case study maths questions

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.

• The 1%: Conquer Your Consulting Case Interview
• Consulting Career Secrets
• Cover Letter & Resume
• McKinsey Solve Game (Imbellus)
• BCG Online Case (+ Pymetrics, Spark Hire)
• Bain Aptitude Tests (SOVA, Pymetrics, HireVue)
• Kearney Recruitment Test
• All-in-One Case Interview Preparation
• Industry Cheat Sheets
• Structuring & Brainstorming
• Data & Chart Interpretation
• Case Math Mastery
• McKinsey Interview Academy
• Brainteasers

Case Interview Math: The Insider Guide

Last Updated on March 27, 2024

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

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

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

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

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

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

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

Why Candidates Struggle with Case Interview Math

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

Let’s start with some positivity.

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

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

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

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

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

Let’s get to it!

Case Math Mastery Course and Drills

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

The Purpose of Case Interview Math

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

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

Identifying problems and quantifying their impact

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

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

Supporting recommendations

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

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

Test of your quantitative skills

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

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

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

A simplified version of reality

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

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

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

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

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

The myth of perfection

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

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

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

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

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

Effective Strategies for Tackling Case Interview Math Questions

Different skill levels, same problem.

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

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

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

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

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

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

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

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

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

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

My approach to every case math problem

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

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

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

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

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

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

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

Exercise caution with mental math

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

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

Typical Case Interview Math Problems and Key Formulas

3 types of case math problems.

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

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

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

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

Case math formulas

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

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

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

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

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

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

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

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

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

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

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

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

Case Interview Math Tips and Tricks

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

Tackle the problems aggressively

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

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

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

Re-learn and practice basic calculus

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

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

Consider the numerical impact in your analysis

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

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

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

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

Express problems quantitatively

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

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

Sanity check everything

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

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

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

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

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

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

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

Keep track of units

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

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

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

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

Use shortcuts in your approach

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

A couple of examples:

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

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

Simplify and round numbers

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

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

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

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

What are the best practices related to rounding?

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

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

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

Take your time

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

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

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

Watch the 0s

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

• scientific notation.

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

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

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

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

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

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

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

Case Interview Math Practice Questions

Practice case math question #1.

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

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

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

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

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

Market size calculation

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

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

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

Revenue projections

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

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

Profitability analysis

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

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

Break-even analysis

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

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

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

Practice case math question #2

Let’s look at another example:

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

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

Average cost per unit:

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

Capacity utilization

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

Recommendation

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

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

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

Mental Math Concepts and Shortcuts

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

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

Basic arithmetic calculations

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

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

Build groups of 10

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

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

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

Go from left to right

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

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

Subtractions

Make it to 10

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

For instance: 4 2 – 2 5

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

Let’s use another example: 3853 – 148

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

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

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

42 – 25 = (42 – 20) – 5 = 22 – 5 = 17

Multiplications

Get rid of 0s

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

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

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

Factor with five

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

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

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

Exchange percentages

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

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

Convert to yearly data

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

Convert percentages

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

Split into 10ths

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

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

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

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

Advanced case math concepts

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

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

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

Fractions, ratios, percentages, and rates

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

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

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

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

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

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

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

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

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

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

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

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

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

Growth rates

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

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

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

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

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

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

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

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

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

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

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

Expected value and outcomes

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

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

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

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

Avoid the Most Common Pitfalls and Mistakes

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

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

What Should You Do When You Get Stuck?

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

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

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

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

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

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

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

How to Prepare for Case Interview Math

General practice recommendations.

Incorporate case interview math practice into your preparation plan .

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

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

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

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

Free practice resources and habits

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

Practice with the following free iOS and Android apps.

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

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

Free case interview math drill generator

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

You can access the tool for free here:

Our Case Interview Math Mastery Academy

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

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

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

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Frequently Asked Questions: Case Interview Math

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

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

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

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

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

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

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

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

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

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

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

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

Struggling with case interview math?

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

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

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Case Interview Math: 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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Hacking the Case Interview

Case interview math includes basic arithmetic, percentages, ratios and proportions, various formulas , data interpretation, and market sizing and estimation. Having strong case interview math skills is essential to pass consulting interviews.

Case interview math will show up in every case interview. If you cannot solve case interview math problems quickly, efficiently, and correctly, you will not be able to pass your consulting interviews and land a consulting job offer.

Consulting math may seem difficult and intimidating, especially for candidates without strong quantitative backgrounds or candidates that have not done math in a long time, but the math itself is very basic and simple.

If math is your weak point in case interviews, we have you covered.

In this comprehensive article, we’ll go through all of the case interview math fundamentals and formulas you need to know. We’ll also cover the five types of consulting interview math problems and essential case interview math tips to help make the math easier and simpler for you.

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

Case Interview Math Fundamentals

You should be familiar with the following case interview math fundamentals:

Percentages

• Ratios and proportions
• Linear Equations

Fractions are numbers expressed as n/d. Examples include: 1/2, 2/3, and 3/4. You should know how to do the following calculations with fractions:

• Simplification : 36/54 = 2/3
• Addition : 1/3 + 1/4 = 4/12 + 3/12 = 7/12
• Subtraction : 1/3 – 1/4 = 4/12 – 3/12 = 1/12
• Multiplication : 3/4 * 4/9 = 12/36 = 1/3
• Division : 3/4 ÷ 9/2 = 3/4 * 2/9 = 6/36 = 1/6

Decimals are numbers in which the positioning of the period or decimal point determines the value of the digits.

Example :   1,542.36 = (1 * 1,000) + (5 * 100) + (4 * 10) + (2 * 1) + (3 * 0.1) + (6 * 0.01)

You should know how to do the following calculations with decimals:

• Addition : 1.21 + 3.5 = 4.71
• Subtraction : 14.2 – 8.05 = 6.15
• Multiplication : 1.12 * 5 = 5.6
• Division : 14.7 ÷ 4.2 = 3.5

You should also be familiar with scientific notation, which is a decimal expressed as the product of a number with only one digit to the left of the decimal point and a power of 10.

Example : 2.731 * 10^2 = 273.1

Percentages are numbers expressed out of 100 or per hundred.

Example : 75% = 75/100 = 0.75

You should know the formula for percent change.

Percent Change = (New Value – Old Value) / Old Value

Example : The price of gas last month was $3 per gallon. This month, the price increased to $3.60 per gallon. By what percentage did the price of gas increase?

Percent Change = ($3.60 - $3) / $3 = 20%

The price increased by 20%.

Ratios and Proportions

A ratio is a relation between two numbers. Examples include:

A proportion is a relation between two equal ratios. Examples include:

• 2 to 3 is to 4 to 6

Example : A factory requires 2 supervisors for every 15 workers. If a factory has 45 workers, how many supervisors are required?

