level but are currently missing one or two steps to make it work; then ask for guidance (e.g., “I need to compare the additional costs and revenues. It is clear to me how to get to the revenue numbers. For the additional costs I am not 100% certain that I am approaching this correctly, would you have any input on that” ).

Interviewers want you to succeed and a little push on the math approach does not automatically lead to a rejection unless it happens more frequently in one case or across cases or you need support in more areas of the case.

How to Prepare for Case Interview Math

General practice recommendations.

Incorporate case interview math practice into your preparation plan .

Regardless of your current quantitative reasoning skills, devote time during your case interview preparation to brush up on your mental and pen-and-paper math skills. If you are struggling with math or are dealing with a couple of insecurities in this area, there is no reason not to practice case math drills for at least two hours per day for a couple of weeks.

At the end of your preparation, shortly before your interviews, you want to be in a state where you can tackle every problem you see flawlessly and swiftly with confidence and without anxiety. Your structuring, overall problem-solving, and chart interpretation can be in the highest percentile of all candidates, but if you do not fix your math issues and insecurities, you will still be rejected.

I want to stress this point because most candidates fail due to issues with their quantitative reasoning, something that is entirely preventable with effort and time. I would even go so far as to say that if you do not feel 100% ready to tackle any section of the case, postpone your interviews if you have the chance until you feel fully ready.

There are several things you could do to get up to speed with mental and pen-and-paper math. The trick is to be confident in your ability to efficiently work through simple math and resilient enough to face external pressures in the process. If you are starting out, (re)-learn and practice basic calculus such as additions and subtractions, multiplications and divisions, averages, percentages, and fractions.

Free practice resources and habits

Get used to numerical reasoning by working with numbers you encounter in your daily life, be it the bar tab, the grocery store receipt, or figures and data you find in the news, especially business reporting. Perform simple arithmetic operations on the numbers you encounter in your head or with pen and paper. Do not use a calculator. Work on some simple business cases. For instance, while waiting at the doctor, calculate how much profit they make a month, a year, etc. The opportunities are endless.

Practice with the following free iOS and Android apps.

• Magoosh Mental Math
• Mental Math Cards Challenge
• Coolmath Games: Fun Mini-Games
• Matix Mental Math Games
• Math Games and Mental Arithmetic

Do all of this in a stressful environment. You want to build stamina and resilience to outside influences and stressors. Use the apps in the crowded and noisy subway, calculate during mock interviews with unpleasant interviewers who stress you out, in front of friends and family, or simply with time limits.

Free case interview math drill generator

Boost your case interview preparation with our Case Interview Math Drill Generator. Seamlessly create tailored math problems designed to boost your speed and accuracy and stand out in the interview process field.

You can access the tool for free here:

Our Case Interview Math Mastery Academy

Our comprehensive preparation strategy encompasses two distinct components designed to enhance your case interview readiness, regardless of your current skill level.

First, our “ Case Interview Math Mastery ” course and drills offer a multi-tiered learning experience. This program is meticulously crafted to support you at any stage of your preparation journey, from mastering basic calculations to tackling advanced numerical challenges, exhibit math problems, solving intricate business problems, and navigating full case math scenarios.

It comes with a 25-part video series and 2000 practice drills. Whether you’re starting to build your foundation or refining your skills, our course is structured to elevate your skills comprehensively.

Second, my personalized coaching sessions provide targeted guidance to further your case interview and problem-solving capabilities. With a track record of over 1,600 interviews, each receiving a five-star rating, and hundreds of offers generated for my clients, this coaching service is a testament to my commitment to excellence.

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Frequently Asked Questions: Case Interview Math

Mastering consulting firm interview math can significantly boost your chances of landing a job at top consulting firms. Preparing for case interviews at top consulting firms like McKinsey, BCG, and Bain involves mastering the art of solving quantitative problems efficiently and accurately. Understanding the basics is crucial, but candidates often have more nuanced questions about improving their performance in case interview math.

Below, we answer the most pressing questions to help you navigate the complexities of case interview math with confidence.

What specific math topics should I review to prepare for consulting case interviews? To excel in consulting case interviews, focus on reviewing arithmetic operations, percentages, ratios, basic algebra, and estimation techniques. Understanding these foundational topics is crucial for analyzing business scenarios and making data-driven decisions. Read through our comprehensive guide to management consulting math to equip yourself with the necessary concepts and learn how to prepare for case interview math with my structured approach.

How can non-quantitative background candidates improve their math skills for case interviews? Candidates from non-STEM or non-business backgrounds should start with basic arithmetic and gradually progress to more complex topics through online courses, practice problems, and math-focused case interview preparation materials. Consistent practice and application of math in real-life scenarios can also enhance proficiency. Elevate your quantitative analysis for consulting interviews by practicing with our curated examples. Practice drills for consulting interview math are essential.

Are there any common mathematical errors to avoid during consulting case interviews? Yes, common errors include incorrect rounding, magnitude errors, note-taking issues, mixing up units of measurement, and overlooking simple arithmetic mistakes. Double-checking your work and practicing mental math in stressful conditions can help avoid these pitfalls.

How do consulting firms evaluate candidates’ mathematical reasoning in case interviews? Consulting firms assess candidates’ ability to logically approach quantitative problems, perform accurate calculations under pressure, and derive meaningful insights from numerical data. Demonstrating clarity in thought process and precision in results is key.

Can you provide examples of complex case interview math problems and how to solve them? Complex case interview math problems often involve multiple steps, such as calculating market sizes, revenue growth over time, or cost optimization strategies. Breaking down the problem into smaller, manageable parts and using a structured approach to solve each part is an effective strategy. For free practice examples, please check this link here . For a professional case interview math course with 2000+ drills, please check out our Math Academy .

How important is speed in solving math problems during consulting case interviews? While directionally correct accuracy is paramount, speed is even more important in case interviews. Being able to quickly perform calculations and reach an outcome in the right ballpark allows more time for analysis and developing recommendations. Practice is essential to improve both speed and accuracy.

What are the best practices for presenting mathematical findings clearly in a case interview? Best practices include summarizing key findings succinctly, explaining the logic behind your calculations, and discussing the implications of your results in the context of the case. Use clear, concise language and structure your response logically. Advanced quantitative problem-solving in case interviews demands not just skill, but also strategic thinking.

How can I practice case interview math under realistic conditions? Simulate the interview environment by practicing math problems under timed conditions, without the use of a calculator, and ideally with a partner or professional case coach to mimic the pressure of real interviews. Online simulators and practice tests can also provide a realistic challenge. Overcoming math challenges in consulting case studies is achievable with the right mindset and tools.

What role does mental math play in consulting case interviews, and how can I improve it? Mental math is crucial for quickly estimating and calculating during discussions. Improving mental math involves regular practice with drills, learning shortcuts, and challenging yourself to do everyday calculations in your head. Mental math is also relevant when talking about strategies for acing McKinsey , BCG , and Bain math tests.

Arming yourself with these insights can dramatically improve your performance in consulting case interviews, setting you on the path to success in your consulting career.

Struggling with case interview math?

Tackling math problems in case interviews can feel overwhelming, but remember, you’re not the only one facing this hurdle. Math, especially under pressure, can be challenging, but it’s a skill that can be honed with practice and the right strategies. If you find yourself puzzled by specific problems or methodologies, don’t hesitate to share your questions below in the comment section. We are happy to help you out!

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Florian spent 5 years with McKinsey as a senior consultant. He is an experienced consulting interviewer and problem-solving coach, having interviewed 100s of candidates in real and mock interviews. He started StrategyCase.com to make top-tier consulting firms more accessible for top talent, using tailored and up-to-date know-how about their recruiting. He ranks as the most successful consulting case and fit interview coach, generating more than 500 offers with MBB, tier-2 firms, Big 4 consulting divisions, in-house consultancies, and boutique firms through direct coaching of his clients over the last 3.5 years. His books “The 1%: Conquer Your Consulting Case Interview” and “Consulting Career Secrets” are available via Amazon.

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Case interview math

An overview of case math problems, why firms use them and how to prepare.

Case math context | Example problems | How to prepare | Practice makes perfect

Case interviews are chock-full of math. This makes sense since consultants do math everyday in their casework and need to be sharp analytically to be effective on the job.

In their interviews, consulting firms use case math to see if candidates are up to snuff in terms of their analytical abilities.

In this article, we walk through why consulting firms use case math in their interviews, the essential case math skills and examples of how they can come up, and some tips for how to prepare.

What case math problems are really testing (Top)

The obvious question for most candidates is "Why am I being tested on these mental math abilities? Won't I have Excel for that when I'm on the job?"

While it's true that you'll have plenty of analytics tools around you to do simple, and more often quite complex, calculations while on the job, that's not what case math is testing for.

Consulting firms use case math to test two things:

• A candidate's ability to quickly confirm or disprove a hypothesis
• A candidate's ability to prioritize analyses by quickly doing math to rule out areas which don't need further analysis

Workflow on a management consulting case is an iterative, hypothesis-driven process. Given a problem, consultants come up with a hypothesis for a solution and then do analysis to confirm or disprove that hypothesis.

For example, let's say the CEO of a large consumer-packaged-goods (CPG) player is looking to improve the profitability of an underperforming product line. The average margin on these products is $100, and they sell about 500,000 units per year. She's set a target of $5 million in profit improvement. The partner on the case's hypothesis is that increasing the profit margin through supplier negotiations will make that happen.

We know from previous experience with a similar client that supplier negotiations would at most improve margins by 2%. Faced with this hypothesis, a good consultant would be able to quickly disprove it, maybe even during the meeting in which it was proposed! A 2% margin increase would produce $2 of marginal profit per unit, and if sales remain steady we would only see a $1 million improvement in profitability.

This quick analysis is a powerful tool because it allows a team to quickly pivot to other aspects of the profitability problem. Maybe we can increase sales through promotional activity in addition to cutting costs through supplier negotiations. Whatever the solution ends up being, we know that supplier negotiations alone won't cut it.

Our quick numerical analysis drove the process forward and helped to prioritize our efforts towards potentially more high-yield solutions. This is exactly what consulting firms need from their consultants, and that's why they test for it in their interviews.

The case math skills you need to master (Top)

Let's get into the nitty gritty of what type of math you'll see in your case interviews. For each skill, we'll walk through examples of how it may appear in a case interview.

Division and multiplication

Big division and multiplication are staples of case interviews. They're an easy way to test a candidate's mettle - it's not everyday you have to multiply or divide two numbers in the millions!

Case problems will throw all sorts of multiplication and division problems at you. You'll get numbers with tons of zeros, odd numbers that can't be easily simplified, and everything in between. The key to solving these will be to use shortcuts, break-up messy numbers into easier to manage chunks, and stay organized.

Let's walk through two ways we can use shortcuts to make multiplication and division way easier.

Case Example: Dealing with a TON of zeros

Let's say our client wants to understand the average productivity of their employees on their manufacturing line. They have 50 workers and on a given day produce 100,000 widgets. How many widgets are produced per employee?

Figuring out how many times 50 goes into 100,000 isn't easy, but what if it doesn't have to be that complicated? Let's remove 3 zeros from 100,000, effectively dividing it by 1,000. 50 goes into 100 twice. Add those three zeros back, we get 2,000. So, (100,000 widgets) / (50 employees) = 2,000!

Case Example: Working with messy numbers

Our client is a massive, and I mean massive pizza joint in New York City. They have 200 pizza ovens that can each produce 125 pizzas per week. What's their total pizza making capacity?

To make this problem easier, we can break up the problem into two parts using the distributive law. Instead of 200 * 125, we can set up the problem as (200 * 100) + (200 * 25).

• (200 * 100) = 20,000
• (200 * 25) = 5,000
• Now, add it back together: 20,000 + 5,000 = 25,000.

So in total, our client can produce a whopping 25,000 pizzas each week.

Percentages calculations

Working with percentages and proportions is all over business analysis, and the use of percentages is a key skill in tons of different case math problems (more on this later…).

Percentage problems aren't hard to conceptualize, they're just the multiplication of a proportion to a given metric. The trick is learning how to do them quickly, or how to structure more complicated questions so that you don't get lost in a sea of numbers.

Let's go through the two basic ways percentages calculations come up in case interviews:

• Easy percentage calculations
• Messy percentage calculations

Case Example: Easy percentages

Simple percentage questions can be quite easy.

If cost of goods sold (COGS) is 10% of a Company A's revenue and they did $160 million in revenues this year, what is the exact level of COGS?

Calculation: 10% of $160 million can be calculated as $160/10, which is $16 million.

Let's say an analyst from Company A approaches us and tells us COGS are actually 15% of revenue. We can break up the calculation into two parts. Instead of directly calculating 15% of $160 million, we can calculate 10% and 5% of $160 million and add them together.

Calculation: We know 10% of $160 million is $16 million, and 5% of $160 million is half of that. So in total, COGS is $24 million.

Case Example: Messy percentages

Okay, let's make it a bit more complicated. As a result of a cost cutting initiative, Company A has reduced COGS to just 13% of revenue. Revenues have remained stable at $160 million. What is the exact level of COGS?

We can use the same technique from before. We'll split up the percentages into easy to manage chunks. Instead of a direct calculation of 13%, we can set it up as ($160 million * 10%) + ($160 million * 3%).

We know 10% of $160 million is $16 million

To further split up our 3% figure, we can set it up as 3*(1% * $160 million).

1% of $160 million is $1.6 million. Multiplied by three, that's $4.8 million

In sum, we get COGS = ($16 million) + ($4.8 million) = $20.8 million.

Breakeven analysis

Breakeven analysis asks an interviewee to determine the amount of sales necessary to recoup a large upfront investment or cost - the breakeven point for a certain product or service. To put it simply, breakevens ask "How many units (or services) do I need to sell to make up for my upfront costs?"

