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How to Write a Case Study on Mathematics

Last Updated: September 15, 2021

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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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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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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.

Mathematics and Case-Based Instruction

• Posted August 1, 2004
• By News editor

An Interview with Senior Lecturer Kay Merseth

Katherine K. Merseth, senior lecturer on education and director of the Teacher Education Program led a team of mathematicians, teachers, and teacher educators, and edited the cases in the recently published Windows on Teaching Math: Cases of Middle and Secondary Classrooms (Teachers College Press). Written for both pre-service and in-service teachers, the book includes 11 cases, each with an objective to improve the teaching and understanding of mathematics at the 7th- through 12th-grade levels and to provide opportunities to examine classroom practice and assess student thinking. These materials adopt the highly successful pedagogical approach of case-based instruction used at the Harvard Business School to examine the work of secondary classroom mathematics teachers. What follows is an interview with Merseth on the advantages of case-based pedagogy in mathematics instruction.

Q: What are the necessary skills to teach mathematics?

A: Solid content knowledge, strong knowledge about pedagogical techniques in mathematics, a deep understanding of the epistemology of mathematics, knowledge about how people learn math, and the ability to listen to and engage the learner. Many consider only the content to be important. In part, this is because there are so many math teachers lacking basic knowledge about the content and for whom math was not a major or focus in college. Many of the current mathematics teachers in secondary schools today have not studied the topics they are teaching since they themselves were middle- or high-school students. In my opinion, cases are a good way to add to the discourse on how we can better understand and teach mathematics.

Q: You note that math students in the U.S. are not performing to the standards that they should be, especially in comparison with other countries. To what do you attribute this gap?

A: Societal beliefs about mathematics, the rote, skills and procedural focus of the curriculum in schools, and the preparation of our teachers are three factors to blame. The common conception is that math is rule-oriented knowledge, static, and a difficult subject only mastered by a few. In reality, mathematics is an interactive problem-solving process involving conjectures, hypotheses, and refutations that flow. Much of the typical curriculum in schools is outdated, repetitious, and unrepresentative of the evolution of the field. Where many textbooks stress computation and procedure, mathematics teaching should focus on understanding and sense-making. Finally, only one out of every two math and science teachers possesses adequate pedagogical training, and thus many tend to teach only in the way they were taught. A change in the mathematical education of our young people is a goal our society can and must achieve.

Q: Why do you think case-based instruction is effective for teaching mathematics?

A: Case discussions are a natural mode of learning about reform-oriented teaching, as they elicit critical examinations of various teaching approaches. The complexities of real classrooms are brought to life by highlighting the multiple dimensions of teaching. Cases and case-based instruction are known as a pedagogical approach that helps participants develop analysis and problem-solving skills. Cases bring practice face-to-face with the content of the subject in a realistic way. It is also important to note that cases provide a safe and engaging way to explore, examine, and analyze pedagogical possibilities as well as specific content.

Q: How did you develop the cases? What was the primary objective in developing them?

A: The set of cases were designed using a theoretical framework that includes four dimensions: subject matter knowledge, pedagogy, student language and understandings, and classroom and school context issues. The style and format of the cases in this book build on the expertise developed by the Harvard Business School. The late C. Roland Christensen of the Business School and the Ed School offered inspiration and encouragement for the exploration of the use of cases in the field of professional education. In order to develop pedagogical skills, case materials need to stimulate and encourage the imagination of the case reader, which hopefully leads to various creative responses presented in the case. In addition, the cases in Windows on Teaching Math cover topics considered "hard to teach" and/or "hard to learn," such as rate, ratios, proportions, limits, independence of events in probability and multiple representations. These materials are cutting-edge and provide marvelous opportunities for teachers to engage in lively discussions about content and classroom teaching techniques in mathematics.

Q: How important is the role of the facilitator in case-based instruction?

A: Case facilitators must wear many hats--they are responsible for the content of the case, for the discussion flow and process, for individual students and their sense of safety to join the discussion. In fact, I believe a safe environment, one where participants feel comfortable raising a range of comments and discussing personal beliefs, is critical to case-based learning. The facilitator is important because he or she must encourage participants to respond to each other, not the "instructor." Finally, the physical environment is important. We've found that case discussions flourish when participants are able to observe each other's facial expressions and body language. A U-shape or circle are ideal as they also allow the facilitator to move throughout the group.

The latest research, perspectives, and highlights from the Harvard Graduate School of Education

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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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An case study examples on mathematics is a prosaic composition of a small volume and free composition, expressing individual impressions and thoughts on a specific occasion or issue and obviously not claiming a definitive or exhaustive interpretation of the subject.

Some signs of mathematics case study:

• the presence of a specific topic or question. A work devoted to the analysis of a wide range of problems in biology, by definition, cannot be performed in the genre of mathematics case study topic.
• The case study expresses individual impressions and thoughts on a specific occasion or issue, in this case, on mathematics and does not knowingly pretend to a definitive or exhaustive interpretation of the subject.
• As a rule, an essay suggests a new, subjectively colored word about something, such a work may have a philosophical, historical, biographical, journalistic, literary, critical, popular scientific or purely fiction character.
• in the content of an case study samples on mathematics , first of all, the author’s personality is assessed - his worldview, thoughts and feelings.

The goal of an case study in mathematics is to develop such skills as independent creative thinking and writing out your own thoughts.

Writing an case study is extremely useful, because it allows the author to learn to clearly and correctly formulate thoughts, structure information, use basic concepts, highlight causal relationships, illustrate experience with relevant examples, and substantiate his conclusions.

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4 best practices for case interview math: part 2

When we last left off in this series of Casing 101 blogs, we had structured a detailed framework that broke the problem into its unit-level variables and presented it to the interviewer (see link here ). But this is just the beginning. From here the interview will move into a series of problem-solving exercises including math problems, interpretation of complicated charts and exhibits, and brainstorming exercises.  

Of these, math typically causes the most anxiety among candidates. Performing multi-step math on paper in front of a stranger, while verbalizing your calculations, is an experience that requires significant practice to get comfortable. But practice alone is not enough; there are certain best practices that will help candidates excel in the math portion of their case interviews. We’ll walk through these now.

Best Practice #1: Structure Your Math Before Doing any Calculations

One of the most common mistakes I see candidates make in mock interviews is their belief that the ability to do raw, complex calculation is all that matters. As a result, when given a math problem, these candidates will spend 2-3 minutes in silence, scribbling a series of equations on their paper that are impossible for the interviewer to follow, before offering an answer without any explanation of how they reached it.  

To be sure, consultants must be able to do complex analysis, often involving challenging math. However, a consultant’s analysis is only worthwhile if they can present it to the client in a clear, concise manner that the client understands and can rely upon to make change in their organization. Accordingly, interviewers are not only testing your ability to do math, but also your ability to clearly lay out your analysis and explain it in an intelligible way.

To that end, before doing any math, I recommend structuring the analysis by laying out on paper the order of operations you will follow to get to the answer. For example, imagine you were given the following information about a hospital, and your interviewer asked you how many surgeries the hospital would need to conduct per year to break even. How would you structure this problem?

When first given data such as the tables above, you should not do any math. Instead, start by examining the exhibit and immediately think about how the data that is given can be used to get to the right answer. While you may not have figured out every necessary step in the series of calculations, you should at least recognize the first few steps. Begin writing out these first steps, outlining the “intermediate answers” you’ll get to as you progress through the calculations. As you work through the steps to get to these “intermediate answers” you should continually recognize and outline the subsequent calculations you need to make until you arrive at the final answer. To make your order of operations as easy to follow as possible for your interviewer, lay out the order of calculations vertically in a table, with a second column that will be filled in once you begin the actual calculation. This format also has the added bonus of helping you avoid getting lost as you perform multi-step calculations.

Best Practice 2: Be Verbal

As mentioned above, your interviewer is not simply testing your ability to do math. He or she also is testing your ability to explain complex analysis to clients in a concise and intelligible manner. Therefore, you should strive to speak to your interviewer as you lay out your calculations and perform the subsequent math.

Accordingly, once you have laid out your full order of calculations in a table, spin the paper to face your interviewer, and explain to them the steps that you will take to arrive at the final answer. Once you have done so and begin performing the calculations, continue to talk out loud as you fill in the numbers in the table you have created. By doing so, your interviewer can follow your work and can course correct you if you make any small math errors, such as multiplying by 100 instead of 1,000. Importantly, a small math error is not a disqualifier, so long as you recover quickly and show structure in your problem-solving.

Best Practice 3: Use Mental Math Where Possible to Fill in Your Calculations

Once you have laid out your calculations in a table and explained your approach to your interviewer, it is time to begin performing the actual calculations. Using the initial data you were given, begin filling in the second column of the table you created above. As discussed, talk to your interviewer as you perform each of these calculations so that they can follow your train of thought.

Where possible, use mental math for your calculations. It becomes tedious when a candidate needs to use pencil and paper for every single calculation. Mental math makes it easier to communicate and will make you seem like a more fluid thinker. Certainly, if you cannot do the math in your head, then you should use a pencil and paper – it’s better to get the right answer with paper than the wrong answer with mental math. But certain calculations, such as $15 million divided by 1,500, you should be able to do in your head.

Below is an example of what your table should look like once you’ve performed all the calculations.

Best Practice 4: Give the “So What?”  

For a consultant, performing calculations that give you the right answer is not enough. You must also be able to take your analysis and contextualize the output, showing that you understand the implications of the analysis you performed. Accordingly, once you have gotten to the right answer, give your interviewer the “so what?” – a brief explanation of what the answer actually tells you about the situation you are analyzing.  

For instance, in this case you might say something like, “We need to perform 2,000 surgeries per year to break even. We currently only perform 1,500 surgeries, so we would need to increase surgeries by one-third just to breakeven, and by even more to be profitable. This is a huge increase and one that might not be realistic. In addition to trying to drive more surgeries to our hospital, we should also evaluate ways to cut costs to achieve profitability."

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How to Study Math in College

Introduction.

Why aren't you getting better grades in mathematics? Do you feel that you have put in all the time on it that can be expected of you and that you are still not getting results? Or are you just lazy? If you are lazy, this material is not intended for you. But if you have been trying and your grades still don't show your ability, or if you have been getting good grades but still feel that the mathematics does not mean very much to you, it is very likely that you do not know how to study effectively. This material aims to help you to study mathematics effectively.

Some of you, may feel that you have successful study methods of your own different from the ones described here. In that case, you need not feel you must change your methods, although you might profit from comparing your methods with these.