We know that the ratio of supervisors to workers is 2 to 15 or 1 supervisor for every 7.5 workers.

To find the number of supervisors required, we can divide 45 workers by 7.5 to get 6. Therefore, 6 supervisors are required.

Statistics 

Only very basic statistics knowledge is needed for case interviews. You should be familiar with four important concepts:

• Mean or average
• Weighted average
• Standard deviation
• Expected value

The most important concept is calculating the mean or average of a set of numbers.

Example : If our three largest competitors are paying $10 per hour, $11 per hour, and $15 per hour for labor. What should we expect our average labor cost to be?

Average = ($10 + $11 + $15) / 3 = $36 / 3 = $12

The average cost of labor is $12 per hour.

You should also be able to calculate a weighted average, which is a calculation that takes into account the varying degrees of importance of numbers in a data set.

Example : Laptops have a profit margin of 20% while repair services have a profit margin of 60%. If 70% of our company’s revenues comes from laptops and 30% of revenues comes from repair services, what is the overall profit margin of the company?

To determine the average profit margin of the company, we take the profit margins of each product or service and multiply it by their respective weights. In this example, the weights are determined by the percentage of the company’s total revenue.

Profit Margin = (20% * 70%) + (60% * 30%) = 14% + 18% = 32%

The overall profit margin of the company is 32%.

You should also understand the concept of a standard deviation, which is a measure of how spread out the distribution of data is from the mean or average. You won’t need to ever calculate this in an interview, but should be familiar with how to interpret it.

A large standard deviation means that data points are far from the mean and are spread out. A small standard deviation means that data points are clustered closely around the mean.

Finally, know how to calculate expected value, which is a predicted value calculated as the sum of all possible values multiplied by the probability of occurrence.

Example : A fashion design company is considering partnering with a national retailer to sell its clothing. If the partnership were to happen, there is a 40% likelihood that the partnership will be a success and the company will achieve $120M in sales in the first year. However, there is a 60% likelihood that the partnership will be less successful, generating only $40M in sales in the first year. What is the expected value of sales in the first year?

Expected value = (40% * $120M) + (60% * $40M) = $72M

The expected value of sales is $72M in the first year.

Linear   equations

A variable is a letter or symbol that represents an unknown quantity.

Only very basic algebra is required for case interviews. You should know how to solve a linear equation that has one unknown.

To solve a linear equation with one unknown, you will need to isolate the unknown variable onto one side of the equation.

Remember that whatever operation you do on one side of the equation, you must do the same operation on the other side of the equation.

Example : A company’s revenues grew by 60% this year compared to last year. If this year’s revenue was $100M, what was the company’s revenues last year?

To solve this question:

• Let x = the company’s revenues last year
• 1.6x = $100M
• The unknown variable is already isolated on one side of the equation, so we just need to divide both sides of the equation by 1.6

The company’s revenues last year were $62.5M.

Case Interview Math Formulas

There are a small number of consulting math formulas you should know for case interviews: profit formulas, investment formulas, operations formulas, market share formulas, and finance formulas.

For a complete guide to all formulas you need to know, check out our article on case interview formulas .

Profit formulas

1. Profit = Revenue – Costs

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

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

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

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

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

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

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

3. Contribution Margin = Price – Variable Cost

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

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

4. Profit Margin = Profit / Revenue

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

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

Investment formulas

5. Return on Investment = Profit / Investment Cost

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

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

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

6. Payback Period = Investment Cost / Profit per Year

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

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

Operations formulas

7. Output = Rate * Time

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

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

8. Utilization = Output / Maximum Output

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

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

Market share formulas

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

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

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

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

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

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

Example 1 : Your company has a 20% market share in the shirts market. Your largest competitor has a 50% market share. Therefore, your relative market share is 20% / 50% = 0.4.

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

Finance formulas

11. Gross Profit = Sales – Cost of Goods Sold

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Types of Case Interview Math Problems

Now that you know the consulting math fundamentals, you can move onto learning the five types of math problems you’ll likely encounter in case interviews:

Profit and Breakeven Questions

Investment Questions

Operations Questions

Charts and Graphs Questions

Market Sizing Questions

For each type of problem, we’ll go through what formulas and concepts you need to know and go through a few practice questions.

Profit questions typically involve calculating revenue, costs, and profit.

Revenue is the income a business generates from its operations. Costs are the expenses the business incurs in running its operations. Profit is the earnings that a business keeps after paying all of its expenses.

There are four basic profit formulas that you should know, which can be put together and simplified into a single comprehensive equation for profit.

Example : A pizza store sells 100,000 pizzas per year at a price of $10 each. Assume that each pizza costs $4 to produce. The store pays $150K per year in rent and hires two employees at a salary of $75K per year. What is the pizza store’s annual revenues, costs, and profit?

To solve this, we simply need to use our various profit formulas.

• Revenue = Quantity * Price = 100,000 pizzas * $10 = $1M
• Variable Costs = Quantity * Variable Cost = 100,000 pizzas * $4 = $400K
• Fixed Costs = $150K + (2 * $75K) = $300K
• Costs = Variable Costs + Fixed Costs = $400K + $300K = $700K
• Profit = Revenue – Costs = $1M - $700K = $300K

From our calculations, we know that annual revenues are $1M, annual costs are $700K, and annual profit is $300K.

Another type of question are profit margin questions. Profit margin measures how much earnings a business keeps relative to its revenues. You need to know the profit margin formula:

• Profit Margin = (Revenue – Costs) / Revenue
• Profit Margin = Profit / Revenue

Example : What is the profit margin of the pizza store in the previous example?

To solve this, we simply need to use the profit margin formula.

Profit Margin = Profit / Revenue = $300K / $1M = 30%

Therefore, the pizza store has a profit margin of 30%.

The third type of question you may be asked are breakeven questions . A breakeven occurs when a company sells enough product such that it has exactly recouped all costs. In other words, breakeven occurs when profit is zero.

To solve for the breakeven point, simply set profit equal to zero and solve for the unknown variable in the equation.

Example : How many pizzas does the pizza store in the previous example need to sell in order to break even?

For this question, the unknown variable we are solving for is quantity. Therefore, set profit equal to zero and solve for this unknown variable.

• Price = $10
• Variable Costs = $4
• Let Q = quantity needed to break even
• Profit = (Price – Variable Costs) * Quantity – Fixed Costs
• Set profit equal to zero
• $0 = ($10 - $4) * Q - $300K

Therefore, the pizza store needs to sell 50,000 pizzas in order to break even.

Investing is the use of cash today in the hopes of creating wealth in the future. For case interviews, examples of investments you’ll see include:

• A company spending money on marketing to generate sales
• A company spending money to build a new product to increase revenues
• A company spending money on infrastructure to decrease costs in the long-term
• A company spending money to acquire a company to increase revenues
• A private equity firm acquiring a company to resell in later years for a profit

Investment questions will typically ask you to calculate a return on investment, known as ROI, and the time required to break even, also known as the payback period.

