case study math drills

Career in Consulting

Case Interview Math: 47 realistic exercises

This is a complete guide to case interview math in 2024.

One thing is 100% sure: you’ll have to solve consulting math problems in all your case interviews at top consulting firms such as McKinsey, BCG, or Bain & Company.

This article is for you if you don’t know anything about case interview math problems or don’t feel confident about your math skills.

But if you feel confident about your case math skills, keep reading because your confidence can be your Achilles’ heel.

You don’t want to miss this guide’s detailed process and actionable tips for approaching consulting case interview math problems and securing an offer in your dream consulting firm .

Let’s jump right into it.

Table of Contents

Understanding case interview math, what is a case interview math problem.

First, let’s zoom out:

At his core, a case interview consists of identifying and solving all the mini-problems necessary to resolve the main problem.

The list of mini-problems necessary to resolve the main problem is called a framework for issue tree.

Case interview math is the practice of solving quantitative problems with the goal of solving a mini-problem of your framework. Quantitative problems involve analyzing data using basic arithmetic, percentage calculations, and business formulas.

Let’s take an example:

Imagine a client asking for your help assessing whether they should enter a new market (this is the main problem)

One of the mini-problems is determining if this strategy would be profitable.

Therefore, a consulting math problem could be a break-even analysis involving estimating how many units a company should sell to make zero profit.

A sample case interview math problem can be a break-even analysis.

As the above image shows, a sample case interview math problem can be a break-even analysis.

For instance:

Sample case interview math problems

Here are other consulting math problems that can be asked in case interviews:

A manufacturer sells a set of headphones for $300. The cost of materials is $20, labor is $10, factory rent is $25K per month, and utilities and other operating costs are $10K per month. How many headphones would they have to sell to make a $7M profit per month?

A watch manufacturer sells watches for $220 each. To produce a watch, the company spends $20 on materials and $15 on labor. It has $0.5M in monthly operating costs. If it sells 3,000 watches per month, what is its monthly profit?

A wood desk factory sells a desk for an average price of $200. To produce each wood desk, the company spends $30 on materials and $40 on labor. They have $0.1M in monthly operating costs. How many wood desks does the factory need to sell monthly to break even?

Apple’s iPhone X retail price is $650. Should Apple reduce the price by $50? Assume COGS = $200, whatever the retail price; with a price cut, the quantity sold would increase from 5M to 6M.

What is the payback period if solar panels cost $5,000 to install and the savings are $100 each month?

Now, the next section is about what your interviewers assess.

Which skills are assessed with case interview math?

Your interviewers are testing if you possess these three consulting core skills :

Case structuring : your capacity to break down a problem into smaller and easier-to-solve problems

Case analytics (or quantitative analysis): your capacity to calculate numbers (using mental math).

Business acumen : your capacity to interpret numbers and derive conclusions from these numbers

To begin with, like every question in case interviews, you must be structured .

And consulting math problems is no exception.

Second, to calculate numbers, you must be able to perform mental calculations, which means that calculators or computers are not allowed.

In the ChatGPT era, using mental math might sound silly.

However, management consulting firms like McKinsey , BCG , or Bain believe a strong correlation exists between candidates’ mental math ability and their chances of becoming successful consultants.

There is no turnaround : you must demonstrate good math skills to pass each consulting interview.

Finally, case interview math isn’t just about crunching numbers—it’s about unraveling the mysteries behind the data to uncover insights that drive real-world solutions.

To conclude this section, I want you to understand this:

The most challenging aspect of consulting math problems is performing calculations (fast) while someone stares at you with a $100k+ consulting job offer at stake.

Exploring the 6 types of case interview math problems

Before we dive into the step-by-step case interview math strategies, it’s essential to know the different consulting math problems you’ll have in your case interviews.

That way, you can prepare more effectively and increase your chances of securing an offer from your dream management consulting firm.