On the other hand, some of you may feel that the suggestions on the following pages are over-ambitious - that they would require more time and effort than you are prepared to give. You will probably be right. We cannot expect to do everything to perfection, but we can do the best we are able. Out of the suggestions offered, you can pick the ones that may help you most, and as you find your work improving, you may be able to try further suggestions. So scoff if you wish at these ambitious suggestions, but then give some of them a try, a fair try, and watch the results. 

How to Study:

• Homework  Tips
• How to Make Your Errors Help You Learn  Tips
• Classwork: How to Make the Most of Your Time in Class  Tips
• How to Use the Textbook  Tips
• How to Review for Tests  Tips
• How to Take Tests  Tips

Back to Top

There is a common misconception that homework is primarily something to eventually hand in to the teacher. Actually, the homework is first and foremost a means of learning fundamental ideas and processes in mathematics, and of developing habits of neatness and accuracy. What is passed in to the teacher is only a by-product of that learning process. The following four-step routine is a suggestion for making your home study effective:

• Get oriented. Take a few minutes to think back, look over your notes, and look over the book to see clearly what ideas you have been working on.
• Line up the ideas. Think about the ideas, laws, and methods in the day's assignment or lesson. Don't forget to familiarize yourself with any new words in your mathematics vocabulary. Try to remind yourself of any warnings about errors to avoid that the teacher might have mentioned. Go through any examples given to be sure you really understand the concepts being illustrated.
• Get the assignment accurately off the blackboard. Have a definite place in your notebook where you write down the assignment or lesson. If you do not understand the assignment, don't hesitate to ask.
• Follow the directions.
• Work neatly and accurately.
• Show your complete work, not just the answer. This will help you and your teacher when you are checking through for errors. 
• Always check back to be sure you have done all simple arithmetic correctly.
• Do the work promptly before you have forgotten all the instructions.
• If you get stuck, don't just give up! Look back at the book and your notes for ideas related to the problem. If your work on a problem seems to be completely confused, it sometimes helps to discard your paper entirely and start a fresh. If you still can't clear your thinking, ask the teacher about the problems as soon as possible.
• Help someone else, if you can. There is no better way to learn a topic than by trying to teach it! Also, it is often helpful to call upon a classmate when you do not understand a problem. Often, they are able to explain the concept to you as well (if not better than) the teacher.

How to Make Your Errors Help You Learn

What do you do when an answer is wrong in your homework, or on a test? Do you throw it away and forget it-and then make the same mistake the next time? If you are wise, you will make those errors teach you something. Here's what you can do:

• Analyze the error to see if you can find what you did wrong.
• If it is a careless error and you really knew how to do the work correctly, make a note of it, and if you find that you keep making careless errors frequently, start working more carefully.
• If you can't find where your error is, ask the teacher or a classmate to help you.
• Keep a page in your notebook entitled, "Warning: Errors to Avoid." On the same page write a description of the corrected way to do that kind of exercise, being sure to emphasize the important idea behind it. 

Classwork: How to Make the Most of Your Time in Class

• Get ready. In the minute or two before the class gets started, think over what you have been working on recently.
• Have all necessary equipment: book, pencils or pens, notebook, homework assignment.
• Take down the assignment promptly and accurately.
• Concentrate. This takes an effort if you are the kind whose mind tends to wander.
• Ask questions when you do not understand.
• Listen to the questions and answers of others in the class. When another pupil is answering a question, think how you would answer the question.
• Take part in the class discussion.
• Do not write at the wrong time. When you are taking notes, be sure you do not miss anything that is said while you are doing so. When taking notes, there are two conflicting things you must try to do. One is to make your notes complete and accurate enough to be valuable to you later. The other is to make your notes brief enough so that you can continue to listen to what is being said in class.

How to Use the Textbook

• Use the index and glossary at the back of the book, especially when you have forgotten the meaning of a word.
• When your book gives an example to illustrate an idea, analyze the example carefully for the ideas behind it instead of just trying to make your exercises look like the example.
• If you can't do an exercise, reread the explanatory material in the book and/or go over your class notes.
• Make the most of the study helps at the end of each chapter.

How to Review for Tests

• Start reviewing far enough in advance so you have time to do a careful unhurried job, and still are able to go to bed early the night before the exam.
• Be sure to go through your notes and the examples that are there. If they don't make sense to you, you haven't taken enough notes!
• If there are some formulas for which you are responsible, make a list of them and then practice saying them, or writing them.
• Use the review materials at the end of each chapter. If you are having trouble on a problem, go back to that section in the book and rework some problems there.
• If you were the teacher, what questions would you ask on the test? Prepare yourself for those questions.
• Since it is said that "practice makes perfect", one of the better ways of studying for a test is to do some problems that were previously assigned to you. Go over your homework to be sure you understand the procedure you used in each section.
• Get a good night's rest the night before the exam!
• DON'T WORRY! 

How to Take Tests

• When you take a test, have the right attitude - take pride in doing the best job you can. Don't try to "get by" with doing as little as possible. Have confidence in your own ability.
• Be serious and concerned enough about the test to do your best, but don't worry to the point of anxiety. Fear alone can make a person do poorly on a test, regardless of his ability and knowledge.
• Have all necessary equipment.
• FOLLOW DIRECTIONS. Read carefully and listen carefully for any special instructions, such as where answers are to be written, any changes or corrections, etc.
• Look over the whole test quickly at the start and, unless you are required to do the questions in the order given, do the ones you are sure of first.
• If you are unable to answer a question, leave it and go on to another, coming back to the hard one later. Often, with a fresh start, you will suddenly see much better what to do.
• Be careful to show clearly what you are doing. Remember that the teacher is not a mind-reader, and your grade may depend on whether or not the teacher can see from your work that you understand what you are doing.
• Work neatly. It makes a good impression on the teacher!
• Check back as you go along for accuracy. Careless errors can make a great deal of difference in your score.
• With the right attitude and careful preparation for a test you probably will do well on the exam.
• Remember: The one or two hours of the test are but brief moments in your life span so DON'T PANIC! 

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How to Study Math

Last Updated: September 28, 2023 Approved

This article was co-authored by Grace Imson, MA and by wikiHow staff writer, Amy Bobinger . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. There are 10 references cited in this article, which can be found at the bottom of the page. wikiHow marks an article as reader-approved once it receives enough positive feedback. This article received 16 testimonials and 86% of readers who voted found it helpful, earning it our reader-approved status. This article has been viewed 490,110 times.

There’s no denying it--math can be tricky! Even if you don't feel like it's your strongest subject, though, you can get better at math if you're willing to put in the work. The best way to do well in math is to practice every day, so set aside plenty of time to study on your own or with a group. With a little determination, you can make real strides in math and it will benefit you in every way.

Do your homework, but don't stop there.

• Solving problems over and over is the best way to get good at math, which is why doing your homework is so important.
• If you're looking for practice questions online, you might check out popular math sites like Khan Academy, Math-Aids, and Corbettmaths. [2] X Research source

Read your textbook actively.

• Reading your math book might not sound like a lot of fun, but it can really help you understand a difficult concept you're struggling with. In addition, your textbook can help you understand why things are done a certain way, helping you go beyond simply memorizing the steps.
• Each night when you're studying, read a few pages ahead of what you've already been assigned. That can make it easier to keep up during class because you'll already be familiar with what the teacher is talking about.

Spend a few minutes studying each day.

• If you spend even 30 minutes a day studying math, you'll be less stressed when it's exam time because you'll already be familiar with the material you're being tested on. [5] X Research source
• When you're studying, read through the notes you took in class, go over any formulas you're using in class, and re-read your textbook.
• Try making flashcards—you might write a problem on one side and the answer on the other, or you might write down a formula on one side and its name on the other.

Show your work on every problem.

• In addition to helping you on exams, showing your work makes it easier to go back and see where you made a mistake if you get the wrong answer.
• For instance, if you're solving "2x = 3+1," don't just skip to "x = 2." Write out "2x = 4," then "2x/2 = 4/2," then "x = 2."

Give extra attention to word problems.

• It can help to draw a picture or diagram illustrating the problem. Include any quantities that are listed in the problem. If the quantities aren't named, use a variable like "x" (or you could make up your own, like using "a" if you're solving a problem about apples).

Check your work once you’re finished.

• Some problems—like equations with variables—can be checked by substituting the solution in place of the variable. For instance, if you solved "2x = 3+1" earlier and got "x = 2," you could check that by working through "2*2 = 3+1," or "4 = 4." Since the answer is true, you know you have the correct answer. [8] X Research source

Refresh your memory by going through older problems.

• Do this even if it seems like what you're studying isn't related to those previous chapters. Chances are, something you learn later will help tie everything together.

Take practice tests to check your skills.

• Ask your teacher if there are practice versions of the test that you can use for study.
• You may also be able to find practice tests online. Just search for something along the lines of "geometry + free practice tests" or "ACT math practice tests."

Think about math as a language.

• If you're struggling to remember what the different symbols and terms mean, keep a vocabulary log where you write down any words or terms you learn, along with the definition.
• You can also use flashcards to help you remember the definitions of different words and symbols.

Focus on comprehension more than memorization.

• If you're really struggling with a certain concept, try asking your teacher to explain it again, re-read that chapter in your textbook, ask a friend for help, or find a math tutor.
• Sometimes, teachers and professors might allow you to bring a list of formulas with you to a test. If you don't understand how and when to use them, though, this won't be much help. [12] X Research source

Use repetition to help you nail down formulas.

• At the same time, say them to yourself over and over. The combination of hearing the formula, seeing it, and writing it may help make it easier to remember.
• You can also use mnemonics, or memory devices, to help you remember concepts or formulas. For instance, you might remember the order of operations (parentheses, exponents, multiplication, division, addition, subtraction) by setting the acronym PEMDAS to a phrase like "Please excuse my dear Aunt Sally."

Participate in class.

• If your teacher asks a question, try to come up with the answer, even if you don't raise your hand—it will be good practice for later.
• Focus on key information when you're taking notes. Write down things like formulas, definitions, and sample problems—but spend more time listening than writing. For instance, if your teacher is explaining why it's important to show your work, you might just write, "Show all steps" instead of copying what they say word-for-word. [15] X Research source

Join a study group.

• Be active in your study group—explaining things to other people can actually help you understand the subject better. [16] X Research source

Get more help if you need it.

• Don’t be embarrassed to ask for help. Math can be difficult, and there’s a lot of information to keep up with. The most important thing is to make sure you understand how to do the work, and anything that helps you do that is a good thing.

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Video . By using this service, some information may be shared with YouTube.