Solving these problems follow a pretty standard format. Determine the marginal profit per unit or sale, and divide your initial investment by that metric. So the formula is:

(Investment) / (Unit revenue - unit cost) = Units required to "break even"

Quick tip: Breakevens often involve big division type problems. Mastering that skill will help a lot when dealing with breakeven calculations.

Case Example: Launching a snazzy new tech product

Tech products have high R&D costs, and a critical goal for technology companies is to recoup that initial investment within a reasonable timeframe after launch. For our product, we are given the following information: 1. Our client expects to spend $1 million in development 2. Each unit costs $100 to produce, and it's sold for $300 So, how many units would we need to sell per year to recoup the initial investment?

So, how many units would we need to sell per year to recoup the initial investment?

Let's apply our formula

• Investment = $1 million
• Unit revenue - Unit costs = $200
• Calculation: ($1 million) / ($200) = 5,000 units

Growth estimations

Growth estimates are a staple in business analysis. Companies are always thinking about and forecasting the future, and to do this they apply estimated growth rates to current metrics to inform where a business is going and how that may affect their strategy.

The simplest growth estimation problems will be one-period estimations. For example, if a business is currently doing $1 million in sales and growth is expected to growth over the next year by 20%, sales in the next year will be ($1 million) * (1 + 20%) = $1.2 million.

More complicated growth estimations will have multiple periods, and really tough problems will have varying growth rates. Let's walk through an example of each type below.

Case Example: Determining future revenues from fixed, multiperiod growth

Let's say that our client is currently doing $10 million per year in revenues, and revenue has historically been growing at a rate of 5% year-over-year. Their investors have asked the CEO to prepare a report on how revenues will grow over the next 2 years. We have been told we can assume growth rates will stay the same.

To determine this, we can use the formula for compound growth:

(Present Value) * (1 + growth rate) ^ (number of periods).

For this problem, the formula would be: ($10 million) * (1 + 5%) ^ (2). (1.05)^2 is equal to 1.1025, and ($10 million) * (1.1025) = $11.025 million.

Alternatively, if you don't want to deal with exponents, you could calculate this in a stepwise fashion.

• Year 1: ($10 million) * (1.05) = $10.5 million
• Year 2: ($10.5 million) * (1.05) = $11.025 million

Case Example: Determining future costs from variable, multiperiod growth

In this case, imagine we were working with the same client, but they now want to know what their revenues will be in 4 years. Importantly, they expect growth to be 10% in years 3 and 4, up from 5% in the first two years.

To solve this problem, we can break up the growth estimates into two steps while using the compound growth formula. In the first step, we'll apply the 5% growth rate to the original revenue figure and project 2 years of revenue growth. Then we'll take the result and do the same calculation with the 10% growth rate for the final two years.

• Step 1: ($10 million) * (1.05)^2 = $11.025 million
• Step 2: ($11.025 million) * (1.10)^2 = ~ $13.34 million

Again, if you don't want to deal with the exponents or the numbers are more complicated, you can do the calculations in each step in a stepwise fashion.

We could also use a simple trick to get a "good enough" estimate of our answer. Instead of figuring out a complex exponent, we can add the compound growth rates together and multiply our original value by that sum. Let's see this in practice:

• Step 1: We have two periods of 5% growth, so we'll multiply ($10 million) * (1.10) = $11 million
• Step 2: In this case we have two periods of 10% growth, so we'll multiply ($11 million) * (1.20) = $13.2 million

Notice that while our answer is not exactly correct, it's within 1% of our answer and is certainly close enough for the purpose of a case interview! Plus, you can do this sort of math way faster. It's a win-win situation. More on this trick later...

Market size calculations

Market math problems are an extension of percentage problems applied to a company's market share or a total market size. They'll come in all shapes and sizes, but always ask you a version of the following question:

If X% of a market is $X, how big is the total market?

Market math problems come in two forms: ones with "easy" numbers, and ones with "messy" numbers. Let's walk through an example of each.

NOTE: For more in-depth market sizing estimate drills, see our overview of on how to approach market sizing estimates .

Quick Tip: Mastering percentages will make these problems a breeze!

Case Example: Market math with "easy" percentages

We know that our client has captured 10% of the market, and currently does $10 million in revenues. What is the full market size?

Mathematically, the formula is: (Revenues) / (Marketshare). In this case, ($10 million) / (0.1) = $100 million total market size.

A far easier way to do this is to recognize that 10% goes into 100% ten times. So, we can multiply our client's revenues by 10 to get the market size. So ($10 million) * (10) = $100 million. This shortcut can be applied to all sorts of "easy" percentages.

• For 5%, the multiplier is 20
• For 20%, the multiplier is 5
• For 33%, the multiplier is 3

You get the idea!

Case Example: Market math with "messy" percentages

Now let's imagine that our client has a market share of 17%, and revenues are still $10 million. What's the total market size?

There's no easy multiplier that we can use here, at least at first... The trick with these messier problems is to make a rough estimation by rounding the market share to an "easy" number. Interviewers usually don't expect an exact answer, and as long as you don't round to aggressively you should be in the clear!

In this case, we could round 17% to 20%. Then, we can use a shortcut to multiply $10 million by 5 to get a total market size of $50 million. Since we rounded up, you can say that the total market size is just north of $50 million - which we know since we rounded the divisor up.

Tips for how to prepare (Top)

Isolate core skills and master them.

We just went over the skills necessary to rock your case math. Just like you did in school, you need to study and master them.

A tool like RocketBlocks makes this process easy. We have tons of content that walks you through the different types of math encountered in a case interview and comprehensive strategies for approaching and solving these problems.

💡 Shameless plug: Our consulting interview prep can help build your skills

Learn mental math short cuts

In an interview, you're expected to be able to apply these math skills quickly in the context of the case. But as we just went over, not all of the math is super simple and there can be plenty of room for error. Learning math shortcuts will make your case math far more efficient and accurate. You'll make less errors, move more quickly through a case, and have more time to apply the results of your analyses to the problem at hand.

Here's are two examples of these types of shortcuts:

The Rule of 72 : The time it takes a metric to double given a certain growth rate can be roughly determined by dividing 72 by that growth rate. So given a 10% year-over-year growth rate for revenue, you can say that will take roughly 72/10 = 7.2 years for revenue to double.

Estimating Compound Growth Rates : Instead of figuring out a complex exponent, you can quickly estimate a compound growth by adding the component growth rates together. So if you are estimating 3-year compound growth of a metric at 5% growth year-over-year, a good estimate would be to apply a 15% growth rate to your metric. This can be very helpful when you don't need an exact answer for the problem at hand.

Note: This technique doesn't work well for super high growth rates or a ton of periods.

Conclusion: Practice makes perfect (Top)

The key to getting good at case math is to do A LOT of practice problems. The more the better. As you do more you'll get faster, see more of the different types of problems, and get used to applying core case math skills in different contexts.

One way to practice is to do lots of mock cases. This will present relevant case math but the downside is you might spend 1hr on a case and only do math for 5 minutes! To excel at the math component, you should do targeted practice on math specifically.

A more targeted way to prepare for case math is to use a tool like RocketBlocks to isolate the skills you're weakest on and gain access to an almost unlimited number of problems. RocketBlocks helps you practice case math in two ways:

• Practice cut and dry case math problems using our Math drills
• Practice case math in the context of applied data analysis in our Charts and Data drills

Bottom line: to get good at case math you have to do a lot of example problems.

Read this next:

• Case interview format 101
• Analytical skills for consulting
• Soft skills for consulting
• 1st vs. 2nd round case interviews
• Consulting recruiting process

See all RocketBlocks posts .

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Consulting & Case Interview Math Practice Guide

The truth is you do not need to be an expert mathematician to work as a consultant, still, you will need to do a lot of calculations when working in this field. 

The math concepts utilized in consulting are not more challenging than those studied from academic math. More exactly, it is different and attempting to solve problems the same way as in school will not be effective. 

This article will explain to you what makes math so important for aspiring consultants and provide you with some crucial math areas in which you need to be proficient in as well as how to best practice consulting math.

Table of Contents

Do you need math in consulting or case interviews?

Math is omnipresent in case interviews….

This industry is known for its complex business problems and challenging strategic decisions, which require a strong foundation in mathematics. Thus, mathematics is a fundamental skill that is essential for success in consulting case interviews .  

Consulting companies usually use case interviews to test candidates' quantitative skills and problem-solving abilities, which means that proficiency in math is a must-have skill for any aspiring consultant.

Candidates who are proficient in math have a significant advantage in the case interview process. They are better equipped to analyze data, create models, and make informed decisions based on quantitative analysis . This is because math skills enable candidates to think logically and analytically, which is crucial in the consulting industry.

In addition to helping candidates solve complex business problems, good math skills can also help them to solve the case more efficiently. By quickly identifying the key data points and using mathematical formulas to analyze them, interviewees can save time and leave more time for insightful ideas and recommendations.

… because it’s always there in real consulting work

Additionally to case interviews, the consulting industry as a whole also places a high value on mathematical ability. Without a strong foundation in math, consultants may struggle to work effectively in their daily duty.

Math skills are essential for performing data analysis and modeling, which are crucial aspects of the consulting job. Consulting firms rely heavily on data-driven insights to deliver value to clients, and math skills are necessary to analyze data, identify patterns, and draw meaningful conclusions that can help clients make informed decisions.

Secondly, to develop engaging talks and reports that successfully illustrate information and suggestions, solid arithmetic skills are also required. 

These abilities are essential since clients rely on consultants to provide them with actionable insights that can drive their business forward. By using mathematical formulas and models, consultants can present complex data in a clear and concise manner , making it easier for clients to understand and act upon.

Furthermore, math skills are crucial for financial analysis, which is another critical aspect of the consulting job. Consultants must be able to analyze financial data, create financial models, and make informed decisions based on quantitative analysis . This requires a strong foundation in math, including knowledge of statistics, probability, and financial mathematics.

7 Types of math you need in consulting

Basic operations (add, subtract, multiply, divide)

Definition: Basic math refers to the fundamental arithmetic operations used in mathematics. It includes addition (+), subtraction (-), multiplication (x), and division (/), which are used to perform simple calculations and solve basic math problems.

These fundamental arithmetic operations are used in various calculations and are necessary for understanding more advanced math concepts. Basic math is used to calculate various metrics, ratios, and other complex work in consulting as well as case interviews. Without a strong foundation in basic math, it would be challenging to perform such calculations accurately and efficiently.

Suppose a consulting project requires calculating the total cost of producing 10,000 units of a product. The cost per unit is $50 for direct materials, $30 for direct labor, and $20 for overhead expenses. To calculate the total cost, we need to use addition and multiplication:

Total cost = (Direct materials cost per unit + Direct labor cost per unit + Overhead cost per unit) x Number of units

Total cost = ($50 + $30 + $20) x 10,000 = $1,000,000

Ratios and percentages

Definition: Ratio is a comparison of two or more quantities, while percentage is a ratio expressed as a fraction of 100. Ratios and percentages are used to express relationships between different variables and are commonly used in finance, statistics, and other fields.

Consultants frequently use ratios and percentages to analyze financial statements, assess market share, and evaluate operational performance. Additionally, they evaluate various scenarios and spot shifts over time using this kind of calculation.

Suppose a consulting project requires analyzing the profitability of a company. We need to calculate the gross profit margin, which is the ratio of gross profit to revenue expressed as a percentage. If the gross profit is $500,000 and the revenue is $1,000,000, we can calculate the gross profit margin as follows:

Gross profit margin = (Gross profit / Revenue) x 100%

Gross profit margin = ($500,000 / $1,000,000) x 100% = 50%

Management accounting formulas and principles

Definition: Accounting math is a set of mathematical principles and methods used in accounting to record, classify, and analyze financial data. It includes the calculation of various financial ratios, such as profit margin, return on investment, and debt-to-equity ratio, which are used to evaluate a company's financial health.

Accounting math involves the use of specific formulas and calculations to prepare financial statements, such as balance sheets, income statements, and cash flow statements. Consultants regularly have to use accounting math to interpret financial data, identify areas for improvement, and develop financial models.

Common accounting math formulas:

Balance Sheet Equation: Assets = Liabilities + Equity

Income Statement Equation: Revenue - Expenses = Net Income

Gross Margin Ratio = (Revenue - Cost of Goods Sold) / Revenue

Debt-to-Equity Ratio = Total Debt / Total Equity

Current Ratio = Current Assets / Current Liabilities

Suppose a consulting project requires analyzing the financial statements of a company. We need to calculate the current ratio, which is a measure of the company's liquidity. If the current assets are $1,000,000 and the current liabilities are $500,000, we can calculate the current ratio as follows:

Current ratio = Current assets / Current liabilities

Current ratio = $1,000,000 / $500,000 = 2

Basic finance formulas and principles

Definition: Finance math refers to the mathematical principles and methods used in finance to analyze and manage financial data. It includes the calculation of financial ratios, such as return on investment, net present value, and internal rate of return, which are used to evaluate investment opportunities and make financial decisions.

Finance math includes advanced financial modeling and analysis techniques, such as discounted cash flow analysis, net present value calculations, and internal rate of return analysis. Consultants will need to use finance math with purposes like evaluate investment opportunities, assess risk, and make strategic recommendations.