Calculate financial ratios

Definition: Financial ratios provide insights into a company’s financial health and performance by comparing different aspects of its financial statements.

Calculating financial ratios requires analyzing data provided by the interviewer and applying mathematical formulas to derive meaningful metrics.

Example: Let’s say you’re presented with the following data for a company and asked to calculate the gross margin:

Revenue: $1,000,000

Cost of Goods Sold (COGS): $400,000

Operating Expenses: $200,000

To calculate the Gross Margin, you would use the formula:

Gross Margin = (Revenue − COGS / Revenue) × 100%

Substituting the values:

Gross Margin = (1,000,000 − 400,000 / 1,000,000) × 100% = (600,000 / 1,000,000) × 100% = 60%

So, the Gross Margin of the company is 60%.

The most common financial ratios that you might calculate in case interviews are :

Profit margin

Gross profit and gross profit margin

Operating profit and operating profit margin

Contribution margin

Solve profitability problems.

Definition: Profitability problems involve assessing a company’s ability to generate profits based on the data provided.

Candidates are typically required to estimate the profit generated by a company and identify factors influencing its profitability.

Example: Suppose you’re given the following information about Company XYZ:

Revenue: $5,000,000

Total Costs: $3,000,000

To calculate the company’s profit, you would subtract total costs from revenue:

Profit = Revenue − Total Costs = $5,000,000 − 3,000,000 = $2,000,000

So, Company XYZ’s profit is $2,000,000.

Assess investment opportunities

Definition: Investment opportunity problems involve evaluating the potential returns and risks of investing in a project or opportunity.

Based on the provided data, candidates must calculate metrics such as Return on Investment (ROI), Payback Period, Net Present Value (NPV), and Breakeven Point.

Example: Imagine you’re tasked with assessing the investment opportunity for a project with the following data:

Initial Investment: $100,000

Annual Cash Inflows: $30,000

Discount Rate: 10%

To calculate the Net Present Value (NPV), you would use the formula:

𝑁𝑃𝑉=∑Cash Inflow (1+Discount Rate)𝑛 − Initial Investment

Substituting the values and summing over the project’s lifetime (let’s assume 3 years for simplicity):

𝑁𝑃𝑉 = 30,000(1+0.10)1 + 30,000(1+0.10)2 + 30,000(1+0.10)3 − 100,000

𝑁𝑃𝑉 = 27,273 + 24,793 + 22,539 − 100,000 = 74,605

So, the investment opportunity’s Net Present Value (NPV) is $74,605.

Assess pricing elasticity

Definition: Pricing elasticity problems involve understanding how price changes affect consumer demand for a product or service.

Candidates must analyze the impact of pricing changes on quantity sold, considering factors such as price sensitivity and market dynamics.

Example: Suppose you’re given data showing the following relationship between price and quantity sold for a product:

Price: $10, $15, $20

Quantity Sold: 100, 75, 50

To analyze pricing elasticity, you would calculate the percentage change in quantity sold relative to the percentage change in price.

For instance, the percentage change in quantity sold when the price increases from $10 to $15 would be:

Percentage Change in Quantity Sold=New Quantity−Old Quantity/Old Quantity×100%

Percentage Change in Quantity Sold=75−100/100×100%=−25%

This indicates a 25% decrease in quantity sold when the price increases from $10 to $15.

Solve operation problems

Definition: Operation problems involve calculating output, efficiency, or other metrics related to a business’s operational processes.

Candidates may be asked to determine the output of a production line, identify bottlenecks, or optimize resource allocation based on given input and productivity rates.

Example: Suppose you’re presented with data regarding a manufacturing plant’s production line:

Input: 1,000 units per hour

Productivity Rate: 90%

To calculate the output of the production line, you would multiply the input by the productivity rate:

Output=Input×Productivity Rate

Output=1,000×0.90=900 units per hour

So, the output of the production line is 900 units per hour.

Answer market sizing questions

Definition: Market sizing questions involve estimating the size of a market or segment based on limited or no information provided.