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• ↑ https://math.osu.edu/undergrad/non-majors/resources/study-math-college
• ↑ https://www.weareteachers.com/best-math-websites/
• ↑ https://www.nova.edu/tutoring-testing/study-resources/forms/study-skills-guide.pdf
• ↑ https://www.educationcorner.com/math-study-guide.html
• ↑ Grace Imson, MA. Math Instructor, City College of San Francisco. Expert Interview. 25 November 2019.
• ↑ https://learningcenter.unc.edu/tips-and-tools/studying-for-math-classes/

About This Article

To study math, start by doing extra problems after you do your homework, which you can find online or in the back of your textbook. When you’ve done the problems, check your work to make sure you understand and have the right answer. Additionally, start each study session with drills of things you've already learned so you can keep those skills fresh. If you’re still struggling with a problem or concept, ask a teacher or tutor to help you. For tips on forming a math study group, read on! Did this summary help you? Yes No

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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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Bonus download, proven strategies for effective leadership and results, fast mental math: rounding numbers.

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

PODCASTS:   If you enjoy our podcasts, we will appreciate it if you visit our  Case Interviews  channel or  Strategy Skills channel or The Business of Management Consulting channel on iTunes and leave a quick review. It helps more people find us.

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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. You can watch the video reference below to learn more about Exhibits in consulting.

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.

Scoring in the McKinsey PSG/Digital Assessment

The scoring mechanism in the McKinsey Digital Assessment

Related product

Comprehensive Math Drills

Ace any math problems in the management consulting recruiting process with 400+ consulting math questions

Six types of charts in case interview are: Bar/Column chart, Line chart, Percentage chart, Mekko chart, Scatter plot chart, Waterfall chart.

Business knowledge is not a mandatory condition to become a consultant. Nevertheless, it still has specific obligations and advantages for consultants.

Case Studies

These are stories about real students here at MFM.

Notice that we celebrate the small things that can’t be measured by any test!

All information shared was provided with permission from parents.

Math Shows Up in this Student’s Play

Ryan wasn’t understanding math the way it was being taught to him in school, mostly through verbal instruction, worksheets, and the computer program, and Prodigy. The school actually had a decent curriculum, but the teachers either weren’t trained in…

Growth Mindset + Multisensory Math Helped This Student Shine!

Logistics played a big part. Finding time in our busy schedule to drive to one more thing was a concern. Comfort level played another big part. Being at home made it far more comfortable for my daughter. Her surroundings were familiar which I…

Multisensory Math Helps this Student with Dyslexia Ace Math Tests!

Ethan hated math! He tried and tried to get it, but it just made him really frustrated! He struggled to understand and remember concepts. Accordingly, I felt like I had to come up with 57 different ways to teach a concept so that maybe one of them…

Multisensory Math Surprised This Student!

Big box tutoring is frustrating! We have tried in person tutoring in the past. To be honest, getting someone to show up consistently has been a struggle, and the big box math tutoring companies (Sylvan, Kumon) do not have the personalized, conven…

ADHD Doesn’t Stop this Cowboy from Learning

Cam had so many tutors in the past that were not helpful or able to teach the content where his ADD mind could understand. He would get extremely frustrated and shut down. We took a break for a while before trying to start again. He was adamant…

[Case Study]

by Adrianne Meldrum

Adrianne, Math Tutor

Adrianne works with Ryan twice a week online for 1 hour.  She uses multisensory math techniques to engage Ryan.

Ryan, Homeschooler

Ryan loves beating his dad at sports and building things with his hands!  He recently won Pyramid Solitaire. 

About this Student:

• Name:  Ryan
• School:  Homeschooled
• Struggles:  Dyslexia
• Strengths:  Building with his hands
• Sports Enthusiast 100% 100%
• Creative Thinker 100% 100%
• Master Builder 100% 100%

*Shared with permission from Ryan’s parents.

What problems was Ryan experiencing before you decided to try MFM?

Ryan wasn’t understanding math the way it was being taught to him in school, mostly through verbal instruction, worksheets, and the computer program, prodigy. The school actually had a decent curriculum, but the teachers either weren’t trained in it or didn’t feel they had time to teach it with fidelity.

He wasn’t making any forward progress , and the gap was quickly widening in his second grade year. For numerous reasons, I pulled Ryan from school to homeschool. He was soon diagnosed with dyslexia. I immediately started attending courses to learn what kind of instruction Ryan needed and how to advocate for him.

Why did you select MFM over regular tutoring available near you?

In the meantime, I found through our local university, an elementary teacher who after retirement, got a PhD in math education. 

Ryan worked with this wonderful, loving teacher for six months. She taught his addition facts through 18 and the decomposition/composition of ten through math games that used concrete, representational, abstract concepts. Ryan regained a love for math again, as well as his confidence. Learning through play is truly magical. This was a needed and valuable step for Ryan.

However, after six months, I knew that while Ryan was making some progress, it was not as much as I felt he is capable of, nor was he instructed in the number of topics and skills a second grader should have to keep up with his class. 

I had taken some math courses over the summer specific to helping dyslexic students, however I was struggling to know where to start with Ryan. I needed to find someone who could either mentor me with implementation of the strategies I had been taught or someone who could teach Ryan the rest of these concepts in a way that a child with dyslexia could understand, remember, and apply. 

When I found MFM, I knew I had found the answer to helping me with both needs–help for Ryan and mentorship for myself.  Because of my training experiences I knew that Math for Middles would offer a full lesson plan of learning activities that covered multiple areas of math and would teach in a way that any student could understand, including students with dyslexia and dyscalculia. 

The math concepts are taught systematically, sequentially, explicitly, using multisensory methods, and works through a concrete, representational, and abstract model. There isn’t a better instructional method than that.

Take a peek at some of the math showing up in Ryan’s play.  Below he was making patterns on his own and also categorizing the LEGO minifigures by “with a cape and without a cape”.

These are important concepts in math!

How did Adrianne help Ryan to experience more success in math?

Adrianne began her instruction with Ryan based off an initial, in-depth, formal assessment that identified Ryan’s strength’s and weaknesses.  Adrianne’s lessons are developmentally appropriate and engaging. He is also getting a full curriculum of skills and math topics.

In just seven weeks, Ryan has stopped reversing his 3’s and 9’s; is developing problem solving strategies; strengthening his logic through sorting and  categorization; applying his knowledge of math facts and addition; and beginning work in subtraction.

Adrianne has also provided an educational therapist to help him work through his issues of anxiety, and ADHD in not only math, but also his literacy. MFM provides a way to treat Ryan as a whole person, not just his math difficulties. Ryan is making wonderful progress and I also find my confidence growing in my skills as well. 

What are your goals for Ryan as he continues in his math studies?

 I would like for Ryan to:

• continue to grow his enjoyment of math and his confidence to do it.
• continue to strengthen his logic, problem solving skills, and ability to identify.
• gain an understanding of math terms and definitions through the use of a visual dictionary and words broken into roots and morphemes, with pictorial or representational examples.
• gain a solid understanding of the base ten system and how it applies to the four basic operations, fractions, and money
• solve problems that have functional applications in real life situations.

My ‘Dream Big’ goal is for him down the road is to eventually be able to use the scientific method to solve math problems– to collect data from a set of problems, find the patterns, analyze it and create a theory of how a set of problems could be solved. I recognize this may not be a goal Ryan wants to do though. I just think it sounds fun to help students learn to think this way.

Growth Mindset + Multisensory Math

Helped this student shine.

Adrianne works with Brittany twice a week online for 1 hour.  She uses multisensory math techniques to engage Brittany.

Brittany, 10th Grader

Brittany has been with us for quite awhile and we have seen her take great strides! She loves playing soccer and hanging with her family.

• Name:  Brittany
• School:  Public
• Struggles:  Math confidence, math concepts (these are a thing of the past!)
• Strengths:  Contributing in math class, growth mindset
• Soccer Diva 100% 100%
• Dedicated Student 100% 100%
• Growth Mindset 100% 100%

*Shared with permission from Brittany's parents.

What problems was Brittany experiencing before you decided to try MFM?

Before we started my daughter had a C and was drowning in Algebra 1 by second trimester.

Logistics played a big part.  Finding time in our busy schedule to drive to one more thing was a concern.

Comfort level played another big part.  Being at home made it far more comfortable for my daughter.  Her surroundings were familiar which I feel played a big part in helping her feel safe to ask questions and be herself.

I LOVE the online setting. I didn’t know what to expect at first, but it is fantastic!

Last year in middle school my daughter didn’t start school until 9am, so Adrianne meet with her twice a week before school.  My daughter could do it in her Jammie’s. We weren’t racing to get somewhere, sitting in an uncomfortable classroom. It’s right is the comfort of your own home.

I have 4 kiddos, so to be able to get that much attention for my daughter without running all over town was golden! Now with busy high school schedules, she meets in the evening after dinner.

Look at these awesome wins!! Check out these texts sent from Brittany and her mom:

Notice that Brittany applies her growth mindset.  She takes a test, looks at the feedback, and decides to do the retake offered to her.

She has changed the way she looks at getting math problems wrong!  Love it ?

How did Adrianne help Brittany to experience success like this?

Adrianne plans interactive lessons that focus specifically on what my daughter needs. We have given Adrianne access to our online text book so she knows what the exact learning targets are for my daughter. Adrianne communicates regularly with my daughter asking for photos of homework and exams so she can help my daughter with exactly what she is working on.

She has taught her how to carefully dissect a problem to break it down into simple steps. She has taught her to use different colored pencils to help break down the information in a visual way.  My daughter now uses this technique in her PreAP Chemistry class as well.

Adrianne shared a play list of “study music” to help her stay alert and focused.  My daughter uses it all the time and has even shared it with her siblings.

She has taught her skills to boost her confidence. She is kind and patient and has a great desire to keep working with my daughter until she is confident she truly understands.

And best of all, Adrianne is quick to laugh when she makes a little math mistake, showing my daughter it is okay, that none of us are perfect and a mistake isn’t the end of the world.

Here’s what Brittany’s teacher had to say about her progress:

What are your goals for Brittany as he continues in her math studies?

Our biggest goal is for our daughter to be comfortable rather than intimidated by math.  She no longer hates it, in fact she says she enjoys math “the way Adrianne teaches it.”

She is confident and already talking about taking PreAP Calculus next year.  That is a huge accomplishment for a girl who as a 7th grader was ready to quit math all together.

*Update on Brittany*

Brittany is ROCKING it in math and has started working with our tutor Piper on chemistry.  She is such a great example of what we want every student to aspire to be:

• sees failure as a means for learning
• hard working
• have a growth mindset
• advocate for their needs

Multisensory Math Helps this Student with Dyslexia Ace Math Tests!

