# Case Interview Math: Comprehensive Guide (2024)

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:

• Fractions

• Decimals

• Percentages

• Ratios and proportions

• Statistics

• Linear Equations

Fractions

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

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

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:

• 2/3

• 2:3

• 2 to 3

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

• 2/3 = 4/6

• 2:3 :: 4:6

• 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

• x = \$62.5M

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 and Breakeven 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

• Fixed Costs = \$150K + (2 * \$75K) = \$300K

• Let Q = quantity needed to break even

• Profit = (Price – Variable Costs) * Quantity – Fixed Costs

• Set profit equal to zero

• \$0 = (\$10 - \$4) * Q - \$300K

• Q = 50K

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

Investment Questions

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 Questions

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.

Charts and Graphs Questions

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.

Market Sizing Questions

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:

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:

• 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

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:

• 1K = 1,000

• 1M = 1,000K = K * K

• 1B = 1,000M = K * M

• 1T = 1,000B = K * B = M * M

For example:

• 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

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• 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.

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