Candidates must make assumptions and use logical reasoning to arrive at a plausible estimate, demonstrating their ability to think critically and analyze market dynamics.

Example: Let’s consider a scenario where you’re asked to estimate the size of the online grocery delivery market in a specific city.

Given the lack of precise data, you would need to make several assumptions based on available information, such as population size, average household spending on groceries, and the percentage of the population likely to use online grocery services.

Assuming the city has a population of 1 million, with an average household spending of $200 on groceries per month, and an estimated 20% of households using online grocery services, you could estimate the market size as follows:

Market Size=Population×Average Household Spending×Percentage of Households Using Online Grocery Services

Market Size=1,000,000×$200×0.20=$40,000,000

So, the estimated size of the online grocery delivery market in the city is $40 million.

Related article : I’ve written a comprehensive guide about market sizing and market math here .

Strategies to Master Case Interview Math Problems

The step-by-step approach to solving case interview math problems.

Solving consulting math problems goes beyond performing calculations .

Indeed, consulting companies want to see if you have the mental agility to become a world-class consultant.

Therefore, these firms expect you to solve consulting case interview math problems using the following process :

Develop a structure

Do the math

Interpret the result

Lead the case

Step 1: structure your approach

The first step consists of mapping out your approach to answer the question.

Often, you need to use a business formula to structure your approach.

Therefore, check the most helpful business concepts and formulas mentioned later in this guide.

Important : you can get unclear case interview math problems. Therefore, if it’s the case, start by asking your interviewers clarifying questions.   

Step 2: do the calculations

This is when you plug the data provided by your interviewers into your formula.

Finally, do a quick sanity check after you’ve performing calculations and found the final result.

Step 3: derive insights (the "so-whats")

Remember this: your job as a consultant is not to calculate data but to interpret data.

You must relate your answer to the problem to be solved and discuss the implications of your calculations for your client or the problem.

These implications are known as the “so-whats” in consulting.

Imagine this:

You did a break-even analysis and found that your client must sell 100,000 units to break even.

Now, you must discuss the implications of these findings .

Is it doable?

Can we reduce the breakeven point?

Step 4: lead the case

Finally, discuss the possible next steps in the case interview , i.e., what is the next information you want to know or analysis to do.

Do not overthink the differences between candidate-led and interviewer-led cases.

Instead, ALWAYS lead the case . You’ll stand out and impress your interviewers (and thank me later).

Time Management Tips for Case Interview Math Problems

Efficient time management is crucial during math sections of consulting case interviews, as you are often under pressure to demonstrate analytical precision swiftly.

Practice with a timer to understand how long different types of questions take you to solve.

Always keep track of your time on each problem and set internal benchmarks to ensure you’re not falling behind.

If you find yourself stuck, it’s wise to move on and return later if time permits rather than getting bogged down and risking the completion of more straightforward questions that could yield quick wins.

Tips for Effective Case Interview Math Practice

Structured and consistent practice is key to excelling in case interview math.

Start by familiarizing yourself with common math problems in consulting cases, such as profitability calculations or market sizing.

Use various resources, from online practice problems to case books, to expose yourself to a wide range of question formats and difficulty levels.

Simulate the interview environment as closely as possible during practice sessions by timing yourself and working in a quiet space.

After each session, critically review your solutions to identify errors or inefficiencies in your approach.

Get 4 Complete Case Interview Courses For Free

You need 4 skills to be successful in all case interviews: Case Structuring, Case Leadership, Case Analytics, and Communication. Join this free training and learn how to ace ANY case questions.

8 math techniques you must know to ace case interview math problems

You don’t need to know complex math concepts to answer consulting math problems.

Knowing how to add, subtract, multiply, and divide is enough.

Okay, knowing some basic business formulas helps, too.

But don’t panic: if you don’t have a business degree, this article is what you need.