Adrianne works with Ethan three times a week online for 1 hour.  She uses multisensory math techniques to engage Ethan.

Ethan, 14 yrs old

Ethan has dyslexia and ADHD.  He’s been homeschooled since 3rd grade.  He has set a personal goal to attend public school.  Ethan is well on his way!

• Name:  Ethan B.
• School:  Homeschooler
• Struggles:  Dyslexia, ADHD
• Strengths:  Friendly, Great at Conversation
• Super Fun 100% 100%
• Funny Guy 90% 90%
• Jamming to Music 100% 100%

*Shared with permission from Ethan’s parents.

What problems was Ethan experiencing before you decided to try MFM?

Ethan hated math! He tried and tried to get it, but it just made him really frustrated! He struggled to understand and remember concepts. Accordingly, I felt like I had to come up with 57 different ways to teach a concept so that maybe one of them would be the one that would work. And once we found a method that worked, it wouldn’t stick. He would wake up the next morning having forgotten what we learned the previous day. It felt like we were living out the movie ‘50 First Dates’!  

We would spend 4 hours a day on one math concept and a few math problems! That’s way too much…but I felt like we had to do it in order to get him caught up.  It was difficult and really draining as a mother.  I felt like a failure and worried that my son wouldn’t be able to survive as an adult…without even basic math skills.

Ethan was really hard on himself. He felt like since he had Dyslexia and also struggled with math that he was not smart. It really affected his self-esteem, and this affected all other areas of his life. It was really hard. 

We had already tried all kinds of different types of programs and tutors.  Although there were good things about each of them, none of them helped Ethan make significant progress. 

When I met Adrianne, we spent time talking about how her program, utilizing multisensory math, was different and how it could help Ethan. I was excited, yet apprehensive, praying it would work, but still not 100% convinced since nothing previously had worked.

But we decided to just give it a try and jumped right in.

How did Adrianne help Ethan experience more success in math?

Adrianne is a G ODSEND!  First, she helped my son relax!  He had shut down due to the previous failures with math, and was almost unwilling to try anymore.  But Adrianne was able to get him to open up and want to try.

She helped him understand how HE was indeed smart, and she helped him find some successes early on.  These early wins made him want to try again!   He started to progress by leaps and bounds! I watched his confidence grow as she worked with him.  He was feeling so much better about himself as a person, which made my momma heart happy!

Adrianne has helped him in so many different ways!  By teaching him how to learn and retain math skills, he is now learning and retaining info in other subjects!  And I don’t have to remind him to do his homework now!  He does it on his own and turns it in without me having to hound him.

AND he now says Math is his favorite subject.  WOW!  I never thought that would happen!  This has made such a huge difference in all of our lives! THANK YOU ADRIANNE! We have been so blessed to have met you!

What are your goals for Ethan as he continues in his math studies?

We are hoping to be able to get Ethan up to grade level in math, so that he can attend and graduate from public high school. He really wants to be able to graduate!  With Adrianne’s continued help, I think he can do it!

We also want him to continue to build his confidence.  Adrianne has been so instrumental in his progress so far… I don’t know what we would do without her!

WE LOVE MFM!

Multisensory Math Surprised this Student!

Ruth, Math Tutor

Ruth works with Margo twice a week online for 1 hour.  She uses multisensory math techniques to engage Margo.

Margo, 6th Grader

Margo loves to dance and travel to interesting places with her family.

• Name:  Margo J.
• Grade:  6th
• Struggles:  Short attention span
• Strengths:  Seeing the big picture
• Dance Diva 100% 100%
• Funny Gal 90% 90%
• Surprised Herself 100% 100%

*Shared with permission from Margo’s parents.

What problems was Margo experiencing before you decided to try MFM?

She was generally confused by the lessons at school and felt overwhelmed. This led to many tear-filled nights due to frustration.

#1-Adrianne!   I was looking for inspiration and stumbled upon a podcast with Adrianne describing exactly what we were experiencing at home. I shared the podcast with my husband, then contacted Math for Middles the following day. 

#2- Big box tutoring is frustrating!  We have tried in-person tutoring in the past. To be honest, getting someone to show up consistently has been a struggle, and the big box math tutoring companies (Sylvan, Kumon) do not have the personalized, convenient approach that we so desperately needed. 

How did Ruth help Margo to experience small wins like this?

Margo’s answer:  “Ruth simplifies everything.”

Margo’s 4th and 5th grade teacher (the same teacher both years) did not teach math. When she started 6th grade, Margo felt hopelessly lost in math.

Ruth helps Margo feel capable and so much more confident! She helps with the daily homework, as well as working to ‘fill in the gaps’ Margo has from those lost years. Most of all, she somehow makes it fun! 

What are your goals for Margo as she continues in her math studies?

  Margo’s answer:  “Pass math class. Ha ha!” 

  My goals for Margo:

  1. To continue to stay on top of her math skills in class

  2. To get a better understanding of math fundamentals AKA-‘filling the gaps’

  3. My lofty goal is to work over the summer on the upcoming 7th grade math skills so that Margo is ahead when she goes back to school in late August. Another positive: she can tutor anywhere, even when we are out of town on vacation.

Ruth works with cameron twice a week online for 45 minutes.  She uses multisensory math techniques to engage Cam.

Cameron, 5th Grader

Cameron loves sports, is friendly and outgoing.  He’s discovering that math can be fun!

• Name:  Cameron W.
• Grade:  5th
• Struggles:  ADHD
• Strengths:  Fun-loving and full of surprises!
• Sports Guy 100% 100%
• Texan at Heart 100% 100%

*Shared with permission from Cameron’s parents.

What problems was Cameron experiencing before you decided to try MFM?

Cam had so many tutors in the past that were not helpful or able to teach the content where his ADD mind could understand. He would get extremely frustrated and shut down. We took a break for a while before trying to start again. He was adamant he did not want to work with another tutor.

After convincing him to try one for a few sessions, he ended up frustrated and shut down again. Each day he fell further and further behind in math class while losing his confidence. We needed to think outside the box.

As his parents we knew the traditional tutor method would not work. He was so resistant from the beginning, we were already set up for failure. We had to get creative. My husband heard of the MFM program on the Tilt Parenting podcast.

He knew immediately it was something we had to try. We knew it was something we could get Cam to buy into.

It was a hit from the start!

He was engaged and focused which we had never seen from traditional in person methods.

How did Ruth help Cameron to experience small wins like this?

Ruth knew exactly how to teach to Cam’s brain . She knew all the tricks and all the methods to get him engaged. She used his interest, like sports, to show math concepts.

Little by little, starting from the ground up, she showed him how to solve problems in a way he understood. She celebrated the small wins along the way, building his confidence. Finally, he started realized he could do it.

 Cameron’s teacher also sent us this message recently:

Just wanted to show you that my favorite child not only multiplied a 3-digit by a 2-digit number, he also neatly rewrote the products for the multiplication problems to add them (mostly) correctly!

I am keeping this photographic evidence! ~ Cameron’s 5th Grade Teacher

As you can see, even his teacher noticed a huge shift for Cameron.

What are your goals for Cameron as he continues in his math studies?

Our goals for Cameron first and foremost are for him to realize he can solve math problems!

Before he was not even attempting because he was convinced he would not succeed, but now he actually tries and does succeed!

We want him to continue building his confidence regarding math. He is about to enter middle school so we know the work will become a lot more challenging.

Our goal is to build a strong foundation for him now to succeed in math as the years go on. We want him to know he has Ruth and the MFM staff on his team to help him in the years to come.

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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• The 1%: Conquer Your Consulting Case Interview
• Consulting Career Secrets
• McKinsey Solve Game (Imbellus)
• BCG Online Case (+ Pymetrics, Spark Hire)
• Bain Aptitude Tests (SOVA, Pymetrics, HireVue)
• Kearney Recruitment Test
• Structuring & Brainstorming Drills
• Chart Interpretation Drills
• Case Math Course and Drills
• McKinsey Interview Academy
• Brainteaser Course and Drills
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Case interview math: the insider guide

Aspiring consultants at McKinsey , BCG , and Bain or any of the Big 4 and other consulting firms will have to go through some numerical problems during their case interviews.

In fact, in roughly 70% of consulting case interviews, candidates need to demonstrate the ability to handle quantitative problems confidently. For top-tier firms such as McKinsey, BCG, and Bain this probability is close to 100%.

The more prestigious, the more likely that you will have to handle consulting math during the cases.

For example, you first have to analyze a problem mathematically before qualitatively investigating the particular reason for the numerical result and then providing a recommendation.

In fact, many candidates fear this part of the interview and our experience as interviewers and case coaches with McKinsey shows that most errors during the case interview happen in the math part.

If you want to go deeper to brush up on your math and chart interpretation skills, we have created a program with detailed insider learning materials, 25 videos, and a guidebook as well as 2,000 case interview math drills (including full case interview examples and case interview practice) that mimick the McKinsey, BCG, and Bain case interview math as well as the aptitude and analytics test math for you here:  the Case Interview Math Mastery .

This article is part of our consulting case interview series. For the other articles, please click below:

• 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)

Why candidates struggle with consulting math

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.

That being said, as with every other element in a case interview (structuring and analytics, 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.

They need to first plot a course of action, i.e. figure out what to calculate and then, second, perform the calculations. Adding to that, there is the stress of the interview.

Read our tips below to learn:

• How to come up with the correct calculations in case interviews (the logic)
• How to perform swift calculations both mentally as well as with pen-and-paper (the calculation)
• How to stay cool under pressure (consultant mindset)

Case Math Mastery Course and Drills

Approach case interview math like a real 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

As discussed above, fact-based decision-making and recommendations in case interviews are often based on numerical results. Coming up with the correct logical approach to quantitative problems and following through with the correct calculations is one of the key skills needed for the case interview but also for every working consultant.

Broadly, quantitative analyses are conducted for two reasons.

Finding the problem and quantifying its impact

Before solving a problem, consultants need to figure out what is wrong. They will run quantitative analyses to get to the root of the problem(s), a process that will also give them insights into potential solutions.

During the case interview, you will have to do the same in an abbreviated task.

Supporting recommendations

Numerical answers are needed to make and support business decisions for the client, and in reality, every measure or recommendation that is proposed by consultants will have a quantitative backing and rationale.

The answers you will receive by conducting this analysis during the case interview will be crucial for your recommendation in the end

Consulting firms check your quantitative skills

You are expected to perform well in such an environment. Basically, consulting firms check whether or not you have what it takes for the daily consulting life before candidates start their McKinsey careers, BCG careers, or Bain careers.