Present Value (PV) = Future Value / (1+interest rate)^number of period

Future Value (FV) = Present Value x (1+interest rate)^number of period

Net Present Value (NPV) = sum of all present values of cash inflows - sum of all present values of cash outflows

Internal Rate of Return (IRR) = the interest rate at which the NPV of an investment is zero

Return on Investment (ROI) = (Gain from Investment - Cost of Investment) / Cost of Investment

Suppose a consulting project requires evaluating investment opportunities. We need to calculate the net present value (NPV) of an investment, which is the difference between the present value of cash inflows and the present value of cash outflows. If the cash inflows for the first year are $50,000 and the cash inflows for the second year are $100,000, and the discount rate is 10%, we can calculate the NPV as follows:

NPV = Cash inflow year 1 / (1 + Discount rate)^1 + Cash inflow year 2 / (1 + Discount rate)^2

NPV = $50,000 / (1 + 10%)^1 + $100,000 / (1 + 10%)^2 = $126,456.83

Basic statistics and probabilities

Definition: Probability is the branch of mathematics that deals with the study of random events and their likelihood of occurring. It involves calculating the probability of different outcomes based on the available information and using this information to make predictions.

Probability is used in consulting to assess the likelihood of different outcomes and events, and to develop risk management strategies. Consultants use probability to analyze market trends, identify potential risks, make forecasts, and develop contingency plans.

Suppose a consulting project requires analyzing customer data to identify patterns. We need to calculate the probability of a customer making a purchase given that they have visited the company's website. If the number of website visitors is 10,000 and the number of customers who made a purchase is 500, we can calculate the probability as follows:

Probability of purchase given website visit = Number of customers who made a purchase / Number of website visitors

Probability of purchase given website visit = 500 / 10,000 = 5%

“Weighted” calculations

Definition: This is a method of calculating a value based on the weights assigned to different variables. It is commonly used in finance and economics to determine the overall performance of a portfolio, and in other fields to calculate averages of different sets of data.

A weighted average is used to calculate the average of a set of numbers, with each number being multiplied by a corresponding weight. This type of math is very useful for consultants since it helps them to analyze financial data, such as revenue growth or customer satisfaction, and to develop performance metrics.

Suppose a consulting project requires analyzing survey data. We need to calculate the overall satisfaction score for a product, which is based on ratings for different features. If the ratings for feature A, B, and C are 3, 4, and 5 respectively, and the weights for these features are 30%, 40%, and 30% respectively, we can calculate the overall satisfaction score as follows:

Overall satisfaction score = Rating for feature A x Weight for feature A + Rating for feature B x Weight for feature B + Rating for feature C x Weight for feature C

Overall satisfaction score = 3 x 0.3 + 4 x 0.4 + 5 x 0.3 = 3.0

Exhibits (chart, tables, diagrams)

Definition: An exhibit is a graphical representation (chart) of data that is used to present information in a clear and easily understandable format. It can be used to display trends, patterns, and relationships between different variables, making it a useful tool for visualizing complex data sets.

Consultants need to use charts and graphs to present complex data and analysis in a clear and concise manner. They use various types of charts, such as pie charts, bar charts, and line graphs, to convey important information to clients and stakeholders. They also help consultants to identify trends and patterns in data, making it easier to draw meaningful conclusions and make informed suggestions.

This part of mathematics is quite diversified since there are numerous types of charts, tables, and diagrams, thus I cannot provide examples for every situation. Learn more at: Six types of chart in case interview

Consulting mental math

Why is mental math in consulting important.

This may also be categorized as a type of mathematics, but I've decided to address it separately to emphasize the significance it is. In the consulting industry, quick math is essential. Although the resulting numbers don't have to be 100% accurate (usually the error margin will be around 5%) , you will have to give a quick result.

In many circumstances, there is not a sufficient amount of time to get out a calculator, indeed, they are not even permitted in tests and case interviews . Hence, Mental math is a crucial component of an interview that frequently gets noticed by interviewers when it helps candidates to solve the questions and show their mental agility. 

When you have become a consultant, mental calculation is even more important since it not only helps us save a ton of time but also builds credibility with people around . You wouldn't want to look sluggish and perplexed in front of your managers, clients or interviewers, would you?

How to do mental math for consulting?

For our Comprehensive Math Drills, we have developed a methodical approach to mental calculations with large numbers, consisting of two main steps: ESTIMATION and ADJUSTMENT. This method is used for multiplication, division, and percentage.

Step 1 - Estimation

• Simplify the large numbers by  taking out the zeroes (e.g. 6,700,000 becomes 6.7 and 000000)
• Round the resulting 1-to-2-digit numbers for easier calculations (e.g.: 6.7 becomes 7)

Step 2 - Adjustment

• Perform simple calculations with the multiplicands
• Adjust in the opposite direction of the previous rounding and put the zeroes back in

Step 3 - Percentages

• To do percentages , multiply the original number with the numerator then divide by 100.

Detail example:

• Multiplication: 1,234 x 5,678

Take out zeroes: 12.34 x 56.78 | 00 00

Round: 12 x 60 | 00 00

Calculate: 720 | 0 000

Adjust and add zeros: 7,200,000 (equal down-rounding and up-rounding roughly cancels each other out)

Accurate result: 7,006,652 | Error margin: 2.7%

• Division 8,509 / 45

Take out zeroes: 85 / 4.5 | 00 / 0

Round: 90 / 4.5 | 00 / 0

Calculate: 20 | 0

Adjust and add zeros: 190 (up-rounding means downward adjustment)

Accurate result: 189.09 | Error margin: 0.48%

• Percentage 70% of 15,940

Convert %: 0.7 x 15,940

Take out zeroes: 7 x 15.9 | One 0 in, three 0 out

Rounding: 7 x 16 | One 0 in, three 0 out

Calculate: 112 | One 0 in, three 0 out

Add zeros: 11,200

Adjust : 11,150 (up-rounding means downward adjustment)

Accurate result: 11,158 | Error margin: 0.07%

For percentage calculations, it is even easier with the “ Zeroes management ” . We know that the final answer will have roughly the same number of digits as the original 15,940, something like 1x,xxx or x,xxx. So when having 112 after step “Calculate”, we know the final answer would be close to 11,200.

Mental Math Tips:

• Write down numbers : it’s always a good idea to have a visual of the numbers themselves on paper. This makes it 100 times easier,  especially with “zeroes management” work.
• Sanity check: always take a very brief moment to ask yourself “is this result logical?”; if 70% of 15,940 equals 111,500, perhaps something is wrong with your “zeroes management”. Sometimes you can compare the outcome with another obvious data
• Shortcut percentages: convert percentages into easy, common calculations (e.g.: 33%, 25%, 20% into /3, /4, /5…) if possible. In fact, know as many of these shortcuts as you can.
• It is better to be long than to be wrong: If your mental math is not good and cannot calculate quickly, do not hesitate to ask for a little more time to get the most accurate answer.

Consulting math practice

Step 1: learn about yourself.

The first step to acing consulting math is to understand yourself, your strengths, your weaknesses and your needs. Assess your current skill level and identify areas where you need to improve . This can be done by reviewing your previous academic performance, past work experiences, and feedback from others.

One of the best ways to learn more about the consulting industry and the type of math skills required is to network with those who have experience in this field. They can provide valuable advice and insights on what to expect during tests/interviews and which math skills are most important. 

This is an extremely important step, but it might be challenging to carry out since many people do not know/have contact with any consultants, former consultants and interviewers. If there are a few people in your network, that's fantastic; if not, you can use our Coaching services to get the most reliable information from current consultants.

Step 2: Developing an actionable plan 

Once you have a clear understanding of yourself, you now need to establish a clear, actionable strategy to improve your consulting math skills . This plan should include a list of resources and activities that will help you promote self-study and focus on the areas that need the most improvement.

There are many resources available for consulting math practice , including online courses, textbooks, and practice mental exercise. You may also want to consider hiring a tutor or attending a community to receive personalized guidance and feedback.

When developing your plan, it is important to set realistic goals and establish a timeline for achieving them. This will help you stay motivated and track your progress along the way.

Step 3: Implement the plan

Now that you have a plan in place, it's time to implement it. Set aside dedicated time each day or week to practice your consulting math skills . Consistency is key, so make sure you stick to your schedule and do not skip any practice sessions.

When practicing, it is important to focus on understanding the underlying concepts rather than simply memorizing formulas and equations. Work through practice problems step-by-step and identify where you might be making mistakes. 

Moreover, reviewing your work and seeking feedback from others can help you improve your approach and increase your accuracy.

Step 4: Adjust the plan to best suit your capacity

Finally, It is crucial to modify your strategy as necessary to accommodate your capabilities. If you find that you are struggling with a particular concept or area, don't be afraid to pivot and adjust your plan accordingly. Consider seeking additional resources or seeking guidance from others who have experience.

At the same time, do NOT get too caught up in perfecting every aspect of consulting math is also matter. Recognize your strengths and weaknesses and focus on improving in areas where you can make the most progress. 

Remember that the goal is not to be perfect but to demonstrate your ability to approach and solve complex problems in a logical and efficient manner.

Common math mistakes in consulting 

Messing up formulas.

The inability to apply the formula errors or mess up with numbers/signs is a common mistake made by candidates. Anyone may make this error because of both internal factors like mathematical confusion and external factors like time pressure.

To avoid this issue, you must first be cautious throughout the procedure, go step-by-step, and carefully examine the data and signs . If you need extra time, kindly request it, keep in mind that the important thing is getting the proper outcome.  

Secondly, practice using different formulas to solve various problems before the interview. Ensure that you have a clear understanding of when and how to apply each formula.

Excess or missing zeroes

Another common math mistake is losing units in calculations. When performing calculations, you must keep track of units to ensure that your answer is meaningful and relevant to the context of the problem. Losing units can make your answer meaningless and confusing, which could lead to wrong conclusions.

In order to avoid making this error, it's essential to label each step of your calculation and remember to carry the units as you work with them . Keeping your calculations well-organized will prevent you from losing track of the units. 

Another tip is try to reduce the unit of each metric as much as possible by assigning it to a term. For example, you can write “42,000,000” to “42M”. This will both ease your calculations and avoid confusion, but remember to add the units back to the final result 

Missing the bigger picture

The math done during a consulting case interview serves as a tool, not an end in and of itself. It is crucial to remember that calculations are part of a more significant business problem that you have to solve . 

Many candidates get wrapped up in calculations, arrive at the correct final number, but forget why they were doing the math and the real purpose of the number they have just found out.

To prevent this mistake, remember the significance of the figures you are calculating in the context of your particular business case. One tactic to use is to write the question asked at the top of your sheet before deep diving into your math. 

As you go through your calculation and as you prepare to present your solution, keep reminding yourself the question you were originally asked and ask yourself if the result you got from your calculation is actually answering it. Then when you explain your answer, do so in a way that clearly shows you understand what your final number means.

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Hacking the Case Interview

Although case interviews do not require any technical math or finance knowledge, there are basic formulas that you should know in order to do well in order to master case interview math .

This article will cover the 26 formulas you should know for case interviews. These formulas are organized into the following categories:

• Profit Formulas
• Investment Formulas
• Operations Formulas
• Market Share Formulas
• Accounting, Finance, and Economics Formulas

If you’re looking for a step-by-step shortcut to learn case interviews quickly, enroll in our case interview course . These insider strategies from a former Bain interviewer helped 30,000+ land consulting offers while saving hundreds of hours of prep time.

Profit Formulas for Case Interviews

1. Revenue = Quantity * Price

Revenue is the amount of money a company brings in from selling its products. This can be calculated by taking the number of units sold and multiplying it by the price per unit.

Example: Your company sells shirts for $20 each. Last year, your company sold 1,000 shirts. So, your total revenue last year was 1,000 * $20 = $20,000.

2. Total Variable Costs = Quantity * Variable Costs

Costs are payments that a company needs to make in order to run and operate its business. There are two different types of costs, variable costs and fixed costs.

Variable costs are costs that directly increase for each additional unit of product made. It represents the cost of raw materials needed to make the product.

Total variable costs are calculated by taking the number of units produced or sold and multiplying it by the raw material cost per product.

Example: It costs your company $5 to purchase the raw materials needed to make a shirt. If your company sold 1,000 shirts last year, the total variable costs are 1,000 * $5 = $5,000.

3. Costs = Total Variable Costs + Fixed Costs

Total costs for the company can be calculated by adding total variable costs and fixed costs.

Fixed costs are costs that do not directly increase for each additional unit of product made. They may include costs such as rent for the building or equipment needed to make the product.

Example: Your company pays annual rent of $10,000. It also leases the equipment it needs to make its shirts for $2,000 a year. Therefore, fixed costs are $10,000 + $2,000 = $12,000. Total variable costs were calculated to be $5,000 from the previous example. So, total costs are $12,000 + $5,000 = $17,000.

4. Profit = Revenue – Costs

Profit is the amount of money the company keeps after paying for all of its costs. Profit is calculated by subtracting total costs from total revenue.

Example: Last year, your shirt company generated revenues of $20,000 and had costs of $17,000. The profit last year was $20,000 - $17,000 = $3,000.

5. Profit = (Price – Variable Costs) * Quantity – Fixed Costs

This formula summarizes the previous four formulas into one concise and simplified equation.

6. Contribution Margin = Price – Variable Cost

Contribution margin represents how much money each product sold brings into the company after accounting for the cost of raw materials needed to make the product.

Example: If your company’s shirts sell for $20 and raw materials cost $5, then the contribution margin is $20 - $5 = $15 per shirt.

7. Profit Margin = Profit / Revenue

Profit margin represents the percentage of revenue that a company keeps as profit after taking into account all of its costs.

Example: Last year, your company generated $20,000 in revenue and had $17,000 in costs. Its profit was $3,000. Therefore, your company’s profit margin is $3,000 / $20,000 = 15%.

Investment Formulas for Case Interviews

8. Return on Investment = Profit / Investment Cost

Companies make investments by spending money in the hopes of earning even more money in the future as a result of the investment. Return on investment, or ROI or short, represents how much additional money a company generates relative to the size of its initial investment.