You will need to know the following two formulas:

• Return on Investment = Profit / Investment Cost
• Payback Period = Investment Cost / Profit per Year

Example 1 : A software company acquires a machine learning startup for $20M. The startup’s expertise in machine learning is expected to improve the software company’s product such that it will generate an additional $4M per year in profit. What is the payback period?

Payback period = Investment Cost / Profit per Year = $20M / $4M = 5 years

It will take 5 years for the software company to recover the cost of the investment.

Example 2 : A private equity firm purchases a roofing tile distributor for $100M. After growing the company’s revenues within the first few years, the company was then sold for $130M. What was the return on investment?

Return on Investment = Profit / Investment Cost = $30M / $100M = 30%

The private equity firm has generated a 30% return on investment over the time period.

Operations problems typically deal with production and the usage of machinery or plants. There are two key formulas you need to know:

• Output = Rate * Time
• Utilization = Output / Maximum Output

Example 1 : If it takes a factory 2 hours to produce 100 cars, how many cars can the factory produce in a full day?

Assuming the factory operates for 24 hours, we can calculate the hourly production rate by taking 100 cars and dividing it by 2 hours. This gives us a rate of 50 cars per hour.

Therefore, the factory’s maximum output in a day is 50 cars per hour times 24 hours. This gives us 1,200 cars per day.

Example 2 : If the factory produces 900 cars per day due to lunch breaks and shift changes, what is the factory’s utilization?

To find utilization, we divide the output by the maximum possible output. 900 cars per day divided by 1,200 cars per day gives us 75%. The factory is operating at 75% efficiency.

There are ten types of charts and graphs you should be familiar with:

• Simple bar chart : compares a single variable
• Stacked bar chart : compares a single variable and shows the segments that make up each bar
• 100% stacked bar chart : shows the percentage each segment makes up in a bar, enabling percentage comparisons across bars
• Pie chart : shows the percentage each segment makes up of a total
• Marimekko / Mekko chart : a more complex version of a pie chart that shows the percentage each segment makes up of a total across two dimensions
• Waterfall chart : shows drivers of change between two comparison points
• Histogram : shows frequency distributions
• Line graph : shows changes in a variable over time
• Scatterplot : shows the relationship between two variables
• Bubble chart : a more complex version of a scatterplot that shows the relationship among many variables

You can watch the video below for a review on how to interpret each of these.

A comprehensive guide to market sizing can be found in our market sizing article.

To summarize, market sizing questions ask you to determine the size of a particular market.

Market size is defined as the total amount of sales of a product or service in one year in a given geography or region. However, it can also be defined as the number of units sold in a year or the total number of customers that would purchase a product or service.

There are two main approaches to solving market sizing questions:

• Top-down approach : start with a large number and then refine and break down the number until you get your answer
• Bottom-up approach : start with a small number and then build up and increase the number until you get your answer

Example : What is the market size of tires for personal vehicles in the United States?

To solve this market sizing question, we can use a top-down approach:

• Start with the United States population
• Estimate the average number of people per household
• Estimate the percentage of households that own cars
• Of those households, estimate the average number of cars owned
• Multiply by four wheels
• Estimate the frequency in which wheels are replaced
• Multiply by the cost of a tire
• Multiply all of these figures to determine the market size of tires for personal vehicles

Let’s start with a United States population size of 320M. We can estimate that the average household has 2.5 people. Therefore, there are 320M / 2.5 = 128M households in the US.

Assume that 75% of US households own a car. That gives us 75% * 128M households = 96M households that own a car

Among households that own a car, the average number of cars owned is about 1.5 cars per household. Therefore, 1.5 cars * 96M households = 144M cars.

Each car has 4 tires, so there are 4 * 144M cars = 576M tires for personal vehicles in the U.S.

Tires are replaced approximately once every six years. Therefore, in a given year, 576M tires / 6 = 96M tires are sold.

If tires cost an average of $100 each, then 96M tires * $100 = $9.6B.

The market size of tires for personal vehicles in the United States is $9.6B.

Case Interview Math Tips

Make sure to follow the consulting math tips below to do your best in your upcoming case interviews and impress your interviewer.

1. Develop a structure before doing math : Do not begin doing any math calculations until you have developed an approach or structure. This will prevent you from making unnecessary calculations and help you avoid making math mistakes.

Additionally, by presenting your structure to the interviewer, you can get confirmation on whether the approach makes sense. Once the interviewer approves of your structure, the rest of the math is simple arithmetic.

2. Round numbers when appropriate : Use round numbers to keep the math easy and reduce the likelihood that you make a calculation error.

For example, if you are making assumptions about the size of the United States population, use 320 million instead of 319 million.

If you are multiplying 199 * 17, see if the interviewer will allow you to round so that you are multiplying 200 * 17.

You don’t want to round too much since this may signal to the interviewer that you are uncomfortable performing math calculations if the numbers are not easy and round. However, rounding occasionally can help simplify the calculations and reduce the likelihood that you make a calculation error.

3. Use abbreviations for large numbers : If you are working with numbers in the thousands, millions, billions, or trillions, use abbreviations rather than writing out all of the zeroes.

For example:

• 10,000 can be expressed as 10K
• 200,000,000 can be expressed as 200M
• 30,000,000,000 can be expressed as 30B
• 4,000,000,000,000 can be expressed as 4T

This makes multiplying and dividing large numbers much easier if you know the shortcuts for multiplying and dividing by these abbreviations. You should know that:

• 1M = 1,000K = K * K
• 1B = 1,000M = K * M
• 1T = 1,000B = K * B = M * M
• 14,000 * 5,000 = 14K * 5K = (14 * 5) * (K * K) = 70M
• 2,000 * 14M = 2K * 14M = (2* 14) * (K * M) = 28B
• 12M * 6M = (12 * 6) * (M * M) = 72T

4. Rule of 72 : The Rule of 72 is a shortcut that lets you estimate how long it would take a market, company, or investment to double in size. Simply divide the number 72 by the annual growth rate to get an estimate for the number of years needed to double in size.

For example, if an investment is expected to grow at 9%, then it would take approximately 72 / 9 = 8 years for the investment to double.

If a market is growing at 12% per year, then it would take approximately 72 / 12 = 6 years for the market size to double.

5. Sense check your numbers along the way : Accidentally missing zeroes or adding extra zeroes during your calculations is the most common math mistake. After each step of a math calculation, you can do a quick sense check to see if your answer is the right order of magnitude.

For example, if you are multiplying 125 million by 24, you should expect your answer to be in the billions because 100 million * 20 = 2 billion.