During the case interview, the math you have to deal with to arrive at a certain conclusion will never be too difficult in nature.

What makes it more challenging is the fact that both planning the steps you need to take and then calculating happens in a stressful environment with the interviewer watching your every step along the way.

Of course, calculators cannot be used.

Being able to solve easy numerical questions, do fast mental math, basic calculus (addition, subtraction, multiply, division, percentages, and fractions) and give reasonable estimates is usually enough to survive a typical interview problem.

Advanced mathematical knowledge is not required. Some of the problems you have to solve may be tricky, include multiple steps, and therefore require a certain degree of logic but the calculations itself should not be too difficult.

How to approach a case interview math question

Different skill levels, same problem.

Some candidates might need to (re)-acquire skills not used since high school, and others need to simply dumb down their approach and get used to basic math again. The latter part is especially true for engineers and other people with a quantitative background.

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

Other than math problems in school, the focus of case interview math is always on the context of the case. Getting to results is only the first step. You need to use your results to make sense of them in the context of the case. In turn, 100% accuracy is not needed.

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

In general, 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 impact of your recommendation, etc.

Many candidates are simply scared of digging into the mathematics of a case. Don’t be that person!

In the following section, we will show you what you need to know to keep you on track during the case interview mental and pen-and-paper math.

Our approach to every case math problem

The challenge of case interview math usually consists of two components . You need to learn an approach that works for every type of case math question, regardless of the case or context. Resources such as Case in Point cannot help you in this regard.

First, you need to derive and lay out a path to get to the desired result. Second, you need to do the calculations to get to the actual numbers.

So, how should you approach quantitative problems during a McKinsey, BCG, or Bain case interview?

• Listen. Carefully and actively listen to what your interviewer tells you
• Clarify. Before you dig into the quantitative problem at hand, slow down: clarify the numbers you heard from the interviewer or extracted from charts or data tables or discuss what additional information you would need. Clarify the desired outcome of your analysis
• Draft your logic. Set up your planned approach to the calculation. For more difficult tasks, you could ask for some time to get your head around the problem with one to two minutes being the upper boundary. Draft your logic on the piece of paper you have been handed by the interviewer
• Communicate your approach. Lead the interviewer through your approach. This way you’ll make sure that mistakes are spotted early.
• Calculate. When the interviewer agrees with your approach, follow through with the calculations – alone and in peace. Again, ask for some time and use the paper
• Sanity check your results. Make sure there are no mistakes in your calculations. Do the numbers make sense?
• Communicate your results top-down. Summarize the result(s) you got in a confident and assertive manner. Do not phrase the answer as a question and really focus on the key results when communicating. There is no need to go through each intermediate step of your calculation. Remember, consulting interview communication should always be top-down and follow the Pyramid Principle
• Come up with a hypothesis. When you get a final result, DON’T STOP THERE. Quickly explain and interpret the numbers. Relate the numbers to the problem at hand. Remember why you set up the calculation in the first place. How does it tie in your planned analysis? How does it impact your hypotheses? Is it the final result or just some intermediate result? It is important to discuss the ‘so what’ of your quantitative analysis.

Be cautious with mental math

A word of caution for mental mathematicians. Write everything down , even when you are doing mental calculations. When you make a mistake while calculating mentally you will face a serious problem. You have no track record of your mistake and would have to re-do all calculations. It is also more difficult to spot your own mistakes in time, which could save you.

Additionally, the interviewer is not able to help you or pick you up from the mistake. At least keep a record of your intermediate results for the interviewer to be able to follow and intervene and for you to quickly go back to find and solve a mistake.

Now that we have discussed the process, let’s discuss the skills needed that help you ace case math.

Typical math problems and key formulas

3 types of case math problems.

90% of case interview math problems fall into one of the following categories and should help you come up with a recommendation on

• Market or segment sizing (e.g., ”how many sports cars can be sold in China in the next 5 years”)
• Operational calculations and decisions (e.g., ”if we reduced the lead time for each production step by 15%, how much time would we save in total?”)
• Investment and other financial and strategic decisions (e.g., ”Investment A would give us 12% annual return and Investment B 5.5% every 6 months – what investment should the client go for?”)

Case math formulas

Market or segment sizing . There is no shame in asking the interviewer for certain numbers in a market sizing exercise (e.g. the population of a specific country), however, to make your life easier you should have some numbers memorized:

• World population
• Population of US, UK, Germany, China, India and other large countries
• Population of countries within your geographical region
• Life expectancy
• Average household size
• Income levels

Operational calculations. To solve operational problems, you would need to come up with the formulas on the spot based on the situation and context of the case. Sometimes, you would work with linear optimization problems to maximize or minimize a certain function (for more, see below).

These two formulas could be a good starting point for your efforts.

• Capacity = total capacity / capacity need per one unit
• Utilization rate = actual output / maximum output

Investment, financial, and strategic decisions. In order to evaluate the financial impact of decisions, these few formulas below are key:

• Profit = Revenue – Cost
• Revenue = Price x Volume
• Cost = Fixed cost (the cost that can’t be changed in the short term) + Variable cost
• Profitability (profit margin) = Profit / Revenue
• Market share = revenue of one Product / Revenue of all products (in one market)
• Growth rate = (New number – Old) / old
• Break-even time = Investment / Profit per specific time frame (e.g., annual)
• Break-even # of sales = Investment / Profit per product
• Return on investment = (Revenue – Cost of investment) / Cost of investment
• NPV (Net present value) = Net cash flow of a period / (1 + discount rate)^number of time periods. The NPV is the present value of the sum of future cash in and outflows over a period of time and used to analyze the profitability of an investment or project

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

McKinsey, BCG, and Bain interviewers want to see highly driven candidates . Show self-initiative. If you hesitate or make mistakes the interviewer will test if this was just an anomaly. They will give you even more calculations in the progress of the case, whereas a candidate that proceeded flawlessly through a calculus process often gets short cuts for the next quantitative parts or whole results readily delivered by the interviewer.

In case you messed up one calculation, don’t mess up the next one! Don’t calculate everything in your head. Use pen and paper to structure the numbers. Choose the fastest and easiest approach to set up your calculations.

Keep calculations organized along the way. Ideally, you find what works for you in the first mock interviews and then apply it consistently (habit-forming).

Re-learn and practice basic calculus

(Re-) learn basic calculus operations and practice until you can do it in your sleep. 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 these calculations both mentally and with pen-and-paper under time-pressure and the vigilant eyes of friends, family, and colleagues. It certainly helps to build resilience and stamina.

Become comfortable thinking quantitatively

Get a feeling for numbers , especially percentages (e.g. be able to instantly estimate percentages in your head, also % of %) and magnitudes. This helps you to interpret results and put them into context and spot more obvious mistakes you made in your calculation.

Be able to interpret your results and demonstrate good business judgment (e.g. if the numbers indicate that the goal of the company seems to be out of reach). Make approximations and estimations quickly and correctly. See the implications of your calculations and conclude correctly (ask the question ‘so what?’).

Relate numbers to each other

Quickly relate numbers and outcomes with each other. You can make a habit out of using equations to describe particular relationships. It both helps your thinking and shows that you are structured in your approach. A brief example:

To improve the over-utilization of train tracks draw up the equation: utilization = demand/ capacity. From this equation, you can instantly see that you need to decrease demand and increase the capacity to improve the utilization situation. Obvious but very effective, such an approach also demonstrates composure and logical thinking.

Sanity check everything

Try to spot your own mistakes before the interviewer does, be vigilant, and sanity-check your approach to the problem and outcome of each calculation. Use your judgment to spot calculation and estimation results that seem out of line in the first place (e.g. 18.3% vs. 183%). Maybe you have done a mistake in the calculations or your assumption was off. In this case, act quickly to re-think and give reasons why your numbers might be off.

Keep track of units

Don’t lose track of your units. Is it kg, tons or USD? Set up the calculation before actually doing them, and 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 labeled.

Be efficient and use shortcuts

Be efficient in your calculations. Most answers rely on multiple assumptions and reasonable estimates anyway, therefore not providing a 100% level of precision. Use shortcuts!

Most of the time, close-to-correct answers are expected. If you come up with a population number use 80mn instead of 82.5mn (state it 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 to 40. Round numbers! Of course, ask the interviewer before you do so. 99% of the time, they will agree. The numbers won’t be 100% accurate either way and are not expected to be. Make plausible shortcuts in your approach and calculation to reach plausible numbers.

Simplify and round numbers

Similar to the above point, try to simplify calculations as much as possible , thereby reducing the number of steps you have to calculate and minimizing the chances of mistakes. As the interviewer, if it is ok to simplify beforehand.

One common way to simplify calculations is to round numbers.

• 83 million Germans become 80 million
• 328 million Americans become 320 or even 300 million
• 365 days in a year become 350 or even 300 days (making some clever assumptions about weekends, etc.)
• USD 983 million in revenue becomes 1 billion

The key with rounding numbers is to know when it is a good time to do so. Sometimes you need precise results, other times you have more leeway.

Some case interview case math questions demand precise results. For example, if you are asked whether or not an investment has an ROI above 12% and you can already spot that it is around that number, it would be a wise decision to calculate with precision.

For instance, if you are asked to size a market, there are a lot of assumptions you will make and there is uncertainty. In such cases, rounding numbers would be the way to go, In general, the more assumptions you use to get to a result, the more you can use the rounding technique.

For the latter, as a rule of thumb, you should round only within a 10% margin; otherwise, you might skew the results and provide a false recommendation based on that.

Also, think about rounding consecutive numbers both up and down sequentially to get a more precise result due to the effects cancelling each other out.

For instance, if you want to calculate the Revenue, which is Volume times Price and the Volume = 9,500 units and the Price = USD 35, calculate with a Volume of 10,000 and a price of USD 30. That will roughly keep you in a 10% margin of the precise result. If you round both up or both down, you would already be around 20% off the precise result.

Take your time

Ask the interviewer for some time to prepare your logic and then again to conduct your calculations. 1 to 2 minutes is fine for the logic and up to 5 minutes are ok for the actual calculations . Do not feel pressured to talk to the interviewer while you are thinking or performing calculations.

Focus on one thing at a time. Then communicate your logic or your result.

Watch the 0s

You would not believe how many candidates fall into this trap. Candidates struggle with large numbers, try to simplify, and then end up losing some 0s along the way. Watch out for 0s that you have trimmed or left out to facilitate calculations.

One way to simplify your calculation and keeping track of the 0s is to use notations such as labels or scientific notations.

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 0s.