ROI is calculated by taking the profit that the company generated from the investment and dividing it by the investment cost.

Example: Your company spent $5,000 on marketing to advertise its shirts. As a result, the company generated an additional $6,000 in profits from selling shirts. This profit does not yet take into account the costs of the marketing campaign.  Therefore, the company has a net increase in profits of $1,000 from its original $5,000 investment. The ROI is $1,000 / $5,000 = 20%.

9. Payback Period = Investment Cost / Profit per Year

Payback period represents how long it would take a company to recoup the money it spent on an investment. It is usually specified in years.

Example: Your company invested in redesigning its shirts for $5,000. As a result, the company expects annual profits to increase by $1,000 for every year going forward. Therefore, the payback period for this investment is $5,000 / $1,000 = 5 years.

Operations Formulas for Case Interviews

10. Output = Rate * Time

The output of production can be calculated by taking the rate of production and multiplying it by time.

Example: The machine that your company uses to produce shirts can produce 5 shirts per hour. If the machine runs for 12 hours, then it will produce 60 shirts.

11. Utilization = Output / Maximum Output

Utilization represents how much a factory or machine is being used relative to its maximum possible output.

Example: The machine that your company uses to produce shirts can produce 5 shirts per hour. Therefore, its maximum capacity in a day is 5 shirts per hour * 24 hours = 120 shirts. If your machine is being used to only produce 60 shirts per day, then it is at 60 / 120 = 50% utilization.

Market Share Formulas for Case Interviews

12. Market Share = Company Revenue in the Market / Total Market Revenue

Market share measures the percentage of total market sales a particular company has. Market shares can range from 0%, no presence in the market, to 100%, complete dominance in the market.

Example: Your company sells shirts and generates $100M in annual revenues. The market size of shirts is $500M. Therefore, your company has a market share of $100M / $500M = 20%. 

13. Relative Market Share = Company Market Share / Largest Competitor’s Market Share

Relative market share compares a company’s market share to the largest competitor’s market share. It measures how strong of a presence a company has relative to the market leader. If the company is the market leader, relative market share measures how much of a lead they have over the next largest player.

Instead of using company market share and the largest competitor’s market share, you can use company revenue and the largest competitor’s revenue. This will give you the same answer.

Example: Your company has a 20% market share in the shirts market. Your largest competitor has a 50% market share. Therefore, your relative market share is 20% / 50% = 0.4.

Example 2: Your company is the market leader and has a 50% market share in the shirts market. Your largest competitor has a 25% market share. Therefore, your relative market share is 50% / 25% = 2.

Accounting, Finance, and Economics Formulas for Case Interviews

These formulas are much less commonly seen in case interviews than the previous formulas. You likely won’t need to use these formulas since they require more technical knowledge of accounting, finance, and economics.

However, you should still be familiar with these formulas in the small chance that one of these concepts shows up in your case interview.

14. Gross Profit = Sales – Cost of Goods Sold

Gross profit is a measure of how much money a company makes from selling its product after taking into account the costs associated with making and sellings its product. These costs are often called the cost of goods sold.

Compared to the previous profit formula, which was simply revenue minus costs, gross profit is always higher since it does not take into account all of the costs of the business.

Example: Your company sold $20,000 of shirts last year. The cost to produce these shirts was $5,000. Therefore, your gross profit is $20,000 - $5,000 = $15,000.

15. Operating Profit = Gross Profit – Operating Expenses – Depreciation – Amortization

Operating profit is calculated by taking gross profit and subtracting all operating expenses and depreciation and amortization.

Operating expenses may include rent, utilities, maintenance and repairs, advertising and marketing, insurance, and salaries and wages. So, operating profit is always less than gross profit.

Depreciation is the spreading of a fixed asset’s cost over its useful lifetime.

For example, let’s say that a company purchases a new machine for $10,000 that it expects to last for 5 years. Instead of stating that it incurred $10,000 in costs in its first year, the company may choose to state that the new machine costs $2,000 per year for the next five years.

Amortization is the spreading of an intangible asset’s cost over its useful lifetime. It is the exact same principle as depreciation except that it deals with intangible assets, or assets that aren’t physical.

For example, let’s say that a company purchases a patent for $10,000 and expects the benefits of the patent to last for 20 years. Instead of stating that it incurred $10,000 in costs in its first year, the company may choose to state that the patent costs $500 per year for the next twenty years.

Example: You sold $20,000 of shirts last year. Cost of goods is $5,000, operating expenses are $10,000, depreciation of a machine is $2,000, and amortization of a patent is $500. Therefore, your operating profit is $20,000 - $5,000 - $10,000 - $2,000 - $500 = $2,500.

16. Gross Profit Margin = Gross Profit / Revenue

This is the exact same formula as the profit margin formula except that gross profit is used. Gross profit margin measures how much money a company keeps from selling its products after taking into account cost of goods sold.

Example: Your company has a gross profit of $15,000 from $20,000 of revenue. Therefore, your gross profit margin is $15,000 / $20,000 = 75%.

17. Operating Profit Margin = Operating Profit / Revenue

This is the exact same formula as the profit margin formula except that operating profit is used. Operating profit margin measures how much money a company keeps from sellings its products after cost of goods sold, operating expenses, depreciation, and amortization is taken into account.

Example: Your company has an operating profit of $2,500 from $20,000 of revenue. Therefore, your operating profit margin is $2,500 / $20,000 = 12.5%.

18. EBITDA = Operating Profit + Depreciation + Amortization

EBITDA stands for earnings before interest, taxes, depreciation, and amortization. It is a financial metric used to measure a company’s cash flow or the amount of cash that a company has generated in a period of time.

To calculate EBITDA, start with operating profit and add back depreciation and amortization expenses.

Example: Your company has an annual operating profit of $2,500. Depreciation expenses are $2,000 and amortization expenses are $500. Therefore, your EBITDA is $2,500 + $2,000 + $500 = $5,000.

19. CAGR = (Ending Value / Beginning Value)^(1/Time Period) – 1

CAGR stands for compounded annual growth rate. It measures how quickly something is growing year after year.

Example: Your company generates $144M in annual revenue. Two years ago, your company only generated $100M. Over this time period, your CAGR was ($144M / $100M)^(1/2) - 1= 20%. In other words, your company grew by 20% each year for two years.

20. Rule of 72

The Rule of 72 is a shortcut used to estimate how long a market, company, or investment would take to double in size. To use it, simply divide 72 by the annual growth rate to get an estimate for the number of years needed to double in size.

Example: Your company is growing steadily at 9% per year. Using the Rule of 72, we’d expect it to take 72 / 9 = 8 years for your company to double in size if it maintains its current growth rate.

21. NPV = Cash Flow / [(1 + Discount Rate)^(Time Period)]

NPV stands for net present value. It measures how much future cash flow is worth today.

Receiving $1,000 right now is not the same as receiving $1,000 five years from now. If you received $1,000 right now, you could invest it and grow your money. Therefore, it is better to receive $1,000 right now than to receive the same amount in the future.

Net present value takes this into account.

Cash flow is the amount of money you expect to receive in the future. Time period is how many years in the future you will receive that amount of money. The discount rate is the return you expect to get from investing your money.

Example: You expect to receive $1,000 five years from now. You expect that you will be able to get 8% annual returns by investing in the stock market. Therefore, the net present value of your future cash flow is $1,000 / [(1 + 0.08)^5] = $680.58.  In other words, receiving $680.58 today would give you the same value as receiving $1,000 five years from now.

22. Perpetuity Formula: Present Value = Cash Flow / Discount Rate

An annuity is a fixed sum of money paid at regular intervals such as every year. Perpetuity is an annuity that lasts forever.

The present value of a perpetuity is calculated by taking the cash flow of each payment and dividing it by the discount rate.

Example: You are expecting to receive $1,000 per year for the rest of your life. You expect that you will be able to get 8% annual returns by investing in the stock market. Therefore, the present value of this perpetuity is $1,000 / 0.08 = $12,500.  In other words, receiving $12,500 today would give you the same value as receiving $1,000 each year for the rest of your life.

23. Return on Equity = Profit / Shareholder Equity

Return on equity , or ROE for shirt, measures how effectively a company is using its assets to create profits. It is calculated by taking profit and dividing by shareholder equity, which represents the net worth of a company.

In other words, shareholder equity is the value of a company’s total assets minus its total liabilities.

Example: Your company’s profit this year is $100M. Shareholder equity, or the net worth of the company is $1B. Your company has a ROE of $100M / $1B = 10%.

24. Return on Assets = Profit / Total Assets

Return on assets , or ROA for short, measures how profitable a company is relative to its total assets. In other words, it shows how efficiently a company is using its assets to generate income.

Assets can be anything that has value that can be converted into cash. This includes cash, property, equipment, inventory, and investments.

Example: Your company’s profit this year is $100M. Your company as $400M worth of assets. Your company has a ROA of $100M / $400M = 25%.

25. Price Elasticity of Demand = (% Change in Quantity) / (% Change in Price)

Elasticity is a measure of how much customer demand changes for a product given a change in the product’s price. In almost all cases, an increase in a product’s price results in a decrease in customer demand. Therefore, price elasticity of demand is usually negative.

Example: Your company has decreased its product’s price by 10%. As a result, the number of units sold has increased by 20%. Therefore, the price elasticity of demand is 20% / -10% = -2.

26. Cross Elasticity of Demand = (% Change in Quantity for Good #1) / (% Change in Price for Good #2)

Cross elasticity of demand measures how much customer demand changes for a product given a change in price of a different product.

If two products are complements, an increase in price of one product will result in a decrease in demand of the other product. Complementary products have a negative cross elasticity of demand.

If two products are substitutes, an increase in price of one product will result in an increase in demand of the other product. Substitute products have a positive cross elasticity of demand.

Example: A competitor has decreased the price of a competing product by 20%. As a result, the demand for your product has dropped by 10%. The cross elasticity of demand is -10% / -20% = 0.5.

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Case interview math (mental math) tools, formulas and tips.

Case Interview Math (Mental Math) Tools, Formulas and Tips

Consulting case interview mental math practice is a must as part of one’s overall consulting case interview preparation. All management consulting firms, and certainly McKinsey, BCG and Bain, expect candidates to be very comfortable with quantitative data, statistics, and the ability to make decisions and client recommendations based on data.

Management consultants at firms like McKinsey, BCG, Bain, Deloitte spend a lot of time working with numbers, charts, calculations, financial models in excel and other math work, often mental math work. So any consulting case interview mental math test, and there are really multiple mental math tests scattered throughout the consulting case interview process is something you have to be well prepared for.

This does not mean that you need to have a math degree to have the right level of consulting case interview mental math skills. But you do need to know what is expected of you and you do need to practice mental math a lot.

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How is consulting case interview mental math different from academic math?

Because management consulting is all about solving difficult problems, usually under extreme pressure, the case interviewer is expecting a candidate to approach math problems in a specific way. In academic settings the most important element of solving math problems is accuracy. Accuracy is also very important for case interview math but management consultants usually work under extreme time pressure. And so answers are often required to be close enough to guide towards the “right” recommendation, versus being 100% accurate.

For example, imagine you are asked to calculate the market size for baby diapers for sensitive skin in Singapore. If this was a problem within an academic setting you would be expected to give an accurate answer correct to the decimal point. In consulting case interview settings you will have to make many educated estimations to arrive at, hopefully, a close enough answer. And then you will be expected to do what we call a sanity check to ensure that your answer actually makes sense.

Let’s take a look at an example from a real McKinsey engagement, mentioned by one of our trainers, Kevin P. Coyne. In case you don’t know, Kevin is a former McKinsey worldwide strategy practice co-leader and he leads The Consulting Offer II , which you can access if you join our Premium membership or FIRMSconsulting Insider level.

In this example, Kevin mentioned serving a large bank and during initial interviews with employees of the bank, Kevin’s team noticed that 100% of the profit for that bank was coming from one business unit. That does not mean that all other business units were operating at a loss. But combined all other business units of that bank had zero profit. So the bank was dependent on this one unit to generate all their profits.

Kevin’s team further uncovered that a lot of clients that the unit served were really old. To give a more accurate answer on how bad the situation was Kevin’s team selected only 1 letter in the alphabet and studied the age of all the clients whose name started from that letter, let’s say it was letter B.

This exercise uncovered that within the next 5 years that bank would lose something like half of its clients. And it does not mean the bank will have those clients for 5 years and then they will disappear. No, the clients will start dying now and within 5 years the client base will be about half smaller than now.

And younger people were not interested in that type of service. Doing the same analyses for all clients within the unit would be cost-prohibitive and will take significantly longer, and the limited analysis conducted was more than enough to understand that the bank was in serious trouble and drastic action was required.

This is a great example of how math in real consulting settings is often focused on getting close enough/good enough answers fast and cheap. And as a great management consultant, you will need to have strong enough business judgment to know what is good enough and what is required and to never waste the client’s money and other resources on unnecessary analyses.

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Strengthening your mental math and written math skills is one of the most important elements of preparing for case interviews.

As part of a case interview process, your mental and written math skills will be tested in multiple ways. If you are strong in academic math you are in a good place. However, the style of math used during case interviews is quite different vs. math problems in the academic context, as we discussed above, and takes time to get comfortable with.

Some examples of what case interview math test can include:

Case interview math test can include word problems . Word problems used as part of case interviews are similar to the type of word-based problems you practiced for as part of your GMAT preparation or preparation for other standardized tests. And such a case interview math test may or may not include a business-based context.