Therefore, I’ll share all the business formulas you need to ace consulting math problems later.

But let’s go back to the math techniques you must know. 

Even if knowing basic arithmetics is enough, you must have some strategies to quickly answer math questions in a stressful context like case interviews.

Let’s jump into the first strategy: divide and conquer.  

The divide and conquer technique

The divide and conquer consists of breaking down complex math calculations into manageable parts with easier-to-compute numbers. 

The math ratios cheat sheet

You need to memorize very few things when preparing for your case interviews.

However, memorizing the following cheat sheet will help improve your mental math skills .

In other words: this is a list of critical numbers to know to fasten your mental calculation skills. 

Now, you can use the last two concepts to fasten your mental skills.

Example : What is 26/14? Give an exact answer.

Percentage calculations

1. calculate a% of b:.

Calculating A% of B involves finding a fraction of B represented by A%.

In consulting math problems, this calculation is essential for understanding proportions and applying them to various scenarios.

Example: Imagine you’re analyzing the cost savings achieved by implementing a new efficiency measure in a manufacturing plant.

The total cost savings amount to $10,000, and you need to calculate 20% of this total.

20% of $10,000 = 0.20 x $10,000 = $2,000

So, 20% of $10,000 is $2,000.

2. Calculate What Percentage of A is B:

Determining what percentage of A a value B represents allows consultants to contextualize data and understand its significance relative to a reference point.

Example: Suppose you’re analyzing the revenue contribution of a specific product line within a company’s overall sales.

If the product line generates $25,000 in revenue out of a total company revenue of $100,000, you can calculate the percentage of total revenue represented by the product line as follows:

Percentage of Total Revenue=(Revenue of Product Line/Total Revenue)×100%

Percentage of Total Revenue=($25,000/$100,000)×100%=25%

So, the product line represents 25% of the company’s total revenue.

3. Calculate Percentage Change:

Calculating percentage change allows consultants to assess the magnitude and direction of changes over time, providing insights into trends and performance metrics.

Example: Consider a scenario where you’re analyzing a retail company’s year-over-year growth in sales revenue.

If the company’s sales revenue increased from $50,000 last year to $60,000 this year, you can calculate the percentage change in revenue as follows:

Percentage Change=(New Value−Old Value/Old Value)×100%

Percentage Change=($60,000−$50,000/$50,000)×100%=20%

So, there was a 20% increase in sales revenue compared to last year.

Averages (or Means)

The mean, also known as the arithmetic average, is a fundamental concept in statistics and mathematics.

It represents the sum of all values in a dataset divided by the total number of values.

In the context of consulting math problems, understanding the mean allows candidates to analyze and interpret numerical data effectively.

Example: Suppose you’re analyzing a retail company’s average revenue per customer.

You’re provided with the following data:

Revenue from Customer A: $500

Revenue from Customer B: $700

Revenue from Customer C: $600 To calculate the mean revenue per customer, add up the revenues and divide by the total number of customers: Mean Revenue = $500+$700+$6003=$18003=$600Mean Revenue = $3$500+$700+$600 = $31800​=$600

So, the mean revenue per customer is $600.

Another example: Consider a scenario where you analyze a tech company’s average response time for customer service inquiries.

Response Time for Inquiry 1: 5 minutes

Response Time for Inquiry 2: 10 minutes

Response Time for Inquiry 3: 15 minutes. To calculate the average response time, add up the response times and divide by the total number of inquiries: Average Response Time = 5+10+153=303=10Average Response Time = 35+10+15 =330 =10.

So, the average response time for customer service inquiries is 10 minutes.

Weighted average

The weighted average is a calculation that considers the importance or weight of each value in a dataset.

It’s particularly useful in consulting math problems where certain data points may have greater significance or influence the overall analysis.

Example: Imagine you’re analyzing the average score of products in a customer satisfaction survey, where each product category represents a different proportion of sales revenue.