One particular way that we find quite useful is to make use of the powers of 10, the scientific notation (e.g. 1.4bn / 70mn = 1.4 * 10 9 / 7 * 10 7 ). You can trim the power of tens and do the simple division. Once you reach a conclusion you are able to immediately see the magnitude of your number (in this case: 0.2 * 10 2 = 20).

How to prepare for case interview mental and pen-and-paper math

There are several things you could do to get up to speed with mental and pen-and-paper math that should suffice for McKinsey, BCG, or Bain interviews. The trick is to be confident in your ability to efficiently do simple math and resilient enough to external pressures in the process.

Get number affine by working with numbers you encounter in your daily life, be it the bar tap or the receipt at the grocery store or figures and data you find in the news (especially business news). Put them into context; create relative numbers and percentages. Try to calculate some simple business cases (e.g. waiting at the doctor: how much profit does he make a month, a year, etc.). The opportunities are unlimited.

Additionally, get some apps for Android or iOS that train your mental math abilities. There are some really fun and entertaining apps out there and we will name a few later in this article.

Do all this in a stressful environment. You want to build stamina and resilience to outside influence and stress. Use the apps in the crowded and noisy subway; calculate in mock interviews, in front of friends and family or simply with a time limit (some apps include this function).

Make sure to be able to recite the 1×1 in your sleep. From there you mostly just add some 0s in case interview calculations.

Basic arithmetic calculations

Learn simple calculus shortcuts and see the examples below as starting points:

Build groups of numbers that add up to ten or multiples of ten.

7+3+12+8+5+5 = 40 (10)+(20)+(10) = 40

Subtraction

Learn quick subtraction by finding out what makes it to ten.

• Reverse the subtraction (5-2 = 3)
• Find what makes it to 10 (3+ 7 = 10)
• Add 1 to the digit on the left of the number you are subtracting

Multiplication

• Multiply any 2 digit numbers within 3 seconds
• Cut the 0s, but be careful to add them again in the end, e.g. change 34x36mn to 34×36 = 1224, then add six 0s –> 1,244,000,000; use the label method we descirbe above
• Break apart multiplications by expanding them. Break one of the terms into simpler numbers, e.g. 18×5 = 10×5+8×5 OR (20-2)x5 = 20×5-10=90
• Exchange percentages and simplify the calculation, e.g. 60×13% = 0.6×13 or 6x 1.3=7.8
• Factor common numbers to simplify your calculations when dealing with multiples of 5, e.g. 17×5 = 17×10 / 2 = 85. The most common numbers to keep in mind are: (5 = 10 / 2; 7.5 = 10×3 / 4; 15 = 10×3 / 2; 25 = 100 / 4; 50 = 100 / 2; 75 = 100×3 / 4
• Convert percentages into divisions, e.g. 33% of 500 = 500*1/3 = 500 / 3 = 167
• Split numbers into tenths, e.g. 60% of 200 = 10% of 200*6 = 120
• Apply factoring and expanding as described for multiplications
• Use the table below to learn the division table and fractions by heart:

Learn the basic case interview math formulas and consulting math tricks.

Other common mathematical operations

For our purpose, we refer to the average as a number expressing the mean value in a set of data, which is calculated by dividing the sum of the values in the set by their number.

In case interviews, calculating the average or a number of averages is very popular, since it is simple, yet demands several calculations to arrive at a result. It is a good pressure test of the candidate.

For example, you might be presented a table with three products, each with different production cost. The produces the same amount per product and you might have to calculate the average production cost across all products.

A variation, which is common, would be 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 in your calculations.

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.

Other common contexts, where you are asked to calculate an average could be

Growth rates

• Geographies and countries
• Product categories and segments

Fractions, ratios, percentages, and rates

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

For instance, fractions can be used to represent a number between 0 and 1, and calculate. 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. Take a look at the fraction table on top to learn these by heart. This will go a long way in your case interview performance.

Ratios are comparisons of two quantities, telling you the amount of one thing in relation to another. If you have 5 apples and 4 oranges, the ratio is 5:4 and you have 9 fruit in total.

In case interviews, one tip is to write ratios as fractions of the total .

Next, percentages are a specific form of ratios, with the denominator always being fixed at 100.

From experience, almost 80% of case interviews will include some reference to or use of percentages. Discussion points such as ”Revenue increased by 15% YTD” or ”Costs are down 4% over the last 6 months” are all too common.

Percentages are also very useful when you want to put things into perspective and state your hypotheses. ”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 4 percentage point increase, but is a 10 percent increase in what is being measured.

Rates are ratios between two related quantities in different units, where the denominator is fixed at 1. 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.

One common type of rate is per unit of time , such as speed or heart rate. Ratios that have a non-time denominator include exchange rates, literacy rates, and many others.

Case interviews will often present you with the following rates:

• Growth rate: the ratio of the change of one variable over a period of time versus the starting level
• Exchange rate: worth of one currency in terms of the other
• Inflation rate: the ratio of the change in the general price level in a given 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 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)

Keep an eye on the timeframe rates are expressed in. This could be annually (p.a.), quarterly, per month, etc. Often, the information is provided for different timeframes or denominators. Convert to the same before conducting your analysis, calculations, or comparisons.

You should be comfortable with calculating growth rates. That is fine for one time period.

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

It gets more tricky when you have to calculate growth over multiple periods. You need to get the compound growth rate first.

• (Increase of 30% in year 1, and 25% in year two): 100 x 1.3 x 1.25 = 100 x 1.625 = 162.5

The latter you can do if you want to calculate growth over 2 to 3 time periods. Everything after that would become tedious. If you want to calculate growth of several periods, it is better to estimate. The common shortcut for this would be to use the growth rate and multiply it with the number of years.

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

If you would use the exact CAGR, you would end up with roughly 137 in our example. The deviation of 5 or roughly 3.5% with your simplification is close enough.

However, be aware that the divergence increases with larger numbers, higher annual growth rates, and the number of years. In a case interview, you could account for that by saying that you would like to add 5-10% to your calculated value.

Optimization problems

In a mathematical optimization problem, you need to either maximize or minimize some function relative to a given set of alternatives. This function is called the objective function.

A typical linear optimization problem would look like this.

A local teddy bear factory wants to optimize its product mix in order to maximize its profit. They produce personalized large and small bears. The profit of a large bear is $25 while the profit of a small bear is $20. Each large bear requires 1kg of material to produce while small bears require 0.65kg each. The daily supply of material is limited to at most 50kg. About 8 bears of either product can be produced per hour. At the moment the family wants to limit their workday to 10 hours.

Your objective function would be to maximize the profit of the factory in this example.

Expected value and outcomes

Sometimes, you will have to compare the impact and success of different recommendations or the expected return of an investment . One way to do this is to work with probabilities and calculate the expected value of 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 and make a decision that is most likely to get to the desired outcome.

For example, if you have to decide between two projects and your analysis shows that Project A will yield USD 50 million with a likelihood of 80% and Project B will yield an outcome of USD 100 million with a likelihood of 20%, you would decide for Project A, with an expected value of USD 40 million (Project B: USD 20 million).

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 of those values for each bundle.

Links to practice resources

Regardless of your skill level, devote some time in your  case interview preparation to brush up your mental and pen-and-paper math skills. If you are struggling with math, there is no reason not to practice case interview math drills 2 hours per day for a few weeks. It is important to have a structured consulting mental practice (as found in our math mastery product).

The basics (free online learning)

If you are just starting out, check out the Khan Academy , which is an excellent source to (re)-learn basic calculus such as additions and subtractions , multiplications and divisions , averages , percentages , and fractions .

Once you have the basics down, you should start practicing using our free Case Interview Math Drill Generator.

Use our 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 Academy

If you want to improve your case interview and problem-solving skills and learn the key habits and tricks that make you succeed in any McKinsey, BCG, and Bain interview, try our case practice . Many students that started their BCG careers, Bain careers, or McKinsey careers, went through our training.

We have created a program with detailed insider learning materials, 25 videos, and a guidebook as well as 2,000 practice drills that mimic the McKinsey, BCG, and Bain case interview math as well as the aptitude and analytics test math for you here:  Case Interview Math Mastery .

Case Interview Math Mastery (Video Course and 2,000 Exercises)

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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 with the goal 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 450 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 years. His books “The 1%: Conquer Your Consulting Case Interview” and “Consulting Career Secrets” are available via Amazon.

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How To Solve A Case Study

Introduction

What is a case study? A case study is a deep study or a detailed examination of a particular case. In politics case studies, it can range from little happenings to huge undertakings like world wars. It is not necessary that a case study only highlights only an individual’s case but it can also highlight groups, belief systems, organizations, or events. Necessarily the case study does did not include only one observation but it may include many observations. In the case of studies projects for research involving several cases are called cross-case research; on the other hand, the study of a single case is called within the case research.

How to solve a case study?

Solving a case study requires deep analyzing skills, the ability to investigate the current problem, examine the right solution, and using the most supportive and workable evidence. It is necessary to take notes, highlight influential facts, and underline the major problems involved. Into days modern times; you can also online case study solutions help by contacting experts on their websites. To make it easier we follow a step-wise procedure to make it understandable. So before you begin Writing the case, follow the step-by-step procedure to get reasonable and desired results.

Step#1: Identify The Case

The first step is about taking notes, highlight the key factors which are being involved, and also introduce the relevant factors which are necessary.

Step#2: Focus Your Analysis

Identify the key problems. Find the reason that why Do they exist? How can they affect the organization or client? Which thing is responsible, and go for their best possible solutions.

Step#3: Realize Possible Solutions

Review all the reading related to the case study course, related discussions, consult it with outside sources, and utilize your experience.

Step#4: Choose The Best Possible Solution

Consider the best and supporting evidence. Its pros and cons, and how realistic it is?. Scan the gathered information again and do not overlook it without focusing on each point.

This is how to solve a case study step by step and easily conclude it while benefiting your clients following these well-researched steps. Additionally following these steps one can also be familiar that how to write case study assignment while getting less confusion.

Examples OF Solving A Case Study

As we know case study involves examining things deeply; for example, we take a case study in medicines. It may be related to an ailment or a patient; a case study in the business sector might cover a broader market; in politics, a case study might range from a narrow happening to a huge undertaking. Let’s discuss How to solve case studies with examples,

Example#1: AnaOwas a woman’s pseudonym of a lady named Bertha. A patient of a famous physical expert Jose Breuer. She was never a patient of another physician Freud. Both physicians Breuer and Joseph, extensively discuss her case. The woman was expecting the symptoms of a disease known as hysteria and it is also found that talking about her issues relieving her a lot and her symptoms. Her case becomes beneficial to understand the fact that therapy of talking has an excellent approach towards mental health.