Case interview math can be tested during a full case . In fact, full cases almost always test math along with other skills. For example, coming back to the example above, you may be asked to estimate the market size for diapers for sensitive skin babies in Singapore as part of a full case of your client considering entering the Singapore market. As part of the case, you may also be asked to work with many graphs and charts, which we refer to as data cases . We cover data cases extensively in The Consulting Offer , our flagship program where we help real candidates prepare for interviews with McKinsey, BCG, Bain, Deloitte, etc. You can track candidates’ preparation at various stages, all the way from networking, editing resume and preparing for standardized tests to getting an offer and deciding if they should accept an offer. You can track Ritika joining McKinsey Chicago, Jen joining Bain Boston, Assel joining McKinsey Europe after 5 years out of the workforce and with no prior work at MBB (never before been done), Sanjeev joining BCG, Alice joining McKinsey NYC and much more.

Mental math is also tested as part of case interview math tests. In fact, it is tested a lot as part of the case interview process. You will be required to do math in your head and very fast. This is often one of the most difficult components of a case interview for candidates. The Consulting Offer will help you prepare.

Standard math such as multiplication, division, fractions, percentages, and other concepts are routinely tested. Case interview math tests are usually baked into a case and math is just a component to finding a solution within a specific business context.

In all examples of case interview math above, speed and relative accuracy matter. And the use of calculators is not allowed. So it is crucial to practice and be ready to handle case interview math tests fast, accurately, and without a calculator.

Consulting case interview math formulas

Revenue  = Volume x Price

Cost  = Fixed cost + Variable cost

Profit  = Revenue – Cost

Profit margin / Profitability  = Profit / Revenue

Return on Investment (ROI)  = Annual profit / Initial investment

Breakeven / Payback Period  = Initial investment / Annual profit

EBITDA  = Earnings Before Interest Tax Depreciation and Amortization. EBIDTA is essentially profits with interest, taxes, depreciation and amortization added back to it. It’s useful when comparing companies across various industries.

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What will help you become faster in doing mental math during consulting case interviews is rounding numbers. For example, ~82 million population of Germany becomes 80 million, ~46.7 million population becomes 45 million. The key to rounding numbers is to round them carefully, in a way that does not distort too much the final answer. A good guideline to follow is not to round by more than 10%. It is also helpful to round both up and down as you are working through the case, so the effects, to some degree, cancel each other out. At the end also make sure you check if your answer actually makes sense.

Fast mental math: dealing with large numbers

The key to dealing with large numbers, like 200 million, for example, is to remove zeros and then add them back later. Use labels (m,k,b) to help you keep track. So if you have 200 million, it becomes 200 m to help you remember that it is millions. 200,000 will be 200k. 10 billion will become 10 b. The key to achieving fast mental case interview math is to simplify. For example, 5 x 30 million becomes 5 x 3 = 15 with 7 zeros.

Fast mental math: break down numbers into smaller parts

When dealing with case interview math, another trick that will help you work through the problem faster is breaking down numbers into smaller parts. For example, 14 x 6 = (10 x 6) + (4 x 6) = 84.

Fast mental math: subtracting from numbers with 1 followed by zeros

This is another trick for faster case interview math. Again, simplify. 1000-536 becomes 999-536+1 = 464.

Fast mental math: group numbers into multiple of 10 (addition)

Another trick for fast case interview math is to group numbers into multiple of 10 (for addition). 3+7 + 4 + 6 +13 +7 +21 becomes 10 + 10 + 20 + 21 = 61.

Other tips to achieve fast case interview math (mostly mental math) during a consulting case interview

Here are a few tips to keep in mind to help you perform better during a consulting case interview when it comes to case interview math (and mostly mental math).

• At the beginning of the case ask your interviewer if it is ok to round numbers. Most of the time they will say yes and it will make math calculations much easier and faster.
• Do not rush. If you make a mistake it will take you even longer to fix it. This is if you even catch your mistake. You may also catch your mistake by the time when the interviewer will not give you an opportunity to fix it. And case interview math mistakes can be very embarrassing and lead to a completely wrong recommendation. Of course, there is a lot of time pressure during consulting case interviews so do not take any longer than you need. You need to find a good balance. This comes with a lot of practice. We provide a lot of opportunities for you to practice case interview math. Some full cases are provided below and you will find more on our YouTube channel. And, of course, you can unlock access to all candidates and seasons of The Consulting Offer when you become Premium member (more details below).
• Do not be afraid to write things down when you feel you need it.
• Keep your writing organized. Let say you are estimating how many cars will be purchased in Germany in 2020. As you are putting down numbers for each element of your equation keep it neat and organized so you don’t get confused and it will also help you avoid silly mistakes.
• Do not state your answer to an interviewer as a question. Be confident in your answer.
• As part of your preparation refresh key math topics like ratios, fractions, percentages, averages, and probability.  Khan Academy  is a great place to refresh your math skills. And you will have more than enough opportunities to practice fast mental case interview math as you go through various candidates and seasons within The Consulting Offer (part of  Premium membership).

Practice consulting case interview math / mental math with full cases

As you work through the cases remember to focus on all elements of good case performance, not just math. People usually underestimate how important other elements of case interview preparation are, including FIT. And only realize after being rejected that the elements they ignored during preparation were the reason for the rejection. Learn from the mistakes of others. Take all elements of case interview preparation seriously.

BUSINESS CASE EXAMPLE #1: MCKINSEY, BAIN, BCG ACQUISITION CASE

This case is a McKinsey style case, of medium level difficulty. It should take you 15-20 minutes to solve this case.

The question is given upfront, at 2:02. The part in black is the part the interviewer would share with you and a part in grey is the part interviewer may share as the case progresses. The interviewer wants to see if the interviewee understands the case and asks the right questions.

The case question is quite explicit but even so we will show you how you can adjust the case and make the case more explicit.

Everything rests on the key question. If anything is not part of the key question, ignore it. Even though lots of information is provided, take time to understand and set up the case.

Always show why information is needed, and show progress so the interviewer is they are willing to provide more information. It is a barter. And always use the case information provided and the appropriate language to push the case forward.

BUSINESS CASE EXAMPLE #2: COMPREHENSIVE MARKET ENTRY CASE

We did this recording a few months after we completed the training with Rafik (TCO I). This is one of the most complex market entry cases we had to put together. It has elements of operations, elements of pricing, elements of costing and, obviously, elements of market entry. And it is probably the most difficult market entry case we can do because most market entry cases that most interviewers focus on have a strong market attractiveness element, market profitability element. But very few people actually look at the operational issues of entering the market. And it does not matter who you are interviewing with: Bain, BCG or McKinsey. The bulk of the focus usually goes towards analyzing the market worthiness but not a lot on the operational issues. So we decided, in this case, to flip it around and give this case a strong operational theme.

BUSINESS CASE EXAMPLE #3: PEPSI’S LOS ANGELES BOTTLING PLANT

Operations cases can be tackled in two ways: strategy and operations and within operations from productivity and the supply chain side. This case uses the supply chain side.

This case is candidate-led. As we mentioned above, candidate-led cases are much harder than interviewer-led cases. That is why we at FIRMSconsutling place so much more emphasis on teaching you how to lead cases vs. relying on the interviewer to lead. This will be considered an operations case. Pay attention to a very insightful brainstorming at 14:50 which includes at least one idea you most likely would not come up with if you were solving this case before watching this video.

What else can I do to improve my case interview math?

Mental math is a muscle. But most of us do not exercise it enough once we leave school. So your case interview preparation needs to include math training.

First refresh your knowledge and ability to calculate basic multiplications, divisions, additions, and subtractions, without a calculator. The Consulting Offer program (a part of Premium membership) includes ongoing opportunities to practice this. We also have many cases available for free on the FIRMSconsulting YouTube channel to get you started.

And there are other tools you can use for case interview math prep.

Khan Academy has some resources that you may find helpful. Here are some helpful links:

• Percentages
• Scientific notation
• Additions and subtractions
• Multiplications and divisions

You will need to regularly practice to get comfortable with mental and written math. Case interview math tests require you to do all math calculations fast and accurately. We recommend working through a few sessions of The Consulting Offer a day to ensure multiple opportunities to practice math and other skills you need to give yourself the highest chance to get an offer from firms like McKinsey, BCG, Bain, Deloitte, etc.

Go through a few sessions every day and you will start feeling more comfortable over time not just with case interview math but with your resume, networking, estimations, brainstorming, answering FIT questions in a way that answers what the interviewer is REALLY asking you.

You will also develop or strengthen the ability to lead and handle difficult cases, and the ability to develop your own framework uniquely tailored to solve a particular case, and much more. View it as an investment in skills that will serve you for the rest of your life vs. just searching for tips and tricks to get an offer from McKinsey, BCG, Deloitte, et al.

Additionally, some candidates found the following tools helpful as supplemental materials along with The Consulting Offer. We have not tested those tools but are sharing them in case you would like to explore them.

Mental math games (Android). This one is similar to the mental math cards challenge app on iOS (below).

Mental math cards challenge app (iOS). This mobile app is a good choice if you are an iOS user.

Magoosh’s Mental Math Practice – Arithmetic Flashcards (iOS + Android). And here is another free math app that uses flashcards. And it allows you to track your progress as you study.

MConsultingPrep: math drills

Preplounge: mental-math (registration required)

Case Interview: calculations   (registration required)

How FIRMSconsulting can help me?

You will need to get comfortable doing calculations fast and accurately. And this comes with a lot of practice. If you will be using The Consulting Offer to prepare for your consulting case interviews you will have what seems to be never-ending opportunities to practice mental and written math as part of the full cases and as part of particular questions such as estimations, etc.

Management consulting jobs are very competitive, and working with FIRMSconsulting can mean the difference between getting an offer, or multiple offers, from your target firms and barely getting an offer from the company you hoped you never would need to settle for. And the latter example is something I, unfortunately, observed many of my MBA classmates settled for.

When it comes to case interview math The Consulting Offer program, all 5 seasons of it and counting, with various candidates, includes everything you need to master not just case interview math, but all key aspects of consulting case interviews.

Don’t miss out by investing your time with general math drills when you can practice real-world case interview math examples while being taught by former consulting partners.

If you want the most comprehensive guidance for consulting case interviews math, and other aspects of case interview preparation, so you go to your interviews confidently, become a Premium or FC Insider level member now. And if you still have questions contact FIRMSconsulting ( [email protected] ) to find out why candidates even from top schools like Harvard, Stanford, and MIT choose us when they need consulting case interview preparation help, and stay with us for years and years once they get coveted jobs at McKinsey, BCG, Bain, Deloitte, etc.

WHAT IS NEXT? If you have any questions about our membership training programs (StrategyTV.com/Apps & StrategyTraining.com/Apps) do not hesitate to reach out to us at support @ firmsconsulting.com. You can also get access to selected episodes from our membership programs when you sign-up for our newsletter above or here . Continue developing your  strategy skills .

Cheers, Kris

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Getting smart collective, impact update, why should we use case studies in the math classroom.

By Julie Martin Have you ever wondered why most students are frustrated by math problems? You can do your best to explain how they should implement the formulas to get to the needed solution, but you realize they are not really interested in listening to your attempts to explain how cool math is. You may be the “math problem master,” but there is a particular real-life problem hidden here: your students are not interested in becoming better at math because they think they don’t need it. Students are easily bored of theory and formulas. They realize that different areas of study depend upon the math you teach. They understand that statistics, psychology, physics, computers and astronomy wouldn’t be possible without math. The problem is they don’t see how these specific formulas you’re introducing in today’s lecture make our world a better place to live in. Pedagogy is a true art. You have to  experiment with different approaches and strategies  until you discover the perfect method of teaching math to the particular group of students you’re dealing with. So here one method that usually works in the math classroom, regardless of the students’ ages: case studies.

Reasons to Use Case Studies in the Classroom

When you are looking for contextual examples that show how math principles find their implementation in real life, try presenting case studies in the classroom. If you’re still not convinced on this method, consider these points:

• Case studies promote critical thinking, problem-solving, collaboration and communication skills within the classroom.
• Reviewing a case study is a student-centered activity. It’s easier to relate to real-life situations presented as examples of why this information is pertinent to their future success. Regardless of the professions they intend to pursue, they will certainly need some math skills for professional progress, and case studies can help make that fact clear.
• Case studies are  great for stimulating discussions . When your students understand that math is an important aspect of the development and marketing processes of a famous company, they will have questions. For example, when they drink a soda during a break between classes, they might wonder which formulas were used to produce the perfectly shaped container that fits in their hands and contains the exact amount of liquid they desire.
• A case study will make your students think. First, expose the general information about the study, and intrigue them to think about the challenges presented. If they need assistance in asking the right questions, you can simplify the problem for them. Then they will be inspired to communicate and analyze the issue and will come to their own conclusions.

How to Use Case Studies in the Classroom

So how can you find a specific case study that will intrigue students’ critical thinking and problem-solving skills? Here are four steps:

• First, analyze the interests of your students. Do they like smartphone apps? Of course they do! Maybe you could use a case study that explains how important math formulas are in the development of these games.
• Once you realize what type of case study you need, it’s time to do a little research. If you can’t find anything interesting online in general, you can request custom case study help . Yes, there are professional case study assignment help services that pair you with a mathematician. He will follow your instructions and develop a custom case study that will be unique, impressive and suitable for your classroom. These services also offer free samples of case studies, so you should definitely explore their offer to see if you can find a study relevant to your needs.
• Once you find the perfect case study, it’s time for action. Group your students in teams, doing your best to make them as equal in math skills as possible. Expose the problem of the case study and ask them what they think, giving them time to gather their thoughts. You could also assign a group project for homework, so the teams have time to brainstorm and can share their solutions during the next class.
• Analyze the team answers and then present the results of the actual case study. Spark a discussion that enables your students to ask questions and express their opinions.