Recommended Case Interview Math Resources

Here are the resources we recommend to learn the most robust, effective case interview math strategies in the least time-consuming way:

• Comprehensive Case Interview Course (our #1 recommendation): The only resource you need. Whether you have no business background, rusty math skills, or are short on time, this step-by-step course will transform you into a top 1% caser that lands multiple consulting offers.
• Hacking the Case Interview Book   (available on Amazon): Perfect for beginners that are short on time. Transform yourself from a stressed-out case interview newbie to a confident intermediate in under a week. Some readers finish this book in a day and can already tackle tough cases.
• The Ultimate Case Interview Workbook (available on Amazon): Perfect for intermediates struggling with frameworks, case math, or generating business insights. No need to find a case partner – these drills, practice problems, and full-length cases can all be done by yourself.
• Case Interview Coaching : Personalized, one-on-one coaching with former consulting interviewers
• Behavioral & Fit Interview Course : Be prepared for 98% of behavioral and fit questions in just a few hours. We'll teach you exactly how to draft answers that will impress your interviewer
• Resume Review & Editing : Transform your resume into one that will get you multiple interviews

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Case Study Questions for Class 9 Maths

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Are you preparing for your Class 9 Maths board exams and looking for an effective study resource? Well, you’re in luck! In this article, we will provide you with a collection of Case Study Questions for Class 9 Maths specifically designed to help you excel in your exams. These questions are carefully curated to cover various mathematical concepts and problem-solving techniques. So, let’s dive in and explore these valuable resources that will enhance your preparation and boost your confidence.

Join our Telegram Channel, there you will get various e-books for CBSE 2024 Boards exams for Class 9th, 10th, 11th, and 12th.

CBSE Class 9 Maths Board Exam will have a set of questions based on case studies in the form of MCQs. The CBSE Class 9 Mathematics Question Bank on Case Studies, provided in this article, can be very helpful to understand the new format of questions. Share this link with your friends.

If you want to want to prepare all the tough, tricky & difficult questions for your upcoming exams, this is where you should hang out.  CBSE Case Study Questions for Class 9  will provide you with detailed, latest, comprehensive & confidence-inspiring solutions to the maximum number of Case Study Questions covering all the topics from your  NCERT Text Books !

Table of Contents

CBSE Class 9th – MATHS: Chapterwise Case Study Question & Solution

Case study questions are a form of examination where students are presented with real-life scenarios that require the application of mathematical concepts to arrive at a solution. These questions are designed to assess students’ problem-solving abilities, critical thinking skills, and understanding of mathematical concepts in practical contexts.

Chapterwise Case Study Questions for Class 9 Maths

Case study questions play a crucial role in the field of mathematics education. They provide students with an opportunity to apply theoretical knowledge to real-world situations, thereby enhancing their comprehension of mathematical concepts. By engaging with case study questions, students develop the ability to analyze complex problems, make connections between different mathematical concepts, and formulate effective problem-solving strategies.

• Case Study Questions for Chapter 1 Number System
• Case Study Questions for Chapter 2 Polynomials
• Case Study Questions for Chapter 3 Coordinate Geometry
• Case Study Questions for Chapter 4 Linear Equations in Two Variables
• Case Study Questions for Chapter 5 Introduction to Euclid’s Geometry
• Case Study Questions for Chapter 6 Lines and Angles
• Case Study Questions for Chapter 7 Triangles
• Case Study Questions for Chapter 8 Quadilaterals
• Case Study Questions for Chapter 9 Areas of Parallelograms and Triangles
• Case Study Questions for Chapter 10 Circles
• Case Study Questions for Chapter 11 Constructions
• Case Study Questions for Chapter 12 Heron’s Formula
• Case Study Questions for Chapter 13 Surface Area and Volumes
• Case Study Questions for Chapter 14 Statistics
• Case Study Questions for Chapter 15 Probability

The above  Case studies for Class 9 Mathematics will help you to boost your scores as Case Study questions have been coming in your examinations. These CBSE Class 9 Maths Case Studies have been developed by experienced teachers of schools.studyrate.in for benefit of Class 10 students.

• Class 9 Science Case Study Questions
• Class 9 Social Science Case Study Questions

How to Approach Case Study Questions

When tackling case study questions, it is essential to adopt a systematic approach. Here are some steps to help you approach and solve these types of questions effectively:

• Read the case study carefully: Understand the given scenario and identify the key information.
• Identify the mathematical concepts involved: Determine the relevant mathematical concepts and formulas applicable to the problem.
• Formulate a plan: Devise a plan or strategy to solve the problem based on the given information and mathematical concepts.
• Solve the problem step by step: Apply the chosen approach and perform calculations or manipulations to arrive at the solution.
• Verify and interpret the results: Ensure the solution aligns with the initial problem and interpret the findings in the context of the case study.

Tips for Solving Case Study Questions

Here are some valuable tips to help you effectively solve case study questions:

• Read the question thoroughly and underline or highlight important information.
• Break down the problem into smaller, manageable parts.
• Visualize the problem using diagrams or charts if applicable.
• Use appropriate mathematical formulas and concepts to solve the problem.
• Show all the steps of your calculations to ensure clarity.
• Check your final answer and review the solution for accuracy and relevance to the case study.

Benefits of Practicing Case Study Questions

Practicing case study questions offers several benefits that can significantly contribute to your mathematical proficiency:

• Enhances critical thinking skills
• Improves problem-solving abilities
• Deepens understanding of mathematical concepts
• Develops analytical reasoning
• Prepares you for real-life applications of mathematics
• Boosts confidence in approaching complex mathematical problems

Case study questions offer a unique opportunity to apply mathematical knowledge in practical scenarios. By practicing these questions, you can enhance your problem-solving abilities, develop a deeper understanding of mathematical concepts, and boost your confidence for the Class 9 Maths board exams. Remember to approach each question systematically, apply the relevant concepts, and review your solutions for accuracy. Access the PDF resource provided to access a wealth of case study questions and further elevate your preparation.

Q1: Can case study questions help me score better in my Class 9 Maths exams?

Yes, practicing case study questions can significantly improve your problem-solving skills and boost your performance in exams. These questions offer a practical approach to understanding mathematical concepts and their real-life applications.

Q2: Are the case study questions in the PDF resource relevant to the Class 9 Maths syllabus?

Absolutely! The PDF resource contains case study questions that align with the Class 9 Maths syllabus. They cover various topics and concepts included in the curriculum, ensuring comprehensive preparation.

Q3: Are the solutions provided for the case study questions in the PDF resource?

Yes, the PDF resource includes solutions for each case study question. You can refer to these solutions to validate your answers and gain a better understanding of the problem-solving process.