Product A (50% of sales revenue): Average score of 4.5 out of 5

Product B (30% of sales revenue): Average score of 4.2 out of 5

Product C (20% of sales revenue): Average score of 4.8 out of 5

To calculate the weighted average score, multiply each average score by its corresponding sales revenue proportion, then sum the results:

Weighted Average Score = (0.5×4.5)+(0.3×4.2)+(0.2×4.8)

Weighted Average Score =(2.25)+(1.26) +(0.96)=4.47

So, the weighted average score of products in the customer satisfaction survey is 4.47 out of 5.

Fasten mental calculations: addition and substraction

Let’s use the above techniques (particularly the divide-and-conquer technique) to strengthen our mental calculation skills. 

Let’s start with addition and subtraction. 

I’ve found this great YouTube video from the Math0genius channel:

Fasten mental calculations: multiplication

Let’s continue with multiplication.

You can watch the following video from the same YouTube channel:

Fasten mental calculations: division

Finally, let’s end with division.

12 business formulas you must know to ace case interview math problems

In the following sections, you’ll learn the consulting math formulas and concepts all aspiring consultants must know to solve consulting math problems .

The income statement (P&L)

An income statement (or Profit&Loss statement) is one of the financial statements used for reporting a company’s financial performance.

In income statements, you’ll find:

COGS : costs of goods sold: it includes the costs of material and labor directly used to manufacture the products; COGS are variable costs

SG&A : selling, general & administrative expenses. It includes:

Selling expenses include the costs to sell (marketing expenses, salaries of sales personnel) and to distribute the products; they’re partly fixed and variable costs

G&A expenses include the costs to manage the company (labor costs for IT, HR, etc. + Rent + Utilities, etc.); G&A expenses are fixed costs

Profit is the financial benefit realized when revenue from a business activity exceeds costs and taxes .

In a case interview, you might be asked to calculate the profit using the profit equation, given some information about revenues and costs.

The profit equation is Profit = Revenues – Costs, where:

Revenues = quantity sold x unit price

Costs = variable costs + fixed costs

Example : What were the profits generated by your client last year if they sold 1,000,000 units for $5 per unit and had variable costs of $2.5m and fixed costs of $1m?

Answer : Its profits last year were (1,000,000 x $5)—($2.5m + $1m) = $5m—$3.5 m = $1.5m.

Gross profit

Gross profit is sales minus the cost of goods sold .

The cost of goods sold (COGS) refers to the costs related to producing a company’s products (or services).

COGS exclude costs related to sales, marketing, and administrative activities.

For example , if a company generates $1m of sales and has COGS of $0.7m, its gross profit is $0.3m.

Gross profit margin

Gross profit margin is the gross profit as a percentage of sales .

The gross profit margin formula is:

Gross profit margin = gross profit/sales.

Example : if a company generates $1m of sales and $0.3m of gross profit, its gross profit margin is 30%.

Another example: Your client sells electronic devices for $16 per unit. The manufacturing cost is $7 per unit. What is the gross margin (in % of selling price)? Give an exact answer.

Operating profit

Operating profit is the gross profit minus the operating expenses .

Operating expenses include costs related to sales, marketing, and administrative activities.

It excludes costs related to taxes.

Example : if a company generates $0.3m of gross profit and has operating costs of $0.1m, its operating profit is $0.2m.

Operating profit margin

Operating profit margin is the operating profit as a percentage of sales .

The operating profit margin formula is:

Operating profit margin = operating profit/sales.

Example : if a company generates $1m of sales and $0.1m of operating profit, its operating profit margin is 10%.

Contribution margin is vital in business and finance, especially in management consulting, where precise financial analysis is paramount.

It represents the portion of sales that exceeds variable costs and contributes to covering fixed costs .

In simpler terms, contribution margin reveals how much revenue from each unit sold is available to cover fixed expenses and generate profit.

To calculate contribution margin, subtract variable costs (such as direct materials and labor) from sales revenue.