Example#2: Phineas Gage was an employee in railways. Phineas experiences a scary accident in which a metal rod stuck his skull, damaging a sensitive portion of his brain; although he recovered after that, he comes up with extreme changes in his behavior and personality.

Example#3: Genie a young beautiful girl faced horrifying abuse and isolation. Genie’s case study allows many researchers to examine whether languages could be learned even after hectic times for developing language had vanished for her. Her case also enables everyone to understand that how interference of scientific researches leads to more abuse of a vulnerable person. One can also consult their mentors to understand which case study to buy and get the valuable guidance.

Benefits and Limitations OF A Case Study

A case study could have both strengths and drawbacks. Here we discuss its good and bad things in the form of bullets. First of all, we get to know about its pros.

• It allows investigators and researchers to attain high-level knowledge.
• Give them a chance to attain valuable information from Unusual and rare cases.
• Allow the individuals for research to develop their hypotheses to explore them at experimental research.

Along with its pros, case studies have their cons too. Let’s discuss them in bullets.

• The case study cannot briefly demonstrate cause and influences.
• It cannot be generalized in public.
• It can also lead to bias.

Bottom Lines

Generally, case studies can be included in many different fields like education, anthropology, psychology, medicines, and political sciences. We have discussed in the article above about case study definition, how to solve it, and also include the examples for your better understanding. Hoping that this article will play its part in building your knowledge about studying a case.

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Why Calculus Remains a Math Flash Point

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Corrected : This story has been updated to reflect Ralph Pantozzi’s full statement. Corrected : A previous version of this story misstated the location of Kent Place School. It is located in Summit, N.J.

Calculus has long been one of the most-debated flash points in high school math.

The course is commonly seen as the pinnacle of the high school progression, a clear signal to college admissions counselors that graduates are ready for postsecondary study. But many in the K-12 field question whether it’s really the best mathematical preparation for all students.

And the course is plagued by inequities— data from the U.S. Department of Education’s office of civil rights has shown that Black and Latino students have less access to calculus in their schools than their white and Asian peers. Some high schools don’t even offer the class.

How to design calculus courses, and who exactly should take them, was the focus of a panel on the subject at the National Council of Teachers of Mathematics Annual Meeting here last week.

“The math education community has been thinking and asking questions about the learning and teaching of calculus for many decades,” said Ralph Pantozzi, a math teacher at Kent Place School in Summit, N.J. “Calculus for whom, and when? What should a calculus course look like and feel like?”

Pantozzi cited NCTM’s most recent position statement on the subject , released in June 2022. It says that calculus can provide important foundations for future studies, “particularly in mathematically intensive fields.”

But it also argues that calculus “should not be the singular end goal of the PK–12 mathematics curriculum at the expense of providing a broad spectrum of mathematical preparation.”

Some experts in the math education field have suggested that high schools offer alternative data science pathways , which could provide students with preparation in statistical analysis that could support them in a wide range of college majors and career fields.

A few states, including Ohio, Oregon, and Utah , have created high school math pathways that encourage students to take different advanced math courses, based on their career interests.

Still, these changes tend to be controversial, given the important role calculus plays in the college admissions process, and the urging from some in postsecondary math education that it is a necessary foundation for incoming college students who are interested in majoring in a STEM field.

Read on for three takeaways about the purposes calculus serves—and doesn’t serve—from the panelists’ conversation.

Calculus is still an important admissions factor for many colleges

Just Equations, a nonprofit that advocates for greater educational equity in math, surveyed college admissions counselors and high school counselors in 2021 and 2022. Of the 1,250 selective four-year colleges and universities Just Equations contacted, 137 responded: 58 percent private institutions and 42 percent public.

“What we learned is that they do indeed look at calculus as the gold standard in this business,” said Melodie Baker, the national policy director at Just Equations.

Eighty percent of admissions officers said that colleges place a priority on calculus, and 53 percent said that having taken calculus gives students an edge in the admissions process.

“While this is often practiced, it’s not an actual policy,” said Baker. Even though college admissions officers hold these beliefs, many colleges don’t explicitly state these preferences in their admissions materials or on their websites, she said.

This opacity in the process can disadvantage lower-income or first generation students, Baker said. “Students who come from wealthy backgrounds are more likely to know the role that calculus plays in college admissions,” she said.

In a separate 2023 survey of high school and college students, mostly in California, Just Equations found that 60 percent of students whose parents went to college agreed with the statement, “Students who take calculus are more likely to be admitted to elite or highly selective colleges.” Only 40 percent of students whose parents had not attended college said the same.

Not all students have access to the course

“What a lot of college admissions counselors do not know is that calculus is truly an issue with access,” said Baker. “Only 50 percent of the high schools in the country even offer calculus. And the ones that do have lower enrollment of Black and Latinx students.”

Data from the U.S. Department of Education’s office of civil rights show that highly segregated schools with majority Black or Latino enrollment are much less likely to offer calculus—only 38 percent of these schools offer the course.

Colleges champion calculus, but it’s unclear whether it’s necessary for all students

Even in states that have created alternative high school pathways, state guidance to students still recommends that those interested in STEM fields take calculus. But what about students who are interested in other disciplines? Do they need the subject too?

Joan Zoellner, the lead of the Launch Years Initiative at the University of Texas at Austin’s Charles A. Dana Center, shared data from the Research, Planning & Professional Development Group for California Community Colleges examining the performance of community college students in their first postsecondary math course .

All of these students were pursuing a business administration degree, and needed to pass a business calculus course to graduate.

Students who had previously taken a high school calculus course did well in business calculus—71 percent of them who took business calculus in their first year passed the class.

But many students who hadn’t taken calculus in high school succeeded in business calculus, too. Sixty-eight percent of students who had taken high school statistics and enrolled directly in college business calculus passed in their first year, as did 63 percent of students who had ended high school with precalculus.

“We don’t really know what the make or break skills, or habits of mind, are for [college] calculus,” said Zoellner.

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HW Discussion Module 8: "After Math" Case Study in Communication...

HW Discussion Module 8: "After Math" Case Study in Communication...

HW Discussion Module 8: "After Math" Case Study in Communication and Emotional Intelligence

33 unread replies.33 replies.

Discussion - Read the Case Study Below and Add to the Discussion (based on Skip Downing's four components of emotional intelligence Chapter ...)

Due:  By November 7, 2023

Case Study in Critical Thinking: "After Math"

When  Professor Bishop  returned midterm exams, he said, "In 20 years of teaching math, I've never seen such low scores. Can anyone tell me what the problem is?" He ran a hand through his graying hair and waited. No one spoke. "Don't you people even care how you do?" Students fiddled with their test papers. They looked out of the window. No one spoke.

Finally, Professor Bishop said, "Okay, Scott, we'll start with you. What's going on? You got a 35 on the test. Did you even  study? "

Scott , age 18, mumbled, "Yeah, I studied. But I just don't understand math."

Other students in the class nodded their heads. One student muttered, "Amen, brother."

Professor Bishop looked around the classroom. "How about you, Elena? You didn't even show up for the test."

Elena , age 31, sighed. "I'm sorry, but I have a lot of other things besides this class to worry about. My job keeps changing my schedule, I broke a tooth last week, my roommate won't pay me the money she owes me, my car broke down, and I haven't been able to find my math book for three weeks. I think my boyfriend hid it. If one more thing goes wrong in my life, I'm going to scream!"

Professor Bishop shook his head slowly back and forth. "Well, that's quite a story. What about the rest of you?" Silence reigned for a full minute.

Suddenly  Michael , age 23, stood up and snarled, "You're a damn joke, man. You can't teach, and you want to blame the problem on us. Well, I've had it. I'm dropping this stupid course. Then I'm filing a grievance. You better start looking for a new job!" He stormed out of the room, slamming the door behind him.

"Okay, I can see this isn't going anywhere productive," Professor Bishop said. "I want you all to go home and think about why you're doing so poorly. And don't come back until you're prepared to answer that question honestly." He picked up his books and left the room. Elena checked her watch and then dashed out of the room. She still had time to catch her favorite reality show in the student lounge.

An hour later, Michael was sitting alone in the cafeteria when his classmates Scott and  Kia , age 20, joined him. Scott said, "Geez, Michael, you really went off on Bishop! You're not really going to drop his class, are you?"

"Already did!" Michael snapped as his classmates sat down. "I went right from class to the registrar's office. I'm outta there!"

I might as well drop the class myself,  Kia thought. Ever since she was denied entrance to the nursing program, she'd been too depressed to do her homework. Familiar tears blurred her vision.

Scott said, "I don't know what it is about math. I study for hours, but when I get to the test, I get so freaked it's like I never studied at all. My mind just goes blank." Thinking about math, Scott started craving something to eat.

"Where do you file a grievance against a professor around here, anyway?" Michael asked.

"I have no idea," Scott said.

"What?" Kia answered. She hadn't heard a word that Michael or Scott had said. All she could think about was how her whole life was ruined because she would never be a nurse.

Michael stood and stomped off to file a grievance. Scott went to buy some French fries. Kia put her head down on the cafeteria table and tried to swallow the burning sensation in her throat.

1.There are five characters in this story. Rank them in order of their emotional intelligence. Give a different score to each character and explain your reason. Refer to the four components of emotional intelligence (emotional self-awareness, emotional self-management, social awareness, relationship management - Downing pages )

Least emotionally intelligent  1 2 3 4 5  Most emotionally intelligent

NEXT, consider these questions in your response.

2. Could any of the characters communicated differently? Did the characters consider the variables present in the communication methods?

3. What emotional choices do you think would have earned more positive results?

4. How can we control our communication strategies? Were the characters good listeners?

5. Do you see barriers to effective communication in this case study? 

Contribute to the discussion by considering all the ideas above. 

Answer & Explanation

1. Ranking the characters in the story in terms of emotional intelligence:

Michael - 2

Professor Bishop -  4

• Scott: Scott demonstrates some level of emotional intelligence as he openly acknowledges his struggles with math and communicates them to the professor. He shows self-awareness about his difficulties and expresses his frustration but doesn't react aggressively.
• Michael: While Michael stands up for his beliefs and takes action, his outburst and aggressive behavior indicate a lack of emotional self-management and social awareness. He doesn't effectively consider the variables present in his communication method.
• Professor Bishop: The professor maintains a relatively calm demeanor and tries to engage with the students. He exhibits emotional self-management and social awareness. However, he doesn't delve deeper into the students' concerns and doesn't provide much support.
• Elena: Elena expresses multiple stressors in her life but doesn't effectively communicate her challenges with the professor. She lacks emotional self-management and social awareness, and her communication focuses on external stressors without addressing her struggles with the class.
• Kia : Kia is the least emotionally intelligent in this context. She is completely absorbed in her own negative emotions and doesn't engage in any form of communication. She doesn't consider variables in the communication methods or express her concerns to others.