Case Studies Bring Math to Reality

Once you’ve introduces case studies to your students and used them in several lessons, you can take things even further. Assign a case study at the end of each unit so students can analyze the formulas and think how they can implement them in a real-life situation. Encourage them to seek out their own relevant case study examples to bring to class and share as well. Case studies are the perfect addition to a cool teacher’s math curriculum. They take mathematical concepts beyond formulas, pages of text and geometrical images. Try introducing case studies into your classroom and watch student engagement increase! For more, see:

• April is Math Awareness Month
• “With Math I Can” – Changing Our Mindsets About Math
• Talking Math: 100 Questions That Help Promote Mathematical Discourse

Julie Martin is a tech researcher from California University.

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Measuring noncommutative algebras - a new Bernstein's inequality

In mathematics, as in footwear, order matters. Putting on your socks before your shoes produces a different result to putting on your shoes before your socks. This means that the two operations 'putting on socks' and 'putting on shoes' are noncommutative . Noncommutative structures are widespread in mathematics, appearing in subjects ranging from group theory to analysis and differential equations.

Mathematics everywhere

We all know that mathematical activity goes on nowadays in a great variety of settings – not just in academia, but across the whole range of industry, education, and beyond.   This diversity in mathematics is by no means new, and yet the study of the history of mathematics has often failed to capture it.   

Holographic duals of Argyres–Douglas theories - a case study by Christopher Couzens

Holography is one of a set of powerful tools which theoretical physicists use to understand the fundamental aspects of nature. The holographic principle states that the entire information content of a theory of quantum gravity in some volume is equivalent (or dual) to a theory living at the boundary of the volume without gravity. The boundary degrees of freedom encode all the bulk degrees of freedom and their dynamics and vice versa.

Inheritance of nontrivial dynamics in chemical reaction networks - a case study by Murad Banaji

Systems of chemical reactions figure prominently in models throughout the natural sciences. Biology in particular gives rise to hugely complicated reaction networks which, somehow, perform intricate and delicate functions. How do we make sense of these complex networks, and try to understand the relationships between their structure and function? 

Understanding lung damage in COVID-19 patients

Mathematical modelling played a key role in describing the spread of the COVID-19 pandemic; now a different kind of maths is helping us understand how immune cells interact in the lungs of patients with severe COVID-19.

In a damaged lung with a massive immune cell infiltrate, as seen with severe COVID-19 infection, it can be difficult to figure out which cells are involved in causing lung injury. 

From natural philosophy to applied mathematics - 400 years of Sedleian Professors

For the first several centuries of its existence, an education at the University of Oxford entailed a basic grounding in a range of different subjects, rather than the specialised study of a single discipline.  The goal was to turn out well-rounded individuals rather than narrow experts.  Nevertheless, the university often tried, wherever possible, to provide advanced instruction in specific areas for those students who were interested. 

Beyond the Riemann hypothesis — primes and smooth numbers

In this case study we'll highlight new world records, going 23% beyond the Riemann Hypothesis. To explain, we start with the (last digit of) prime numbers : \[ {\color{blue}{\bf 2}}, {\color{green}{\bf3}}, {\color{orange}{\bf5}}, {\color{red}{\bf7}}, 1{\bf1}, 1{\color{green}{\bf3}}, 1{\color{red}{\bf7}}, 1{\color{purple}{\bf9}}, 2{\color{green}{\bf3}},\ldots \] After some thought, we may realize that no such last digit may be even (after ${\color{blue}{\bf 2}}$ itself), else the whole number is even; nor may ${\color{orange}{\bf5}}$ appear again.

Multi-scale modelling of viral outbreak risks

During the COVID-19 pandemic, mathematical modelling played a major role in informing public health policy responses.  A key question for public health policy makers is whether the introduction of a virus into a population is likely to lead to sustained transmission. This is critical for understanding the epidemic and/or pandemic potential of a novel virus – notably, for example, following the first detected COVID-19 cases in Wuhan, China.

Mathematical modelling of cleaning

Many modern challenges facing industry and government relate to cleaning: removing pollution from water or industrial waste gases, or decontaminating the environment after chemical spills. Mathematicians in Oxford are collaborating with various industrial and government partners to model specific cleaning challenges, deepening understanding of these processes and working towards optimal cleaning solutions.

Plants with teeth: Oxford Researchers reveal the mechanics of insect-eating pitcher plants

Researchers at the University of Oxford’s Botanic Garden and in Oxford Mathematics have shown that the shape, size and geometry of carnivorous pitcher plants determines the type of prey they trap. The results have been published today in the Proceedings of the National Academy of Sciences (PNAS) .

Exploiting matrix structure for faster algorithms

In many modern applications, a key bottleneck is the solution of a matrix problem of the form Ax=b where A is a large matrix. In numerical weather prediction, such systems arise as a sub-problem within data assimilation algorithms. In this setting, finding the most likely initial condition with which to initialise a forecast is equivalent to finding the (approximate) solution x.

Developing an understanding of the mechanisms behind the changes in clotting induced by oral contraceptives

Generalized Symmetries from String Theory - a research case study by Dewi Gould

Symmetry underpins all physics research. We look for fundamental and beautiful patterns to describe and explain the laws of nature. One way of explaining symmetry is to ask: "what is the full set of operations I can do to my real-world experiment or abstract theory written on paper that doesn't change any physical measurements or predictions?'' There are simple symmetries we are perhaps already familiar with. For example, lab-based physics experiments usually don't care if you wait an hour to do the experiment or if you rotate your apparatus by 90 degrees.

Leaping beetles inspire miniature jumping robots

Inspired by jumping insects, Oxford Mathematicians have helped develop a miniature robot capable of leaping more than 40 times its body length - equivalent to a human jumping up to the 20th floor of a building. The innovation could be a major step forward in developing miniature robots for a wide range of applications.

Hilbert's 16th Problem and Mathematical Chemistry

In 1900, David Hilbert posed his list of 23 mathematical problems. While some of them have been resolved in subsequent years, a few of his challenging questions remain unanswered to this day. One of them is Hilbert's 16th problem, which asks questions about the number of limit cycles that a system of ordinary differential equations can have. Such equations appear in modelling real-world systems in biological, chemical or physical applications, where their solutions are functions of time.

The beauty of interdisciplinary research - a research case study by José Moran

What do quantum mechanics, genetics, economic choice theory and ant behaviour have in common? Naturally, the first answer is that they have nothing at all to do with each other! Nonetheless, finding bridges between different fields is the bread and butter of the field of complex systems , an offshoot of statistical physics and other fields.

Computable foundations for geometric topology

Algebraic topology is the study of the continuous shape of spaces by the discrete means of algebra. The beginning of modern algebraic topology can be traced back to an insight of Pontryagin in the 1930s which relates the global smooth geometry of manifolds to algebraic invariants associated to the local symmetries of those manifolds - this relation converts something smooth and geometric (called a manifold, potentially endowed with further structure) to something algebraic that can be written down with symbols and formulas, i.e.

Mathematical ecology in a more-than-human world - a case study by Siddharth Unnithan Kumar

Ecology, by my definition today, is the study of the phenomena, relationships, patterns and processes which constitute this living and breathing world which we all call home. Its reach is boundless, and the angles through which to look at ecological processes are diverse and span all academic disciplines. It is emerging as the central focus of the present day.

Case Interview Math - Know THIS Before Your Consulting Interview

• Last Updated January, 2024

Rebecca Smith-Allen

Former McKinsey Engagement Manager

Why is Case Math Important?

Steps to Minimize Mistakes

4 Types of Case Math Problems

Extracting Data From Charts

Case Math Examples

Critical Numbers to Know

Market Sizing Math

Links to Resources

Are you nervous about solving math problems in the middle of your consulting interview? 

Don’t worry. You’re not alone. 

Luckily, typical case interview math problems aren’t hard. They’re basic algebra – addition, subtraction, multiplication, division, and fractions/percentages. They can also involve extracting data from tables and interpreting it. All the math you need you covered by age 13–and if you need to brush up, we’ve got you covered.

You’ll find consulting math easier if you know what type of problems to expect and follow our 4-step approach to minimizing case interview math mistakes . 

If this is your first visit to MyConsultingOffer.org – head to  Case Interview Prep   for an intro to the management consulting interview process. 

If you’re familiar with what a consulting case interview is, and want a preview of the math you’ll need to succeed in consulting interviews, you’re in the right place. 

Let’s get started!

Why Do Consulting Case Interviewers Care about Math Skills When Everyone Has a Calculator in their Phone?

Why do case math because a consulting team’s time is expensive..

Nail the case & fit interview with strategies from former MBB Interviewers that have helped 89.6% of our clients pass the case interview.

Consulting Math: 4 Types of Problems

1. market sizing.

Examples: What is the size of the market for laundry detergent in Latin America? How many pizza pies are consumed each day in Pittsburgh?

2. Financial calculations

Examples: What are XYZ company’s profits? What is XYZ company’s profit margin? What is XYZ company’s market share? What is XYZ company’s growth rate?

3. Investment analysis

Example: How many years would it take to break-even on an investment? What is the return on investment?

4. Operations problems

Example: What is the capacity (of a factory, machine or worker)? What is the utilization rate?

Watch this video to learn how to master case interview math. For reference, I’ll also provide key formulas and examples below.

Case Interview Math Examples

Financial calculations.

Here are the key formulas you need to know:

• Profits = Revenue – Costs
• Profit margin = Profits/Revenue
• Market share = Revenue for XYZ company/Revenue for all companies in the market
• Growth rate = (New –  Old) /Old 

So Revenue growth rate = ( this year’s revenue – last year’s revenue) / last year’s revenue

Here are some examples of financial math calculations:

• A firm with $100 million in revenue and $80 million in costs has profits of $20 million.
• A firm with $20 million in profits and $100 million in revenue has a 20% profit margin.
• A firm with $100 million in revenue in an industry with $400 million revenues has a 25% market share.
• A firm that made $100 million this year and $90 million last year had an 11% growth rate.

Investment Analysis

• Break-even:  Investment cost / annual profit = years to break even
• Return on investment (ROI) = (Revenue – investment costs) / Investment cost

Here are some examples of investment calculations:

• A company spends $10 million in costs to enter a business with annual profits of $1 million has a 10-year payback.
• A company that will earn  $12 million on a business that it paid $10 million to enter had a 20% ROI.

Operations Problems

• Capacity in units = total capacity / capacity required to make one unit 
• Utilization rate = actual output / maximum possible output

Here are some examples of operations math calculations:

• If a machine that can produce a widget every 5 minutes, it has a capacity of producing 96 widgets in an 8-hour shift. 8 hours * (60 minutes/hour) / 5 minutes = 96.
• If the widget machine is left unmanned while the operator is at lunch (1 hour) and during breaks (2, 15 minutes each), then the machine will only make 78 widgets. 78 widgets / its capacity of 96 means it ran at a utilization rate of 81.25%.

My friend attended college with me. She was a non-American student and I knew she wanted to marry someone who could give her American citizenship, so I started with the US population – 320 million .

Then I cut the population in half to focus on men since she was interested only in men. 320 million / 2 = 160 million.

To rule out men who were too old or too young for her, I cut the population of men into age groups. The average life expectancy in the U.S. is 80 years. I wanted to focus on only men 20-30, so 1/8 of the U.S. male population. 160 million / 8 = 20 million.

I also wanted to focus on men that were taller than her, but actually, my friend is pretty short. Almost all men in the right age range are taller than her so there was no need to factor that into my calculation. Men 20-30 who are taller than Davis’s very short friend = 100% or 20 million.

Next, I looked at attractiveness: both the portion of the target male population that my friend would find attractive and the percentage of those guys who would find her attractive. 

This friend’s pretty picky. She was only interested in guys who were 9’s or 10’s on a scale of 1 to 10, so about 20% of the population. 20 million men aged 20-30 * 20% who are really hot = 4 million.

I decided to be conservative and assume only 10% of the population my friend thought was attractive would also find her attractive. 10 % of the 4 million hot guys age 20-30 would find her attractive = 400,000.

My friend wanted a guy that had a good income, so I decided to focus on people attending or who’d graduated from college. In fact, she was really picky, so I decided to focus on top colleges like the Ivy League schools. About 1% of the U.S. male population attends highly competitive schools like the Ivy League. 400,000 * 1% = 4,000.

Finally, I assumed many of the guys who were in the right age range and who my friend found attractive and who found her attractive and who went to a top college might already be taken. I cut the number I’d come up with in half to reflect the highly eligible guys who were already off the market. 4,000 hot, really smart guys age 20-30 / 2 = 2,000.

My “so what” based on this analysis was that my friend might want to be a little less picky. All her requirements brought the number of eligible bachelors down from 160 million to only 2,000. But she thought that a group of 2,000 guys was plenty for her to find Mr. Right in, so she decided to keep her standards high.

4 Steps to Minimizing Mistakes in Case Interview Math

It’s hard to do math under the pressure of an interview. These steps are the key to keeping your math accurate.

1 – Be clear on what the calculation will tell you.

Don’t just start doing calculations. Know what business problem you’re trying to solve and how your math will give you insight so you can make a decision. Share this with your interviewer.

2 – Structure your approach.

Before you do any calculation, walk through the steps you’ll take to answer the question with your interviewer. There are frequently several steps in consulting math problems.

3 – Do the calculation step-by-step.

Once you have your approach to the case math problem, do one calculation at a time to ensure accuracy.

4 – Explain the “so what?”

Don’t wrap up your answer with just a number, explain what the number means in the context of the business problem you were trying to solve. What would you recommend to the client?

Here’s an example of Consulting Math:

A manufacturer of high-quality wood outdoor furniture is considering extending its product line to include Adirondack chairs. It only wants to enter the market if it can make at least $5 million a year.

Data provided: 

The size of the North America market for Adirondack chairs is estimated to be $1 billion. The top 4 players hold 20% market share between them. Profit margins average 20%. 