Understanding contribution margin is crucial for making informed pricing strategies, product mix, and overall profitability decisions.

Let’s envision a consulting math problem that embodies the concept of contribution margin in a case interview scenario at a top consulting firm.

Suppose you’re tasked with analyzing the financial performance of a client’s product line and advising on potential pricing adjustments.

Here’s the scenario:

Company XYZ sells a product for $50 per unit.

Variable costs per unit, including materials and labor, amount to $20.

Fixed costs associated with production, such as rent and salaries, total $100,000 per month.

To calculate the contribution margin per unit, subtract variable costs from the selling price:

Contribution Margin per Unit=Selling Price−Variable Costs

Contribution Margin per Unit=$50−$20=$30

Now, we can determine the contribution margin ratio by dividing the contribution margin per unit by the selling price:

Contribution Margin Ratio=Contribution Margin per Unit/Selling Price×100%

Contribution Margin Ratio=$30/$50×100%=60%

This means that for every dollar of sales revenue generated, $0.60 contributes to covering fixed costs and generating profit .

In this scenario, understanding the contribution margin allows consultants to assess the impact of potential price changes on the company’s profitability.

By optimizing the contribution margin, businesses can make strategic decisions to enhance financial performance and achieve sustainable growth.

Breakeven analysis

A break-even analysis tells you how many units a company must sell to reach its breakeven point, i.e., to make $0 of profit .

In other words, a breakeven point is when the costs equal revenues.

Given that:

Revenue = Price x Quantity

Costs = Fixed costs + (variable costs x quantity)

Therefore, the breakeven point is:

Breakeven = (fixed costs)/(price – variable costs) = (fixed costs)/(unit contribution margin)

Example: A wood desk factory sells a desk for an average price of $200. To produce each wood desk, the company spends $30 on materials and $40 on labor.

They have $0.1M in monthly operating costs. How many wood desks must the factory sell monthly to break even?

Return on investment (ROI)

The return on investment (ROI), expressed as a %, is the ratio between the profit generated by an investment and the initial capital invested .

The higher the ROI of an investment, the better the investment is performing.

An ROI is easy to calculate and helps benchmark and comparison purposes.

The ROI formula is as follows: (gain from the investment)/(the capital invested)

Example : An investor purchased a property for $500K. Two years later, the investor sells the property for $750K. What is the ROI of this investment?

Answer : The ROI is ($750K – $500K) / $500K = 50%, meaning each $1 invested earned $1.5.

Payback period

The payback period, expressed as a number of years, is the amount of time it takes to recover the cost of an investmen t.

In other words, the payback period is the length of time an investment reaches a breakeven point.

Shorter paybacks mean more attractive investments

The payback period formula is as follows:

Payback period = (cost of the investment)/(net annual cash inflows)

For instance , if a company invests $300,000 in a new production line, and the production line then produces a positive cash flow of $100,000 per year, then the payback period is 3.0 years ($300,000 initial investment ÷ $100,000 annual payback).

Market share

A market share represents the percentage of the total market a company has .

And a market share can be measured in value or volume.

Example : if your client generates $1m of sales in a market estimated at $10m, its (value) market share is 10%.

Another example : your client sells 10,000 units per year, and the market is estimated at 1 million units sold per year.

Therefore, your client’s market share (in volume) is 1%.

Pricing elasticity

Pricing elasticity measures consumers’ responsiveness to a change in a product’s price; higher price elasticity suggests that consumers are more responsive to price changes .

Pricing elasticity is defined as:

For instance :

A company sells 300 million cigarettes at $8 per pack

The price elasticity for cigarettes is -0.3, which means that for every 1% rise in price, the quantity sold decreases by -0.3%

If the company raises prices from $8 to $10 per pack (+25% change in price), they can expect a drop of 22.5 million in sales (-7.5% change in quantity sold)

Practice drills for case interview math (with answers)

You can practice using the following case interview math practice drills in this chapter. 