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CBSE Class 9 Mathematics Case Study Questions

Table of Contents

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If you’re looking for a comprehensive and reliable study resource and case study questions for class 9 CBSE, myCBSEguide is the perfect door to enter. With over 10,000 study notes, solved sample papers and practice questions, it’s got everything you need to ace your exams. Plus, it’s updated regularly to keep you aligned with the latest CBSE syllabus . So why wait? Start your journey to success with myCBSEguide today!

Significance of Mathematics in Class 9

Mathematics is an important subject for students of all ages. It helps students to develop problem-solving and critical-thinking skills, and to think logically and creatively. In addition, mathematics is essential for understanding and using many other subjects, such as science, engineering, and finance.

CBSE Class 9 is an important year for students, as it is the foundation year for the Class 10 board exams. In Class 9, students learn many important concepts in mathematics that will help them to succeed in their board exams and in their future studies. Therefore, it is essential for students to understand and master the concepts taught in Class 9 Mathematics .

Case studies in Class 9 Mathematics

A case study in mathematics is a detailed analysis of a particular mathematical problem or situation. Case studies are often used to examine the relationship between theory and practice, and to explore the connections between different areas of mathematics. Often, a case study will focus on a single problem or situation and will use a variety of methods to examine it. These methods may include algebraic, geometric, and/or statistical analysis.

Example of Case study questions in Class 9 Mathematics

The Central Board of Secondary Education (CBSE) has included case study questions in the Class 9 Mathematics paper. This means that Class 9 Mathematics students will have to solve questions based on real-life scenarios. This is a departure from the usual theoretical questions that are asked in Class 9 Mathematics exams.

The following are some examples of case study questions from Class 9 Mathematics:

Class 9 Mathematics Case study question 1

There is a square park ABCD in the middle of Saket colony in Delhi. Four children Deepak, Ashok, Arjun and Deepa went to play with their balls. The colour of the ball of Ashok, Deepak,  Arjun and Deepa are red, blue, yellow and green respectively. All four children roll their ball from centre point O in the direction of   XOY, X’OY, X’OY’ and XOY’ . Their balls stopped as shown in the above image.

Answer the following questions:

Answer Key:

Class 9 Mathematics Case study question 2

• Now he told Raju to draw another line CD as in the figure
• The teacher told Ajay to mark  ∠ AOD  as 2z
• Suraj was told to mark  ∠ AOC as 4y
• Clive Made and angle  ∠ COE = 60°
• Peter marked  ∠ BOE and  ∠ BOD as y and x respectively

Now answer the following questions:

• 2y + z = 90°
• 2y + z = 180°
• 4y + 2z = 120°
• (a) 2y + z = 90°

Class 9 Mathematics Case study question 3

• (a) 31.6 m²
• (c) 513.3 m³
• (b) 422.4 m²

Class 9 Mathematics Case study question 4

How to Answer Class 9 Mathematics Case study questions

To crack case study questions, Class 9 Mathematics students need to apply their mathematical knowledge to real-life situations. They should first read the question carefully and identify the key information. They should then identify the relevant mathematical concepts that can be applied to solve the question. Once they have done this, they can start solving the Class 9 Mathematics case study question.

Students need to be careful while solving the Class 9 Mathematics case study questions. They should not make any assumptions and should always check their answers. If they are stuck on a question, they should take a break and come back to it later. With some practice, the Class 9 Mathematics students will be able to crack case study questions with ease.

Class 9 Mathematics Curriculum at Glance

At the secondary level, the curriculum focuses on improving students’ ability to use Mathematics to solve real-world problems and to study the subject as a separate discipline. Students are expected to learn how to solve issues using algebraic approaches and how to apply their understanding of simple trigonometry to height and distance problems. Experimenting with numbers and geometric forms, making hypotheses, and validating them with more observations are all part of Math learning at this level.

The suggested curriculum covers number systems, algebra, geometry, trigonometry, mensuration, statistics, graphing, and coordinate geometry, among other topics. Math should be taught through activities that include the use of concrete materials, models, patterns, charts, photographs, posters, and other visual aids.

CBSE Class 9 Mathematics (Code No. 041)

Class 9 Mathematics question paper design

The CBSE Class 9 mathematics question paper design is intended to measure students’ grasp of the subject’s fundamental ideas. The paper will put their problem-solving and analytical skills to the test. Class 9 mathematics students are advised to go through the question paper pattern thoroughly before they start preparing for their examinations. This will help them understand the paper better and enable them to score maximum marks. Refer to the given Class 9 Mathematics question paper design.

QUESTION PAPER DESIGN (CLASS 9 MATHEMATICS)

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Class 9 is an important milestone in a student’s life. It is the last year of high school and the last chance to score well in the CBSE board exams. myCBSEguide is the perfect platform for students to get started on their preparations for Class 9 Mathematics. myCBSEguide provides comprehensive study material for all subjects, including practice questions, sample papers, case study questions and mock tests. It also offers tips and tricks on how to score well in exams. myCBSEguide is the perfect door to enter for class 9 CBSE preparations.

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14 thoughts on “CBSE Class 9 Mathematics Case Study Questions”

This method is not easy for me

aarti and rashika are two classmates. due to exams approaching in some days both decided to study together. during revision hour both find difficulties and they solved each other’s problems. aarti explains simplification of 2+ ?2 by rationalising the denominator and rashika explains 4+ ?2 simplification of (v10-?5)(v10+ ?5) by using the identity (a – b)(a+b). based on above information, answer the following questions: 1) what is the rationalising factor of the denominator of 2+ ?2 a) 2-?2 b) 2?2 c) 2+ ?2 by rationalising the denominator of aarti got the answer d) a) 4+3?2 b) 3+?2 c) 3-?2 4+ ?2 2+ ?2 d) 2-?3 the identity applied to solve (?10-?5) (v10+ ?5) is a) (a+b)(a – b) = (a – b)² c) (a – b)(a+b) = a² – b² d) (a-b)(a+b)=2(a² + b²) ii) b) (a+b)(a – b) = (a + b

MATHS PAAGAL HAI

All questions was easy but search ? hard questions. These questions was not comparable with cbse. It was totally wastage of time.

Where is search ? bar

maths is love

Can I have more questions without downloading the app.

I love math

Hello l am Devanshu chahal and l am an entorpinior. I am started my card bord business and remanded all the existing things this all possible by math now my business is 120 crore and my business profit is 25 crore in a month. l find the worker team because my business is going well Thanks

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CBSE Class 9 Maths Case Study Questions PDF Download

Download Class 9 Maths Case Study Questions to prepare for the upcoming CBSE Class 9 Exams 2023-24. These Case Study and Passage Based questions are published by the experts of CBSE Experts for the students of CBSE Class 9 so that they can score 100% in Exams.

Case study questions play a pivotal role in enhancing students’ problem-solving skills. By presenting real-life scenarios, these questions encourage students to think beyond textbook formulas and apply mathematical concepts to practical situations. This approach not only strengthens their understanding of mathematical concepts but also develops their analytical thinking abilities.

