Consulting Case Interview Math: Everything You Need to Know
Consulting 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 case interview math may seem difficult and intimidating, especially for candidates without strong quantitative backgrounds, but the math itself is very basic and simple.
In this article, we’ll go through all of the math concepts and fundamentals you need to know for consulting case interviews. Then, we’ll go through the five types of consulting case interview math problems. Finally, we’ll cover consulting math tips to help make the case interview math easier and simpler for you.
Consulting Case Interview Math Fundamentals
You should be familiar with the following math fundamentals:
- Ratios and proportions
- Linear Equations
Fractions are numbers expressed as n/d. Examples include: 1/2, 2/3, and 3/4. You should know how to do the following calculations with fractions:
- Simplification: 36/54 = 2/3
- Addition: 1/3 + 1/4 = 4/12 + 3/12 = 7/12
- Subtraction: 1/3 – 1/4 = 4/12 – 3/12 = 1/12
- Multiplication: 3/4 * 4/9 = 12/36 = 1/3
- Division: 3/4 ÷ 9/2 = 3/4 * 2/9 = 6/36 = 1/6
Decimals are numbers in which the positioning of the period or decimal point determines the value of the digits.
Example: 1,542.36 = (1 * 1,000) + (5 * 100) + (4 * 10) + (2 * 1) + (3 * 0.1) + (6 * 0.01)
You should know how to do the following calculations with decimals:
- Addition: 1.21 + 3.5 = 4.71
- Subtraction: 14.2 – 8.05 = 6.15
- Multiplication: 1.12 * 5 = 5.6
- Division: 14.7 ÷ 4.2 = 3.5
You should also be familiar with scientific notation, which is a decimal expressed as the product of a number with only one digit to the left of the decimal point and a power of 10.
Example: 2.731 * 10^2 = 273.1
Percentages are numbers expressed out of 100 or per hundred.
Example: 75% = 75/100 = 0.75
You should know the formula for percent change.
Percent Change = (New Value – Old Value) / Old Value
Example: The price of gas last month was $3 per gallon. This month, the price increased to $3.60 per gallon. By what percentage did the price of gas increase?
Percent Change = ($3.60 - $3) / $3 = 20%
The price increased by 20%.
Ratios and Proportions
A ratio is a relation between two numbers. Examples include:
- 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.
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.
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.
Types of Consulting Case Interview Math Problems
Now that you know the consulting case interview math fundamentals, you can move onto learning the five types of math problems you’ll likely encounter in consulting 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.
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: 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: A private equity firm purchases a roofing tile distributor for $100M. After growing the company’s revenues within the first few years, the company was then sold for $130M. What was the return on investment?
Return on Investment = Profit / Investment Cost = $30M / $100M = 30%
The private equity firm has generated a 30% return on investment over the time period.
Operations problems typically deal with production and the usage of machinery or plants. There are two key formulas you need to know:
- Output = Rate * Time
- Utilization = Output / Maximum Output
Example: 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: 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
- Stacked bar chart
- 100% stacked bar chart
- Marimekko / Mekko chart
- Pie chart
- Waterfall chart
- Line graph
You can watch the video below for a review on how to interpret each of these.
Market Sizing Questions
Market sizing questions ask you to determine the size of a particular market.
Market size is defined as the total amount of sales of a product or service in one year in a given geography or region. However, it can also be defined as the number of units sold in a year or the total number of customers that would purchase a product or service.
There are two main approaches to solving market sizing questions:
- Top-down approach: start with a large number and then refine and break down the number until you get your answer
- Bottom-up approach: start with a small number and then build up and increase the number until you get your answer
Example: What is the market size of tires for personal vehicles in the United States?
To solve this market sizing question, we can use a top-down approach:
- Start with the United States population
- Estimate the average number of people per household
- Estimate the percentage of households that own cars
- Of those households, estimate the average number of cars owned
- Multiply by four wheels
- Estimate the frequency in which wheels are replaced
- Multiply by the cost of a tire
- Multiply all of these figures to determine the market size of tires for personal vehicles
Let’s start with a United States population size of 320M. We can estimate that the average household has 2.5 people. Therefore, there are 320M / 2.5 = 128M households in the US.
Assume that 75% of US households own a car. That gives us 75% * 128M households = 96M households that own a car
Among households that own a car, the average number of cars owned is about 1.5 cars per household. Therefore, 1.5 cars * 96M households = 144M cars.
Each car has 4 tires, so there are 4 * 144M cars = 576M tires for personal vehicles in the U.S.
Tires are replaced approximately once every six years. Therefore, in a given year, 576M tires / 6 = 96M tires are sold.
If tires cost an average of $100 each, then 96M tires * $100 = $9.6B.
The market size of tires for personal vehicles in the United States is $9.6B.
Consulting 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 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.
- 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
- 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.
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