Now apply the 4-step approach for minimizing mistakes.

Be clear on what the calculation will tell you. “I’d like to calculate likely market share and profit our client could expect if it entered the market for Adirondack chairs in order to see if the opportunity is large enough to meet their criteria of $5 million profits.”

Structure your approach. “ To do this calculation, I’d first look at the likely market share our client could achieve based on the share of the top 4 players in the market. Then I’d calculate the revenues that they could expect by multiplying the market share by the size of the market. I’d then use the industry’s average profit margin to calculate the level of profit they could expect.”

Do the calculation step-by-step. “ The top 4 players in the market for Adirondack chairs have a combined market share of 20%. 20% market share /4 players = an average market share of 5%. Based on the client’s success in the wood outdoor furniture market, we expect they could achieve this level of market share as well.”A 5% market share in a market with $1 billion in annual revenue would give the client $1 billion * .05 or $50 million in revenue.$50 million in revenue for a product with a 20% profit margin would give the client $10 million in expected revenue.”

Explain the “So what?” “ Based on these calculations, the client should consider entering the market for Adirondack chairs because the expected annual revenue of $10 million exceeds their minimum criteria of $5 million.”

More case interview math: extracting data from charts.

In some cases, particularly when a case interview math problem is one step in a multi-step case question, an interviewer might hand you a chart with key information on it. These charts often have more pieces of data on them than you need. You need to find the right data. Or, you may need to manipulate the data to find what you need.

For example, a table might provide a company’s costs and revenues when the information you need is profit margin. Or, it might provide the revenues of the top 4 companies in the market and a 5th number that totals the revenue of all other players in the market, but you might need market share for player #3.

Don’t let all the number on the chart throw you. You’ll still be using the same basic formulas to analyze these numbers. 

Instead, take a minute to understand the information the chart provides and explain it to your interviewer. Then proceed with the case interview math problem as described above: be clear about what you’re looking for in the calculation, walk the interviewer through your approach, do the actual calculation, and then provide the “so what.” Using this approach will make interpreting data tables in an interview straightforward.

Critical Numbers to Know for Case Math Problems

Links to bain, bcg & mckinsey resources on case interview math.

Want to hear more about the importance of case interview math straight from the top management consulting firms? Bain and BCG list quantitative analysis as part of what they test for. McKinsey provides examples of the type case math in its case interviews in this practice test .

On this page, we’ve covered the typical types of case interview math problems you’ll see, explained how to extract data from charts, and provided tips on minimizing your consulting math mistakes. These skills will help boost your confidence in management consulting interviews and ensure you ace your case.

People who are working on their consulting math skills typically find the following other pages helpful:

• MECE – Mutually Exclusive, Collectively Exhausting ,
• McKinsey Case Interview,  and
• Consulting Written Case Interviews .

We hope this page has increased your confidence in tackling consulting math problems! Comment below on the types of consulting math problems you find the toughest. We’ll provide tips from our coaches.

Help with Case Study Interview Preparation

Thanks for turning to My Consulting Offer for advice on case study interview prep. My Consulting Offer has helped almost 85% of the people we’ve worked with to get a job in management consulting.  For example, here is how Brenda was able to get a BCG offer when she only had 1 week to prepare…

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If you want to learn more about how to ace your case interviews, apply to work with us.  We’ll show you how you get an offer without spending hundreds of hours preparing.

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How to Write a Case Study on Mathematics

Last Updated: September 15, 2021

wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. To create this article, volunteer authors worked to edit and improve it over time. This article has been viewed 27,107 times.

The aim of a well-written case study in Mathematics is to provide guides and teachers with important information on any number of situations that are problematic. These could be classroom specific or something huger. As a student, your case study should articulate your research goals very carefully and then go on to give good methods and a proper conclusion. Writing a math case study could be difficult but it will help you earn good grades and ensure that your guides and teachers know that you're taking your courses seriously.

Here's how you can write a good case study and ensure that you get the marks you deserve:

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Book review: Alan Schoenfeld, Heather Fink, Sandra Zuñiga-Ruiz, Siqi Huang, Xinyu Wei, and Brantina Chirinda (2023) Helping students become powerful thinkers: case studies of teaching for robust understanding

Routledge. 10.4324/9781003376903 272 pp. Hardback: ISBN 9781032450629. $204.75. Paperback: ISBN 9781032441689. $77.25. e-Book: ISBN 9781003375197 $69.74

• Published: 26 March 2024

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Congenial conversations are generally agreeable discussions in which participants focus on politeness and privacy and avoid fault lines and conflict. Collegial conversations are characterized by deep collaboration where participants display an inquiry habit of mind, use relevant data, and develop relationships based on trust and mutual respect.

Blömeke, S., Jentsch, A., Ross, N., Kaiser, G., & König, J. (2022). Opening up the black box: Teacher competence, instructional quality, and students’ learning progress. Learning and Instruction, 79 , 101600.

Article   Google Scholar  

Charalambous, C. Y., Hill, H. C., Chin, M. J., & McGinn, D. (2020). Mathematical content knowledge and knowledge for teaching: Exploring their distinguishability and contribution to student learning. Journal of Mathematics Teacher Education, 23 (6), 579–613.

Guberman, D., & Leikin, R. (2021). Challenging mathematics teachers to implement open-ended problem solving: The interplay between teachers’ practices and their views. International Journal of Mathematical Education in Science and Technology, 52 (8), 1165–1184.

Google Scholar  

König, J., Santagata, R., Scheiner, T., Adleff, A. K., Yang, X., & Kaiser, G. (2022). Teacher noticing: A systematic literature review of conceptualizations, research designs, and findings on learning to notice. Educational Research Review, 36 , 100453.

Murray, E., Durkin, K., Chao, T., Star, J. R., & Vig, R. (2018). Exploring connections between content knowledge, pedagogical content knowledge, and the opportunities to learn mathematics: Findings from the TEDS-M Dataset. Mathematics Teacher Education and Development, 20 (1), 4–22.

Murray, E., Kim, Y, & DiNapoli, J. (in preparation). Knowledge construction in professional learning communities: Advancements in frame analysis .

Schoenfeld, A., Fink, H., Sayavedra, A., Weltman, A., & Zuñiga-Ruiz, S. (2023a). Mathematics teaching on target: A guide to teaching for robust understanding at all grade levels . Routledge. https://doi.org/10.4324/9781003376903 .

Schoenfeld, A., Fink, H., Zuñiga-Ruiz, S., Huang, S., Wei, X., & Chirinda, B. (2023b). Helping students become powerful mathematics thinkers: Case studies of teaching for robust understanding . Routledge. https://doi.org/10.4324/9781003375197 .

Star, J. R. (2016). Improve math teaching with incremental improvements. Phi Delta Kappan, 97 (7), 58–62.

Suppa, S., DiNapoli, J., Thanheiser, E., Tobias, J. M., & Yeo, S. (2020). The impact of high-stakes accountability on teachers’ use of standards-based mathematics. AERA Open, 6 (2), 1–16.

van Es, E. A., Hand, V., Agarwal, P., & Sandoval, C. (2022). Multidimensional noticing for equity: Theorizing mathematics teachers’ systems of noticing to disrupt inequities. Journal for Research in Mathematics Education , 53 (2), 114–132.

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UC stirs furious debate over what high school math skills are needed to succeed in college

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Briana Hampton, a San Gabriel High School junior, is determined to get into a four-year university to achieve her dream of becoming a social worker or psychiatrist. But she feared she would fail a third-year math course heavy on advanced algebra.

To meet her math requirement, she opted instead for an introductory data science course, approved a few years ago by the University of California as an alternative to advanced algebra. She loves the challenge of learning how to code, conduct surveys and analyze data on topics relevant to her life — sleep hours, stress levels, snacks consumed. She’s also boasting a B average in the class, compared to the Ds and Fs earned in her first-year math class.

“I’ve always struggled with math, but I heard that [data science] was like a really good class and something new and easier than algebra,” Briana said.

But the data science option is gone, at least for now. Last month UC notified California high schools that three of the most popular data science courses no longer count toward the advanced math requirement because the classes fail to teach the upper level algebra content all incoming students must know.

The decision has ratcheted up math anxiety and fomented confusion among high school students throughout California as they chart their high-stakes path for coveted UC admissions. California high schools offering data science classes — about 435 across the state — are also uncertain over how to revise curriculum and counsel their students.

UC first approved a data science course submitted in 2013 by Los Angeles Unified as a move to expand math options for students. At the time, the decision drew little notice. Over the years, efforts spread to increase data science in schools. After the subject was given a more prominent place in California’s new state framework for math instruction adopted last year, UC reviewed its years-old decision.

Faculty math experts concluded the course and two others were too weak on algebra and nixed them as an advanced math alternative. The decision also applies to the California State University system, which shares coursework admission requirements with UC.

The turnabout has unleashed furious debate over what high school math skills are needed to succeed in college — and how best to deliver them equitably to a diverse range of students. Currently, UC and CSU require for admission a three-year sequence of algebra, geometry and advanced algebra — or the equivalent integrated math courses. Both also recommend a fourth year of mathematics beyond those foundational skills.

Where data science fits in and what advanced math courses need to include to meet admission requirements is hotly contested. The UC Board of Regents will take up the issue at its meeting Wednesday and has the authority to overrule the faculty decision.

Opponents of UC’s reversal argue that data science courses give students essential skills to extract meaning from the modern world’s information deluge. They also offer an alternative path to college for those who may struggle with algebra and don’t plan to pursue calculus and STEM majors. About 44% of high school seniors fail to complete two semesters of advanced algebra, according to UC.

Denise Jaramillo, Alhambra Unified School District superintendent, said she was concerned by the UC decision because many of her students have taken the now-disallowed data science courses and have gone on to succeed in college. The district has used a course developed by UCLA since 2017.

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“Limiting student options and choices has never been an equitable practice in schools or the classroom, and doing so has the potential of placing historically marginalized groups at a great disadvantage after high school,” Jaramillo said in a statement.

But supporters of the UC decision counter that all students should be equipped with advanced algebra skills and not tracked into set pathways at such young ages — especially Black, Latino, female and others who are underrepresented in the high-demand, high-paying fields of science, technology, engineering and math.

“Students from underrepresented groups are the most vulnerable to make misinformed pathway choices in high school that could lead them away from preparedness for quantitative majors,” said Jelani Nelson, a UC Berkeley professor of electrical engineering and computer sciences.

Nelson, who is Black, has been particularly outspoken in the debate because he said “misleading marketing” from some data science courses claim they prepare students for STEM coursework and data science majors. But, he said, hundreds of faculty members, along with Elon Musk and industry leaders from OpenAI, Apple, Microsoft, NVIDIA, Google Research and other firms, agree that advanced algebra is essential for those fields.

Jennifer Chayes, dean of UC Berkeley’s new College of Computing, Data Science and Society , said “thousands” of students change their minds about majors once they come to campus. Many intend to study such fields as environmental science, health or criminal justice — then discover an affinity and aptitude for data science and other STEM disciplines and need the math skills for them.

“People are pitting this as data science versus algebra, but that’s really not right,” Chayes said. “It’s data science using algebra 2 that we’re really going to want to do.”

Nelson added that efforts are underway to develop data science courses that include advanced algebra — including one by Bootstrap, an educational nonprofit — and he and other STEM professionals welcome such moves.

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James Steintrager, chair of the UC Academic Senate, said the faculty organization is open to appropriate high school data science courses counting for admission in some way.

“There’s nothing wrong with data science, per se,” he said. “But if it’s not rigorous enough, you’re not preparing UC students, including diverse UC students, for the variety of careers, STEM careers in particular, that they could have. They’re not going to thrive. So you’re really doing those students a disservice.”

A letter to Board of Regents Chair Rich Leib signed by more than 230 school, district and county education leaders castigated UC for the flip-flop, saying they were shut out of the decision, yet have to manage the impact, including a “significantly” reduced ability to teach 21st century data and statistical skills. In another letter Monday to Leib, a University of Chicago data science advocate said the bitter California conflict has reverberated nationally.

The uncertainty over the future of data science courses may cause districts to pause or shut down programs — as San Diego Unified has done — cut resources for teacher professional development, and lead students to believe the field is different from math, said Zarek Drozda, executive director of Data Science 4 Everyone. His group concurred that the current data science courses don’t cover advanced algebra, but believed they can and should be allowed as a fourth-year math option.

Fewer than 400 of about 250,000 applicants to UC last year had taken data science or statistics instead of an algebra 2-type course. CSU data were not available. But as both public university systems collectively enroll the top one-third of California high school graduates, it is likely that the overwhelming majority of applicants take advanced algebra or its equivalent. About two-thirds of California high school students have taken algebra 2 by 11th grade, according to Michal Kurlaender, a UC Davis professor of education policy.

Hollywood High School, for instance, directs all students to complete the three-year math sequence ending in advanced algebra and take data science as a fourth year math option. Securing approval for data science and statistics courses to count for that fourth year of math recommended by UC and CSU is the lobbying focus by the Campaign for College Opportunity and other equity advocates.

But the ramifications of any UC action on the issue could be far-reaching even for students who don’t plan to apply to UC and CSU. Los Angeles Unified, for instance, requires all students to complete that college-prep coursework to graduate. In a districtwide memo issued last spring, L.A. Unified told schools that the data science course it had piloted could substitute for advanced algebra.

LAUSD initiated data science courses

In 2013, LAUSD became the first in California to seek and receive UC approval to offer a data science course.

L.A. schools Supt. Alberto Carvalho called the situation an “age-old dilemma in mathematics regarding the theory of mathematics versus the application.” The data science course includes “sufficient algebraic thinking and operations to satisfy the practicality of mathematics, but may, in fact, fall short of a true course in algebra that provides the A through Z of all algebraic concepts,” he said Tuesday.