Table of Contents

CBSE Class 9th MATHS: Chapterwise Case Study Questions

Inboard exams, students will find the questions based on assertion and reasoning. Also, there will be a few questions based on case studies. In that, a paragraph will be given, and then the MCQ questions based on it will be asked. For Class 9 Maths Case Study Questions, there would be 5 case-based sub-part questions, wherein a student has to attempt 4 sub-part questions.

Chapterwise Case Study Questions of Class 9 Maths

• Case Study Questions for Chapter 1 Number System
• Case Study Questions for Chapter 2 Polynomials
• Case Study Questions for Chapter 3 Coordinate Geometry
• Case Study Questions for Chapter 4 Linear Equations in Two Variables
• Case Study Questions for Chapter 5 Introduction to Euclid’s Geometry
• Case Study Questions for Chapter 6 Lines and Angles
• Case Study Questions for Chapter 7 Triangles
• Case Study Questions for Chapter 8 Quadrilaterals
• Case Study Questions for Chapter 9 Areas of Parallelograms and Triangles
• Case Study Questions for Chapter 10 Circles
• Case Study Questions for Chapter 11 Constructions
• Case Study Questions for Chapter 12 Heron’s Formula
• Case Study Questions for Chapter 13 Surface Area and Volumes
• Case Study Questions for Chapter 14 Statistics
• Case Study Questions for Chapter 15 Probability

Checkout: Class 9 Science Case Study Questions

And for mathematical calculations, tap Math Calculators which are freely proposed to make use of by calculator-online.net

The above  Class 9 Maths Case Study Question s will help you to boost your scores as Case Study questions have been coming in your examinations. These CBSE Class 9 Maths Case Study Questions have been developed by experienced teachers of cbseexpert.com for the benefit of Class 10 students.

Class 9 Maths Syllabus 2023-24

UNIT I: NUMBER SYSTEMS

1. REAL NUMBERS (18 Periods)

1. Review of representation of natural numbers, integers, and rational numbers on the number line. Rational numbers as recurring/ terminating decimals. Operations on real numbers.

2. Examples of non-recurring/non-terminating decimals. Existence of non-rational numbers (irrational numbers) such as √2, √3 and their representation on the number line. Explaining that every real number is represented by a unique point on the number line and conversely, viz. every point on the number line represents a unique real number.

3. Definition of nth root of a real number.

4. Rationalization (with precise meaning) of real numbers of the type

(and their combinations) where x and y are natural number and a and b are integers.

5. Recall of laws of exponents with integral powers. Rational exponents with positive real bases (to be done by particular cases, allowing learner to arrive at the general laws.)

UNIT II: ALGEBRA

1. POLYNOMIALS (26 Periods)

Definition of a polynomial in one variable, with examples and counter examples. Coefficients of a polynomial, terms of a polynomial and zero polynomial. Degree of a polynomial. Constant, linear, quadratic and cubic polynomials. Monomials, binomials, trinomials. Factors and multiples. Zeros of a polynomial. Motivate and State the Remainder Theorem with examples. Statement and proof of the Factor Theorem. Factorization of ax2 + bx + c, a ≠ 0 where a, b and c are real numbers, and of cubic polynomials using the Factor Theorem. Recall of algebraic expressions and identities. Verification of identities:

RELATED STORIES

and their use in factorization of polynomials.

2. LINEAR EQUATIONS IN TWO VARIABLES (16 Periods)

Recall of linear equations in one variable. Introduction to the equation in two variables. Focus on linear equations of the type ax + by + c=0.Explain that a linear equation in two variables has infinitely many solutions and justify their being written as ordered pairs of real numbers, plotting them and showing that they lie on a line.

UNIT III: COORDINATE GEOMETRY COORDINATE GEOMETRY (7 Periods)

The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations.

UNIT IV: GEOMETRY

1. INTRODUCTION TO EUCLID’S GEOMETRY (7 Periods)

History – Geometry in India and Euclid’s geometry. Euclid’s method of formalizing observed phenomenon into rigorous Mathematics with definitions, common/obvious notions, axioms/postulates and theorems. The five postulates of Euclid. Showing the relationship between axiom and theorem, for example: (Axiom)

1. Given two distinct points, there exists one and only one line through them. (Theorem)

2. (Prove) Two distinct lines cannot have more than one point in common.

2. LINES AND ANGLES (15 Periods)

1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so formed is 180O and the converse.

2. (Prove) If two lines intersect, vertically opposite angles are equal.

3. (Motivate) Lines which are parallel to a given line are parallel.

3. TRIANGLES (22 Periods)

1. (Motivate) Two triangles are congruent if any two sides and the included angle of one triangle is equal to any two sides and the included angle of the other triangle (SAS Congruence).

2. (Prove) Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence).

3. (Motivate) Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence).

4. (Motivate) Two right triangles are congruent if the hypotenuse and a side of one triangle are equal (respectively) to the hypotenuse and a side of the other triangle. (RHS Congruence)

5. (Prove) The angles opposite to equal sides of a triangle are equal.

6. (Motivate) The sides opposite to equal angles of a triangle are equal.

4. QUADRILATERALS (13 Periods)

1. (Prove) The diagonal divides a parallelogram into two congruent triangles.

2. (Motivate) In a parallelogram opposite sides are equal, and conversely.

3. (Motivate) In a parallelogram opposite angles are equal, and conversely.

4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal.

5. (Motivate) In a parallelogram, the diagonals bisect each other and conversely.

6. (Motivate) In a triangle, the line segment joining the mid points of any two sides is parallel to the third side and in half of it and (motivate) its converse.

5. CIRCLES (17 Periods)

1. (Prove) Equal chords of a circle subtend equal angles at the center and (motivate) its converse.

2. (Motivate) The perpendicular from the center of a circle to a chord bisects the chord and conversely, the line drawn through the center of a circle to bisect a chord is perpendicular to the chord.

3. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the center (or their respective centers) and conversely.

4. (Prove) The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

5. (Motivate) Angles in the same segment of a circle are equal.

6. (Motivate) If a line segment joining two points subtends equal angle at two other points lying on the same side of the line containing the segment, the four points lie on a circle.

7. (Motivate) The sum of either of the pair of the opposite angles of a cyclic quadrilateral is 180° and its converse.

UNIT V: MENSURATION 1.

1. AREAS (5 Periods)

Area of a triangle using Heron’s formula (without proof)

2. SURFACE AREAS AND VOLUMES (17 Periods)

Surface areas and volumes of spheres (including hemispheres) and right circular cones.

UNIT VI: STATISTICS & PROBABILITY

STATISTICS (15 Periods)

 Bar graphs, histograms (with varying base lengths), and frequency polygons.