Want the answers?

Sign up using this form , and you’ll receive the answers to all the case interview math practice drills. 

Mental calculations: addition and substraction

• Calculate: 457+238
• Calculate: 891−364
• Calculate: 1,245+987
• Calculate: 3,568−1,234
• Calculate: 6,789+2,345
• Calculate: 9,876−5,432
• Calculate: 12,345+6,789
• Calculate: 15,678−9,876
• Calculate: 23,456+8,765
• Calculate: 29,876−12,345

Mental calculations: multiplication

• Calculate: 12×14
• Calculate: 23×15
• Calculate: 35×18
• Calculate: 48×22
• Calculate: 56×29
• Calculate: 67×31
• Calculate: 72×35
• Calculate: 85×39
• Calculate: 96×42
• Calculate: 109×47

Mental calculations: division

• Calculate: 294÷7
• Calculate: 512÷8
• Calculate: 735÷5
• Calculate: 896÷4
• Calculate: 1,023÷9
• Calculate: 1,298÷6
• Calculate: 1,587÷3
• Calculate: 1,824÷8
• Calculate: 2,045÷7
• Calculate: 2,387÷5
• Calculate the Gross Margin given Revenue of $15,000 and Cost of Goods Sold (COGS) of $7,500.
• Estimate the Profit Margin for a company with $50,000 in Net Income and $500,000 in Revenue.
• Determine the Return on Investment (ROI) for an investment of $10,000 that generates a profit of $2,000. 
• Calculate the Operating Margin for a company with $75,000 in Operating Income and $300,000 in Revenue.
• Calculate the Payback Period for an investment of $50,000 that generates annual cash flows of $10,000.
• Determine the Net Present Value (NPV) for an investment with an initial cost of $100,000 and cash flows of $30,000 annually for 5 years, assuming a discount rate of 10%.
• Calculate the Return on Investment (ROI) for an investment of $80,000, generating a net profit of $20,000.
• Analyze the impact on quantity sold if the price of a product increases from $10 to $12, given that the amount sold decreases from 1,000 units to 800 units. 
• Determine the price elasticity of demand if a 10% decrease in price leads to a 15% increase in quantity sold. 
• Analyze the impact on revenue if the price of a product increases from $20 to $25 and the quantity sold decreases from 500 units to 400 units. 

Solve operations problems

• Calculate the output of a production line that processes 500 units per hour with an efficiency rate of 80%. 
• Determine the total cost of producing 1,000 units with a variable cost per unit of $5 and fixed costs of $2,000. 
• Calculate the capacity utilization rate of a factory that can produce 1,200 units per day but currently produces 800 units per day. 

McKinsey case interview math problems

Do you want to practice with four real-life case interview math questions recently used in McKinsey interviews?

I have prepared a 2-hour video presenting detailed solutions to these questions. 

Fill out the following form, and I’ll send you the video.

After you sign up, you’ll get the answers to all the above case interview math practice drills. 

Case interview math: final words

This is my guide to consulting case interview math in 2024.

Which concepts or formulas from today’s guide have you heard about for the first time?

After reading this guide, do you feel more confident about your consulting math skills?

Or maybe you have a question about something from this article.

Either way, let me know by leaving a quick comment below.

Leave a Comment Cancel Reply

Your email address will not be published. Required fields are marked *

You need 4 skills to be successful in all case interviews: Case Structuring, Case Leadership, Case Analytics, and Communication. Enroll in our 4 free courses and discover the proven systems +300 candidates used to learn these 4 skills and land offers in consulting.

Case Interview Math (formulas, practice problems, tips)

(Article updated 17th October 2024)

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

Becoming confident with math 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 math formulas that are used extensively in case interviews. 

Let’s dive in!

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

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

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

Here’s a summarised list of the most important math 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 math 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 math 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 math guide, just click on the link below.