School board member Nick Melvoin said the debate underscores the need for better early math instruction. Board members Jackie Goldberg and Tanya Ortiz Franklin support data science, but not at the expense of mathematical rigor.

The district has yet to provide information requested three weeks ago about how many schools currently offer the introductory data science course and how many students had taken it in lieu of advanced algebra.

Robert Gould, a UCLA teaching professor and undergraduate vice chair in the Department of Statistics and Data Science, led efforts to create coursework as part of a National Science Foundation-funded project to help students develop computational thinking and increase engagement in STEM studies. Initial four- to six-week programs that were developed using data in algebra and biology were too short for meaningful learning, Gould said.

But the move to establish new statewide learning standards, part of a national effort, offered an opportunity to create a year-long course because the new “common core” put more emphasis on statistics. UC approved the data science course in the statistics category. The university listed statistics as an allowable substitute for algebra 2, according to a 2013 UC document.

Data science scrutinized

Gould said the 2020 release of the initial draft of the state guidance for math instruction, known as the California Math Framework, intensified scrutiny because of an emphasis on data science courses. He joined a UC committee that recommended data science courses should be allowed to count for third- and fourth-year math — and the UC faculty board overseeing admission approved the recommendation in October 2020. A year later, a faculty board statement on math included sample course sequences that showed data science and statistics following algebra and geometry instead of algebra 2.

But CSU faculty objected.

In March 2023, the CSU Academic Senate approved a resolution saying Gould’s course represented “inadequate preparation for college and career readiness” that could put 11th-graders at risk in state math testing and “in turn threatens to increase the number of students entering the CSU who are identified as needing extra support to succeed.”

Rick Ford, who headed a CSU faculty committee on academic preparation at the time, said concerns about racial and ethnic disparities in math were well-meaning but misplaced.

“A major issue that I share with many of my equity-minded mathematics education colleagues is that too often we are seeing folks wanting to change standards or curriculum to address the ethnic and racial differentials in success in foundational high school mathematics courses,” he said in an email. “Instead, we should be focusing our efforts on building the proper support necessary to improve those success rates.”

The issue came to a head in July 2023, as state education officials prepared to vote on the new math framework that included data science as an alternative pathway. Amid ferocious lobbying on all sides, the UC faculty board voted to reverse the earlier approval of data science courses.

A working group, appointed to delve into the issues, affirmed the turnabout last month and UC then notified high school counselors.

Gould said he disagrees with conclusions that his course fails to prepare students for college. Still, he said he would do “what it takes” to revise the course as needed to make sure students can continue to gain exposure to data and statistical skills. Today, 7,681 California students are taking his course in 89 schools, along with thousands more in other states and countries.

Inside a data science class

Nearly 100 of those students attend San Gabriel High — and judging by a recent visit, love the data science course.

On this day, they are delving into data on whether females or males die more often in horror slasher films. Leah Ulloa Ruiz, the teacher, leads her class through data files of 485 characters in 50 films. She peppers them with questions and directs them to use coding to find answers.

Ruiz said many of her students say they hate math or don’t understand it. But data science lights them up, gives them confidence and teaches them the importance of data and statistical skills. The UC decision to not count data science as an advanced mathematics course for admission, she said, is wrong.

“I really think it’s a shame and a step backwards,” she said. “A student can be successful in life with a career, job and money without taking algebra 2 or integrated math 3.”

Two of her students, Noah Minchaca and Valentina He, both struggled with algebra-heavy integrated math courses. Valentina failed her second-year course and had to repeat it in the summer, while Noah managed to pass it but said he spent most of the time in class with his head on his desk, napping. He dropped out of his third-year course after two weeks.

“You can’t understand it,” Noah said of his integrated math course. “You keep seeing Ds on your tests and ask, ‘Am I dumb?’ Your self-esteem is blown. And it’s useless, just equations.”

“Yeah, polynomials — you can’t use it in the future,” Valentina said.

But the students said the data science course has taught them skills they expect to use in the future. Noah already has been accepted into an architecture program with Glendale Community College and San Diego State University, while Valentina is thinking about a trade school for training as a medical assistant.

Briana, meanwhile, won’t have to take the third-year integrated math course she skipped in favor of data science since the UC ruling won’t affect students until next year. That’s a relief since she’s planning to apply to both CSU and UC, although her first choice is a historically Black college. But she feels for students who follow her.

“That really sucks,” she said. “I know a lot of kids here are really bad at math and struggle with math. And they’re kind of being forced into a class that I’m not sure it’s gonna help them succeed.”

Staff writer Howard Blume contributed to this report.

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Teresa Watanabe covers education for the Los Angeles Times. Since joining the Times in 1989, she has covered immigration, ethnic communities, religion, Pacific Rim business and served as Tokyo correspondent and bureau chief. She also covered Asia, national affairs and state government for the San Jose Mercury News and wrote editorials for the Los Angeles Herald Examiner. A Seattle native, she graduated from USC in journalism and in East Asian languages and culture.

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• Our Mission

11 Real World Math Activities That Engage Students

Bridging the gap between abstract math concepts and real life experiences can make the subject accessible and relevant for kids.

During a unit on slope, José Vilson’s students just weren’t getting it, and their frustration was growing. The former middle school math teacher began brainstorming creative ways to illustrate the concept. “I kept thinking, ‘My students already understand how this works—they just don’t know that they know,’” Vilson writes in a recent article for Teacher2Teacher . “How can I activate knowledge they don’t believe they have?”

Then he thought about a hill a couple of blocks from school that his students “walk up every day to get to the subway.” He tacked up paper and began sketching stick figures on the hill. “One was at the top of the hill, one was halfway up, one was near the bottom skating on flat ground, and one was on a cliff,” writes Vilson, now the executive director of EduColor. “Which of these figures will go faster and why?” he asked his students. “That got my kids laughing because, of course, my stick figures weren’t going to hang in the MoMA.” Still, his sketch got them thinking and talking, and it provided a simple stepping stone that “gave that math relevance and belonging in their own lives,” Vilson concludes. 

“It’s not unusual for students to walk into our classrooms thinking that math belongs to people who are smarter, who are older, or who aren’t in their immediate circle,” Vilson writes. “But every time I teach math in a way that’s accessible and real for my students, I’m teaching them: ‘The math is yours.’”

To build on Vilson’s idea, we posted on our social channels asking teachers to share their favorite strategies for connecting math to students’ experiences and lives outside of school. We received hundreds of responses from math educators across grade levels. Here are 11 teacher-tested ideas that get students seeing and interacting with the math that surrounds them each day.

Hunt for clues

Coordinate systems can feel abstract to some students—but using coordinates to navigate a familiar space can solidify the concept in a relevant and fun way. “Before starting a unit on coordinates, I make gridded maps of the school—I make them look old using tea staining —and send my students off on a treasure hunt using the grid references to locate clues,” says Kolbe Burgoyne, an educator in Australia. “It’s meaningful, it’s fun, and definitely gets them engaged.”

Budget a trip

Students enjoy planning and budgeting for imaginary trips, teachers tell us, offering ample opportunities to practice adding, subtracting, and multiplying large numbers. In Miranda Henry’s resource classroom, for example, students are assigned a budget for a fictional spring break trip; then they find flights, hotels, food, and whatever else they’ll need, while staying within budget.

Math teacher Alicia Wimberley has her Texas students plan and budget a hypothetical trip to the Grand Canyon. “They love the real world context of it and start to see the relevance of the digits after the decimal—including how the .00 at the end of a price was relevant when adding.” One of Wimberley’s students, she writes, mixed up his decimals and nearly planned a $25,000 trip, but found his mistake and dialed back his expenses to under $3,000.

Tap into pizza love

Educators in our audience are big fans of “pizza math”—that is, any kind of math problem that involves pizza. “Pizza math was always a favorite when teaching area of a circle,” notes Shane Capps. If a store is selling a 10-inch pizza, for example, and we know that’s referring to its diameter, what is its total area? “Pizza math is a great tool for addition, subtraction, multiplication, word problems, fractions, and geometry,” another educator writes on our Instagram. There are endless pizza-based word problems online. Here’s a simple one to start, from Jump2Math : “The medium pizza had six slices. Mom and Dad each ate one slice. How much pizza is left?”

Break out the measuring cups

Lindsey Allan has her third-grade students break into pairs, find a recipe they like online, and use multiplication to calculate how much of each ingredient they’d need in order to feed the whole class. The class then votes on a favorite recipe, and they write up a shopping list—“which involves more math, because we have to decide, ‘OK, if we need this much butter for the doubled recipe, will we need three or four sticks, and then how much will be left over?’” Allan writes. “And then it turns out students were also doing division without even realizing!” 

Sometimes, a cooking mistake teaches students about proportions the hard way. “Nobody wants a sad chocolate chip cookie where you doubled the dough but not the chocolate chips,” adds teacher Holly Satter.

Heading outdoors is good for kids’ bodies , of course, but it can also be a rich mathematical experience. In second grade, kids can head out to measure perimeters, teacher Jenna McCann suggests—perhaps of the flower boxes in the school garden. If outdoors isn’t an option, there’s plenty of math to be found by walking around inside school—like measuring the perimeter of the tables in the cafeteria or the diameters of circles taped off on the gym floor.

In Maricris Lamigo’s eighth-grade geometry class, “I let [students] roam around the school and take photos of things where congruent triangles were applied,” says Lamigo. “I have students find distances in our indoor courtyard between two stickers that I place on the floor using the Pythagorean theorem,” adds Christopher Morrone, another eighth-grade teacher. In trigonometry, Cathee Cullison sends students outside “with tape measures and homemade clinometers to find heights, lengths, and areas using learned formulas for right and non-right triangles.” Students can make their own clinometers , devices that measure angles of elevation, using protractors and a few other household items.

Plan for adult life

To keep her math lessons both rigorous and engaging, Pamela Kranz runs a monthlong project-based learning activity where her middle school students choose an occupation and receive a salary based on government data. Then they have to budget their earnings to “pay rent, figure out transportation, buy groceries,” and navigate any number of unexpected financial dilemmas, such as medical expenses or car repairs. While learning about personal finance, they develop their mathematical understanding of fractions, decimals, and percents, Kranz writes.

Dig into sports stats

To help students learn how to draw conclusions from data and boost their comfort with decimals and percentages, fourth-grade teacher Kyle Pisselmyer has his students compare the win-loss ratio of the local sports team to that of Pisselmyer’s hometown team. While students can struggle to grasp the relevance of decimals—or to care about how 0.3 differs from 0.305—the details snap into place when they look at baseball players’ stats, educator Maggierose Bennion says.

March Madness is a great source of real world data for students to analyze in math class, says sixth-grade math teacher Jeff Norris. Last March, Norris decorated his classroom like a basketball court, then had his students do basic statistical analysis—like calculating mean, median, and mode—using March Madness data, including individual game scores and the total win rate of each team. “We also did some data collection through our own basketball stations to make it personally relevant,” Norris says; students lined up in teams to shoot paper balls into a basket in a set amount of time, recorded their scores in a worksheet, and then examined the scoring data of the entire class to answer questions about mean, median, mode, range, and outliers.

Go on a (pretend) shopping spree

“My students love any activities that include SHOPPING!” says Jessie, a sixth-grade teacher who creates shopping-related problems using fake (or sometimes real) store ads and receipts. Her students practice solving percentage problems, and the exercise includes opportunities to work with fractions and decimals.

To get students more engaged with the work, math educator Rachel Aleo-Cha zeroes in on objects she knows students are excited about. “I make questions that incorporate items like AirPods, Nike shoes, makeup, etc.,” Aleo-Cha says. She also has students calculate sales tax and prompts them to figure out “what a 50% off plus 20% off discount is—it’s not 70% off.”

Capture math on the fly

Math is everywhere, and whipping out a smartphone when opportunities arise can lead to excellent content for math class. At the foot of Mount Elbert in Colorado, for example, math teacher Ryan Walker recorded a short word problem for his fourth- and fifth-grade students. In the video, he revealed that it was 4:42 a.m., and it would probably take him 249 minutes to reach the summit. What time would he reach the summit, he asked his students—and, assuming it took two-thirds as long to descend, what time would he get back down?

Everyday examples can be especially relatable. At the gas station, “I record a video that tells the size of my gas tank, shows the current price of gas per gallon, and shows how empty my gas tank is,” says Walker. “Students then use a variety of skills (estimation, division, multiplying fractions, multiplying decimals, etc.) to make their estimate on how much money it will cost to fill my tank.”

Connect to social issues

It can be a powerful exercise to connect math to compelling social issues that students care about. In a unit on ratios and proportions, middle school teacher Jennifer Schmerler starts by having students design the “most unfair and unjust city”—where resources and public services like fire departments are distributed extremely unevenly. Using tables and graphs that reflect the distribution of the city’s population and the distribution of its resources, students then design a more equitable city.

Play entrepreneur

Each year, educator Karen Hanson has her fourth- and fifth-grade students brainstorm a list of potential business ideas and survey the school about which venture is most popular. Then the math begins: “We graph the survey results and explore all sorts of questions,” Hanson writes, like whether student preferences vary with age. Winning ideas in the past included selling T-shirts and wallets made of duct tape.

Next, students develop a resource list for the business, research prices, and tally everything up. They calculate a fair price point for the good they’re selling and the sales quantity needed to turn a profit. As a wrap-up, they generate financial statements examining how their profits stack up against the sales figures they had projected.

HELP OTHER TEACHERS OUT!

We’d love this article to be an evolving document of lesson ideas that make math relevant to kids. So, teachers, please tell us about your go-to activities that connect math to kids’ real world experiences.