To crack case study questions, Class 9 Mathematics students need to apply their mathematical knowledge to real-life situations. They should first read the question carefully and identify the key information. They should then identify the relevant mathematical concepts that can be applied to solve the question. Once they have done this, they can start solving the Class 9 Mathematics case study question.

Benefits of Practicing CBSE Class 9 Maths Case Study Questions

Regular practice of CBSE Class 9 Maths case study questions offers several benefits to students. Some of the key advantages include:

• Deeper Understanding : Case study questions foster a deeper understanding of mathematical concepts by connecting them to real-world scenarios. This improves retention and comprehension.
• Practical Application : Students learn to apply mathematical concepts to practical situations, preparing them for real-life problem-solving beyond the classroom.
• Critical Thinking : Case study questions require students to think critically, analyze data, and devise appropriate solutions. This nurtures their critical thinking abilities, which are valuable in various academic and professional domains.
• Exam Readiness : By practicing case study questions, students become familiar with the question format and gain confidence in their problem-solving abilities. This enhances their readiness for CBSE Class 9 Maths exams.
• Holistic Development: Solving case study questions cultivates not only mathematical skills but also essential life skills like analytical thinking, decision-making, and effective communication.

Tips to Solve CBSE Class 9 Maths Case Study Questions Effectively

Solving case study questions can be challenging, but with the right approach, you can excel. Here are some tips to enhance your problem-solving skills:

• Read the case study thoroughly and understand the problem statement before attempting to solve it.
• Identify the relevant data and extract the necessary information for your solution.
• Break down complex problems into smaller, manageable parts to simplify the solution process.
• Apply the appropriate mathematical concepts and formulas, ensuring a solid understanding of their principles.
• Clearly communicate your solution approach, including the steps followed, calculations made, and reasoning behind your choices.
• Practice regularly to familiarize yourself with different types of case study questions and enhance your problem-solving speed.Class 9 Maths Case Study Questions

Remember, solving case study questions is not just about finding the correct answer but also about demonstrating a logical and systematic approach. Now, let’s explore some resources that can aid your preparation for CBSE Class 9 Maths case study questions.

Q1. Are case study questions included in the Class 9 Maths Case Study Questions syllabus?

Yes, case study questions are an integral part of the CBSE Class 9 Maths syllabus. They are designed to enhance problem-solving skills and encourage the application of mathematical concepts to real-life scenarios.

Q2. How can solving case study questions benefit students ?

Solving case study questions enhances students’ problem-solving skills, analytical thinking, and decision-making abilities. It also bridges the gap between theoretical knowledge and practical application, making mathematics more relevant and engaging.

Q3. How do case study questions help in exam preparation?

Case study questions help in exam preparation by familiarizing students with the question format, improving analytical thinking skills, and developing a systematic approach to problem-solving. Regular practice of case study questions enhances exam readiness and boosts confidence in solving such questions.

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CBSE Class 9th Maths 2023 : 30 Most Important Case Study Questions with Answers; Download PDF

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CBSE Class 9 Maths exam 2022-23 will have a set of questions based on case studies in the form of MCQs. CBSE Class 9 Maths Question Bank on Case Studies given in this article can be very helpful in understanding the new format of questions.

Each question has five sub-questions, each followed by four options and one correct answer. Students can easily download these questions in PDF format and refer to them for exam preparation.

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CBSE Class 10 Maths Case Study Questions for Chapter 3 - Pair of Linear Equations in Two Variables (Published by CBSE)

Cbse's question bank on case study for class 10 maths chapter 3 is available here. these questions will be very helpful to prepare for the cbse class 10 maths exam 2022..

Case study questions are going to be new for CBSE Class 10 students. These are the competency-based questions that are completely new to class 10 students. To help students understand the format of the questions, CBSE has released a question bank on case study for class 10 Maths. Students must practice with these questions to get familiarised with the concepts and logic used in the case study and understand how to answers them correctly. You may check below the case study questions for CBSE Class 10 Maths Chapter 3 - Pair of Linear Equations in Two Variables. You can also check the right answer at the end of each question.

Check Case Study Questions for Class 10 Maths Chapter 3 - Pair of Linear Equations in Two Variables

CASE STUDY-1:

1. If answer to all questions he attempted by guessing were wrong, then how many questions did he answer correctly?

2. How many questions did he guess?

3. If answer to all questions he attempted by guessing were wrong and answered 80 correctly, then how many marks he got?

4. If answer to all questions he attempted by guessing were wrong, then how many questions answered correctly to score 95 marks?

Let the no of questions whose answer is known to the student x and questions attempted by cheating be y

x – 1/4y =90

solving these two

x = 96 and y = 24

1. He answered 96 questions correctly.

2. He attempted 24 questions by guessing.

3. Marks = 80- ¼ 0f 40 =70

4. x – 1/4 of (120 – x) = 95

5x = 500, x = 100

CASE STUDY-2:

Amit is planning to buy a house and the layout is given below. The design and the measurement has been made such that areas of two bedrooms and kitchen together is 95 sq.m.

Based on the above information, answer the following questions:

1. Form the pair of linear equations in two variables from this situation.

2. Find the length of the outer boundary of the layout.

3. Find the area of each bedroom and kitchen in the layout.

4. Find the area of living room in the layout.

5. Find the cost of laying tiles in kitchen at the rate of Rs. 50 per sq.m.

1. Area of two bedrooms= 10x sq.m

Area of kitchen = 5y sq.m

10x + 5y = 95

Also, x + 2+ y = 15

2. Length of outer boundary = 12 + 15 + 12 + 15 = 54m

3. On solving two equation part(i)

x = 6m and y = 7m

area of bedroom = 5 x 6 = 30m

area of kitchen = 5 x 7 = 35m

4. Area of living room = (15 x 7) – 30 = 105 – 30 = 75 sq.m

5. Total cost of laying tiles in the kitchen = Rs50 x 35 = Rs1750

Case study-3 :

It is common that Governments revise travel fares from time to time based on various factors such as inflation ( a general increase in prices and fall in the purchasing value of money) on different types of vehicles like auto, Rickshaws, taxis, Radio cab etc. The auto charges in a city comprise of a fixed charge together with the charge for the distance covered. Study the following situations:

Situation 1: In city A, for a journey of 10 km, the charge paid is Rs 75 and for a journey of 15 km, the charge paid is Rs 110.

Situation 2: In a city B, for a journey of 8km, the charge paid is Rs91 and for a journey of 14km, the charge paid is Rs 145.