Download free pdf case interview math cheat sheet

2. Case interview math practice questions

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

Doing math 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 math 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 math 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 math 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 math 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% female 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 math 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 math 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 math apps and tools

In the case math 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 math calculations will probably take some more practice. 

Mental math 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 math 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 math muscle in order to become as fast and accurate as possible.

• Preplounge's math tool . This web tool is very helpful to practice additions, subtractions, multiplications, divisions and percentages. You can both sharpen your precise and estimation math with it.
• Victor Cheng's math 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 math 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 math tips and tricks

4.1. calculators are often not allowed in case interviews.

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

Calculators are not usually 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 math 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 math, 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. Practising case interview maths

Sitting down and working through the math 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 practice making calculations under interview conditions.

5.1 Practise with peers

If you have friends or peers who can do mock interviews with you, that's an option worth trying. It’s free, but be warned, you may come up against the following problems:

• It’s hard to know if the feedback you get is accurate
• They’re unlikely to have insider knowledge of interviews at your target company
• On peer platforms, people often waste your time by not showing up

For those reasons, many candidates skip peer mock interviews and go straight to mock interviews with an expert.

5.2 Practise with experienced MBB interviewers

In our experience, practising real interviews with experts who can give you company-specific feedback makes a huge difference.

Find a consulting interview coach so you can:

• Test yourself under real interview conditions
• Get accurate feedback from a real expert
• Build your confidence
• Get company-specific insights
• Learn how to tell the right stories, better.
• Save time by focusing your preparation

Landing a job at a top consulting company often results in a $50,000 per year or more increase in total compensation. In our experience, three or four coaching sessions worth ~$500 make a significant difference in your ability to land the job. That’s an ROI of 100x!

Click here to book case interview coaching with experienced MBB interviewers.

Related articles:

Case interview math

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

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

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

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

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

What case math problems are really testing (Top)

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

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

Consulting firms use case math to test two things:

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

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

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

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

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

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

The case math skills you need to master (Top)

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

Division and multiplication

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

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

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

Case Example: Dealing with a TON of zeros

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

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

Case Example: Working with messy numbers

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

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

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

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

Percentages calculations

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

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

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

• Easy percentage calculations
• Messy percentage calculations

Case Example: Easy percentages

Simple percentage questions can be quite easy.

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

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

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

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

Case Example: Messy percentages

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

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

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

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

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

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

Breakeven analysis

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

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

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

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

Case Example: Launching a snazzy new tech product

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

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

Let's apply our formula

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

Growth estimations

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

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

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

Case Example: Determining future revenues from fixed, multiperiod growth

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

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

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

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

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

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

Case Example: Determining future costs from variable, multiperiod growth

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

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

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

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

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

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

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

Market size calculations

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

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

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

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

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

Case Example: Market math with "easy" percentages

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

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

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

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

You get the idea!

Case Example: Market math with "messy" percentages

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

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

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

Tips for how to prepare (Top)

Isolate core skills and master them.

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

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

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

Learn mental math short cuts

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

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

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

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

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

Conclusion: Practice makes perfect (Top)

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

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

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

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

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

Read this next:

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

See all RocketBlocks posts .

Get interview insights in your inbox:

New mock interviews, mini-lessons, and career tactics. 1x per week. Written by the Experts of RocketBlocks.

P.S. Are you preparing for consulting interviews?

Real interview drills. Sample answers from ex-McKinsey, BCG and Bain consultants. Plus technique overviews and premium 1-on-1 Expert coaching.

Launch your career.

• For schools
• Expert program
• Testimonials

Free resources

• Behavioral guide
• Consulting guide
• Product management guide
• Product marketing guide
• Strategy & BizOps guide
• Consulting case library

Interview prep

• Product management
• Product marketing
• Strategy & Biz Ops

Resume advice

• Part I: Master resume
• Part II: Customization
• Focus: PM resumes
• Focus: Consulting resumes
• Focus: BizOps resumes