2. Yes, the characters could have communicated differently. They did not effectively consider the variables present in the communication methods. For example, Michael's aggressive outburst didn't lead to a productive conversation, and Elena's focus on external stressors rather than her academic challenges didn't effectively communicate her difficulties. The characters could have engaged in more constructive and empathetic conversations.

3. More positive results could have been achieved if the characters had engaged in empathetic and open communication. For instance, if Michael had expressed his concerns calmly and constructively, he might have had a more productive conversation with the professor. Elena could have directly communicated her academic struggles and sought support. Positive results could be achieved by focusing on emotional self-awareness and relationship management.

4. To control communication strategies, individuals should practice emotional self-awareness, which involves understanding their own emotions and how they affect their communication. Good listening skills are essential, as they allow individuals to understand others' perspectives and respond effectively. In this case, the characters didn't exhibit strong listening skills.

5. Barriers to effective communication in this case study include:

• Lack of emotional self-awareness: Most characters struggle to express their emotions effectively.
• Poor emotional self-management: Michael's outburst is an example of inadequate emotional self-management.
• Limited social awareness : Characters often fail to consider the feelings and perspectives of others.
• Inadequate relationship management : The characters don't effectively manage their relationships, which could have been improved with more empathetic and constructive communication.

n ranking the characters in terms of emotional intelligence, it's evident that Professor Bishop emerges as the most emotionally intelligent in this scenario. He demonstrates emotional self-management by maintaining a calm demeanor, social awareness by attempting to engage with the students, and relationship management by addressing their concerns, even though the conversation doesn't delve deeply into the issues. Professor Bishop's ability to remain composed in the face of student frustration showcases his emotional intelligence.

Following Professor Bishop is Scott, who displays some level of emotional intelligence by openly acknowledging his academic struggles and expressing his frustration. Scott's self-awareness about his difficulties and his willingness to communicate them is a step toward addressing the problem constructively. Michael, although he stands up for his beliefs, falls short in terms of emotional self-management and social awareness due to his aggressive outburst. Elena's communication centers on external stressors, lacking emotional self-management and social awareness. Lastly, Kia exhibits the least emotional intelligence as she withdraws into her own negative emotions and fails to engage in any form of communication, highlighting the absence of emotional self-awareness and relationship management.

In this context, the characters could have communicated more effectively by practicing emotional self-awareness and self-management, which would involve acknowledging their emotions and considering how these emotions impact their communication. Constructive conversations, empathy, and active listening could have led to more positive outcomes. The absence of these elements, as well as barriers such as a lack of self-awareness, emotional self-management, and social awareness, hindered the effectiveness of communication in this case study.

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https://educationhub.blog.gov.uk/2023/11/08/get-number-confident-and-unlock-your-potential/

How to boost your maths skills for free

Maths skills are vital for everyday life and work. Adults with a good grasp of numbers have a greater chance of getting a job and having higher wages.

If you want to improve your confidence with numbers, there is a range of help available, and much of it supported by the government and free of charge.

Being number-confident can help you take control and be better prepared to deal with everyday tasks, such as managing personal or work finances, working out how to split a bill between friends or even baking.

It can also prepare you for further study at college, university or work.

What help is available to gain a maths qualification?

Many adults across the country aren’t confident with numbers. If you have not already achieved a maths GCSE or equivalent qualification you can gain one for free.

Our free maths courses allow people to gain either a GCSE or a Functional Skills Qualification. You can find a course near you on the National Careers Service website .

Shaunna 31, completed a free maths course at her local college funded by the government and has never looked back.

“I was bullied at school which led to me leaving at 15 with no formal qualification in maths,” she told the Education Hub.

“After my daughter was born, I realised I wanted to get into teaching, but my maths skills were poor, and I had an irrational fear of doing maths.

“The maths course I completed was a great stepping-stone as it gave me the skills and the confidence boost I needed to go on to further study. Also, I now enjoy maths which is something I never thought possible.

“Following this course, I went on to achieve an English, Science and Maths GCSE, and have recently finished university this year receiving the Dean's Award for my contribution to primary science.

“With the confidence I have gained from improving my numeracy skills as an adult, I can now continue my learning journey and I plan to complete a postgraduate certificate in education (PGCE) qualification soon.”

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Our Multiply programme also helps people improve their basic numeracy through free courses that fit around your life. They boost opportunities for people to take the first steps into maths learning.

Courses are available on a part time or intensive basis.

This allows people to take a formal qualification that will open doors for them and help them to progress and secure well-paid jobs.

If you’re interested in accessing free support with maths and boosting your earning potential, you can take our Multiply quick quiz or have a look at what math courses available near you here.

You may also be interested in:

• Maths to 18: Is maths A Level being made compulsory for 16 to 18-year-olds?
• Studying maths to 18 – what you need to know
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Computer Science > Computation and Language

Title: do llms exhibit human-like response biases a case study in survey design.

Abstract: As large language models (LLMs) become more capable, there is growing excitement about the possibility of using LLMs as proxies for humans in real-world tasks where subjective labels are desired, such as in surveys and opinion polling. One widely-cited barrier to the adoption of LLMs is their sensitivity to prompt wording -- but interestingly, humans also display sensitivities to instruction changes in the form of response biases. As such, we argue that if LLMs are going to be used to approximate human opinions, it is necessary to investigate the extent to which LLMs also reflect human response biases, if at all. In this work, we use survey design as a case study, where human response biases caused by permutations in wordings of ``prompts'' have been extensively studied. Drawing from prior work in social psychology, we design a dataset and propose a framework to evaluate whether LLMs exhibit human-like response biases in survey questionnaires. Our comprehensive evaluation of nine models shows that popular open and commercial LLMs generally fail to reflect human-like behavior. These inconsistencies tend to be more prominent in models that have been instruction fine-tuned. Furthermore, even if a model shows a significant change in the same direction as humans, we find that perturbations that are not meant to elicit significant changes in humans may also result in a similar change, suggesting that such a result could be partially due to other spurious correlations. These results highlight the potential pitfalls of using LLMs to substitute humans in parts of the annotation pipeline, and further underscore the importance of finer-grained characterizations of model behavior. Our code, dataset, and collected samples are available at this https URL

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arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .

Stanford engineer's study show CO2 pipelines costlier, dirtier than wind and solar

“Mathematics,” once explained Edward Frenkel , a renowned mathematician and author, “directs the flow of the universe, lurks behind its shapes and curves, [and] holds the reins of everything from tiny atoms to the biggest stars.”

Another explanation notes that “Math is the only place where truth and beauty mean the same thing.”

That elegant, beautiful truth comes to mind when reading a just-published “case study” that compares the cost and net carbon dioxide (CO2) output of the planned 2,000-mile Summit CO2 pipeline to the wind- and solar-based electricity that could fuel “battery-electric vehicles,” or BEVs.

Farm and Food: Republicans elected a speaker. Now for the really hard, chaotic part.

At its heart, the new study asks, what gives the better environmental and financial return — billions spent on a CO2 pipeline that encourages more ethanol use or investing the same amount on solar and wind generators to power BEVs?

The resulting math, presented by the study’s author, Mark Z. Jacobson , a civil and environmental engineer at Stanford University, is detailed, compelling and irrefutable in its conclusion: Don’t spend another penny on Summit’s five-state, CO2 pipeline.

Jacobson's focus is on E85, the 85-percent-ethanol-to-gasoline blend now being pushed by farm groups as the way to maintain production of the corn-based fuel even as BEVs rise in popularity. (The U.S. Department of Agriculture forecasts that 35% of the 2023/24 U.S. corn crop, or 5.3 billion bushels, will be used to make ethanol.)

The higher ethanol blend, however, doesn’t change the hard math underlying the colossal investment and environmental costs of CO2-generating ethanol plants, ethanol’s use or any of the three proposed CO2-carrying pipelines, Jacobson writes.

“This study concludes that investing in wind turbines to provide electricity to BEVs is far more beneficial in terms of consumer cost savings, CO2 emissions, land use, and air pollution than making the same investment in a plan to capture CO2 from ethanol refineries, pipe the CO2 to an underground storage facility, and use the ethanol to produce E85 for FFVs,” or flex fuel vehicles.

Moreover, Jacobson continues, quoting his revealing math, “The fuel savings alone,” an estimated “$66.9-$111 billion over 30 years”–“is 12-20 times the $5.6 billion investment in [the] Summit [pipeline] project.”

And just to drive home his rural bona fides, Jacobson offers a comparison of the costs to operate competing models of the ever-popular, four-wheel-drive, Ford F-150 — the eight-cylinder flex fuel version that costs “$48,290” — versus its electric twin that costs $21,705 more or “$69,995.”

“Even with this upfront cost difference … the net fuel cost saving to drivers over 30 years (of the electric F-150) is still … 7-12 times Summit’s investment [cost],” he notes.

Environmental costs between the two technologies — wind and solar versus ethanol — show even bigger differences because wind and solar electric generation are zero-carbon emitters and easily beat almost any blend of any fossil or bio-based fuel.

A key element of ethanol’s argument, however, is Big Ag’s insistence that it’s a “green” fuel that, at worst, is carbon neutral and, at best, is carbon negative. Still, no outside-of-ag scientist supports ag’s contention and neither does the Stanford engineer. His math shows more brown and no green.

“With respect to air pollution, tailpipe emissions from E85 vehicles may increase the level of ozone throughout most of the United States in comparison with tailpipe emissions from gasoline vehicles.”

“Moreover,” he adds, “the production, transport, and refining of corn to produce ethanol creates air pollution that may exceed the upstream pollution from gasoline.”

Market Basket: Taste of the tropics: Bululu offers Venezuelan cuisine and treats at Town & Country

The engineer’s hammering math dives into other aspects of the bad bargain that is CO2 pipelines. For example, photosynthesis “is only 1% efficient” while solar panels are “20-23% efficient” and therefore need “only 1/20th of the land to produce the same energy as a biofuel crop.”

The clear bottom line to the Summit pipeline — and, really, any CO2 pipeline — Jacobson says, is as obvious as one-plus-one: Don’t bury CO2 pipelines; instead, bury their very idea. Fast.

Alan Guebert is an agricultural journalist. See past columns at  farmandfoodfile.com . © 2023 ag comm