Refer situation 1

1. If the fixed charges of auto rickshaw be Rs x and the running charges be Rs y km/hr, the pair of linear equations representing the situation is

a) x + 10y =110, x + 15y = 75

b) x + 10y = 75, x + 15y = 110

c) 10x + y = 110, 15x + y = 75

d) 10x + y = 75, 15x + y = 110

Answer: b) x + 10y = 75, x + 15y = 110

2. A person travels a distance of 50km. The amount he has to pay is

Answer: c) Rs.355

Refer situation 2

3. What will a person have to pay for travelling a distance of 30km?

Answer: b) Rs.289

4. The graph of lines representing the conditions are: (situation 2)

Answer: (iii)

Also Check:

CBSE Case Study Questions for Class 10 Maths - All Chapters

Tips to Solve Case Study Based Questions Accurately

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CBSE 12th Standard Maths Subject Matrices Case Study Questions With Solution 2021

By QB365 on 21 May, 2021

QB365 Provides the updated CASE Study Questions for Class 12 Maths, and also provide the detail solution for each and every case study questions . Case study questions are latest updated question pattern from NCERT, QB365 will helps to get  more marks in Exams

QB365 - Question Bank Software

12th Standard CBSE

Final Semester - June 2015

Case Study Questions

(ii) The cost incurred by the organisation on village Y is

(iii) The cost incurred by the organisation on village Z is

(iv) The total number of toilets that can be expected after the promotion in village X, is

(v) The total number of toilets that can be expected after the promotion in village Z, is

(ii) The combined sales of Urad in September and October, for farmer Shyam is

(iii) Find the decrease in sales of Mung from September to October, for the farmer Shyam.

If  \(A=\left[a_{i j}\right]_{m \times n} \text { and } B=\left[b_{i j}\right]_{m \times n}\)  are two matrices, then  \(A \pm B\)  is of order m x n and is defined as  \((A \pm B)_{i j}=a_{i j} \pm b_{i j}\) , where i = 1,2, , m and  j = 1,2, ..., n If  \(A=\left[a_{i j}\right]_{m \times n} \text { and } B=\left[b_{j k}\right]_{n \times p}\)  are two matrices, then AB is of order m x p and is defined as  \((A B)_{i k}=\sum_{r=1}^{n} a_{i r} b_{r k}=a_{i 1} b_{1 k}+a_{i 2} b_{2 k}+\ldots . .+a_{i n} b_{n k}\)   Consider  \(A=\left[\begin{array}{cc} 2 & -1 \\ 3 & 4 \end{array}\right], B=\left[\begin{array}{ll} 5 & 2 \\ 7 & 4 \end{array}\right], C=\left[\begin{array}{ll} 2 & 5 \\ 3 & 8 \end{array}\right] \text { and } D=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\)   Using the concept of matrices answer the following questions.  (i) Find the product AB. 

(ii) If A and B are any other two matrices such that AB exists, then

(iii) Find the values of a and c in the matrix D such than CD - AB = 0.

(iv) Find the values of band d in the matrix D such that CD - AB = 0.

(v) Find B + D.

(iii) If the trust fund obtains an annual total interest of Rs.3200, then the investment in two bonds is

(iv) The total amount of interest received on both bonds is given by

(v) If the amount of interest given to old age home is Rs.500, then the amount of investment in bond Y is

*****************************************

Cbse 12th standard maths subject matrices case study questions with solution 2021 answer keys.

(i) (c) : Let Rs. A, Rs. B and Rs.C be the cost incurred by the organisation for villages X, Y and Z respectively. Then A, B, C will be given by the following matrix equation. \(\left[\begin{array}{ccc} 400 & 300 & 100 \\ 300 & 250 & 75 \\ 500 & 400 & 150 \end{array}\right]\left[\begin{array}{l} 50 \\ 20 \\ 40 \end{array}\right]=\left[\begin{array}{c} A \\ B \\ C \end{array}\right]\)   \(\Rightarrow\left[\begin{array}{l} A \\ B \\ C \end{array}\right]=\left[\begin{array}{c} 400 \times 50+300 \times 20+100 \times 40 \\ 300 \times 50+250 \times 20+75 \times 40 \\ 500 \times 50+400 \times 20+150 \times 40 \end{array}\right]\)   (ii) (c) (iii) (b) (iv) (c) :  Total number of toilets that can be expected in each village is given by the following matrix. \(\begin{array}{l} X \\ Y \\ Z \end{array}\left[\begin{array}{ccc} 400 & 300 & 100 \\ 300 & 250 & 75 \\ 500 & 400 & 150 \end{array}\right]\) \(\left[\begin{array}{c} 2 / 100 \\ 4 / 100 \\ 20 / 100 \end{array}\right]\)   \(\begin{array}{l} X \\ Y \\ Z \end{array}\) \(\left[\begin{array}{c} 8+12+20 \\ 6+10+15 \\ 10+16+30 \end{array}\right]\) = \(\begin{array}{l} X \\ Y \\ Z \end{array}\) \(\left[\begin{array}{c} 40 \\ 31 \\ 56 \end{array}\right]\)   (v) (d)

(i) (a) :  \(A B=\left[\begin{array}{cc} 2 & -1 \\ 3 & 4 \end{array}\right]\left[\begin{array}{ll} 5 & 2 \\ 7 & 4 \end{array}\right]\)   \(=\left[\begin{array}{cc} 10-7 & 4-4 \\ 15+28 & 6+16 \end{array}\right]=\left[\begin{array}{cc} 3 & 0 \\ 43 & 22 \end{array}\right]\)   (ii) (c) (iii) (b) : We have, CD - AB = 0 \(\Rightarrow\left[\begin{array}{ll} 2 & 5 \\ 3 & 8 \end{array}\right]\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]-\left[\begin{array}{cc} 3 & 0 \\ 43 & 22 \end{array}\right]=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]\)   \(\Rightarrow\left[\begin{array}{ll} 2 a+5 c & 2 b+5 d \\ 3 a+8 c & 3 b+8 d \end{array}\right]-\left[\begin{array}{cc} 3 & 0 \\ 43 & 22 \end{array}\right]=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]\)   \(\Rightarrow\left[\begin{array}{cc} 2 a+5 c-3 & 2 b+5 d \\ 3 a+8 c-43 & 3 b+8 d-22 \end{array}\right]=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]\)   By equality of matrices, we get 2a + 5c - 3 = 0  ...(i) 3a + 8c - 43 = 0 ...(ii) 2b + 5d = 0 ...(iii) 3b + 8d - 22 = 0 ...(iv) Solving (i) and (ii), we get a = -191, e = 77 (iv) (c) : Solving (iii) and (iv), we get b = -110, d = 44 (v) (d) : Wehave,  \(B+D=\left[\begin{array}{ll} 5 & 2 \\ 7 & 4 \end{array}\right]+\left[\begin{array}{cc} -191 & -110 \\ 77 & 44 \end{array}\right]\)   \(=\left[\begin{array}{cc} -186 & -108 \\ 84 & 48 \end{array}\right]\)

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