do you dread the math portion of case interviews perhaps you don't come from a quantitative or analytical background or maybe your math skills are Rusty from not having done math in a while I'm a former Bane manager and interviewer and I'm going to give you a comprehensive review of case interview math this is the only video you need to watch to learn case interview math by the end of this video you'll know all of the math fundamentals you need to solve any case interview in just about 20 minutes we'll cover the six fundamental math Concepts you need to know how to solve the five types of case interview math questions and five case interview math tips to put you in the top 10% of candidates one of these math tips is our number one tip that helps most candidates cut their math mistakes in half instantly when they use it so make sure you watch Until the End case interview math will show up in every Consulting 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 case case interview math may seem difficult and intimidating especially for candidates without strong quantitative backgrounds but the math itself is very basic and simple let's get started right away the six fundamental case interview math concepts you should be familiar with the following math fundamentals fractions decimals percentages ratios and proportions basic statistics and algebraic equations let's quickly cover each of these fractions fractions are numbers expressed as one number the numerator divided by another the denominator examples include 1/2 2/3 and 3/4s you should know how to do the following calculations with fractions simplification 36 over 54 = 2/3 addition 1/3 + 1/4 = 4 12th + 3 12th which = 7 12th subtraction 1/3 - 1/4 = 4 12th - 3 12th = 1 12th multiplication 3 over 4 * 49 = 12 36 or 1/3 division 3 4 / 9 /2 = 34 * 2 9 which = 6 36 which = 1 6 decimals decimals are numbers expressed as a whole number and fractional part a decimal point separates these two components here's an example 15423 = 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 for example 2.73 1 * 10 ^ 2 = 273.15 equal 75 over 100 or 0.75 you should know the formula for percent change percent change equals the new value minus old value ID by the 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 equals $3.60 minus $3 ID by $3 which equals 20% the price increased by 20% ratios and proportions a ratio is a relation between two numbers it can be expressed in one of three different ways shown here a proportion is a relation between two equal ratios examples include the following example a factory requires two 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 one supervisor for every 7 .5 workers to find the number of Supervisors required we can divide 45 workers by 7.5 to get six therefore six supervisors are required statistics only very basic statistics knowledge is needed for case interviews you should be familiar with four important Concepts meanor average weighted average standard deviation and expected value the most important concept is calculating the mean or average of a set of numbers example if our three large competitors are paying $10 per hour $1 per hour and $15 per hour for labor what should we expect our average labor cost to be the average equals 10 + 11 + 15 / 3 which is the same as $36 divid by3 which equals $12 so 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 Revenue 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 so profit margin equals 20% * 70% plus 60% * 30% which equals 14% + 18% or 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 you 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 and the company will achieve $120 million in sales in the first year however there was a 60% likelihood that the partnership will be less successful generating only $40 million in sales in the first year what is the expected value of sales in the first year the expected value is 40% * $120 million plus 60% * $40 million which gives us $72 million the expected value of sales is $72 million in the first year algebraic 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 for example a company's revenues grew by 60% this year compared to last year year if this year's Revenue was $100 million what was the company's revenues last year to solve this question let x equal the company's revenues last year and we have 1.6x equal $100 million 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 this gives us x = 62.5 million the company's revenues last year were $62.5 million if you're finding this video helpful please take 3 seconds to give this video a like And subscribe to the channel so that you don't miss any of our new Consulting case interview content let's move on to the five types of case interview math questions now that you know the Consulting case interview math fundamentals you can move on to learning the five types of math problems you'll likely encounter in Consulting case interviews these are profit and break even questions investment questions operations questions charts and graphs questions and 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 break even 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 to know which can be put together and simplified into a single comprehensive equation for profit profit equals Revenue minus costs Revenue equals quantity time price costs equal total variable costs plus total fixed costs and total variable costs equals quantity time variable costs putting these together we get the formula profit equals price minus variable costs times quantity minus total fixed costs example a pizza store sell sells 100,000 pizzas per year at a price of $10 each assume that each Pizza costs $4 to produce the store pays $150,000 per year in rent and hires two employees at a salary of $75,000 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 equals quantity time price or $ 100,000 pizzas time $10 which equals a million variable costs equals quantity time variable cost or $ 100,000 pizzas * $4 which equals $400,000 fixed costs equal $150,000 +2 * $775,000 which equal $300,000 total costs equals variable costs plus fixed costs or $400,000 plus $300,000 which equal $700,000 and finally profit equals Revenue minus costs or $1 million minus $700,000 which equals $300,000 from our calculations we know that annual revenues are1 million annual costs are 700,000 and annual profit is $300,000 another type of question are profit margin questions profit margin measures how much earnings a business keeps relative to its Revenue you need to know the profit margin formula which is profit margin equals Revenue minus costs ided by revenue or profit divided by 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 equals profit divid revenue or 300,000 ID a million which equals 30% therefore the pizza store has a profit margin of 30% the third type of question you may be asked are Break Even questions a break even occurs when a company sells enough product such that it has exactly recouped all costs in other words break even occurs when profit is zero to solve for the break even 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 we know price equals $10 variable costs equal $4 fixed costs are $300,000 and Q is the quantity needed to break even so plugging this into our profit equation profit equals price minus variable costs time quantity minus fixed costs and we set this to zero so 0 = $10 - $4 * Q - $300,000 and so Q = $50,000 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 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 or a private Equity Firm acquiring a company to resell in later years for a Prof 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 equals profit divide by investment cost and payback period equals investment cost divided by profit per year example a software company acquires a machine learning startup for $20 million the startup's expertise in machine learning is expected to improve the software company's product such that it will generate an additional $4 million per year in profit what is the payback period payback period equals investment cost divided by profit per year or $20 million divided by $4 million which is 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 $100 million after growing the company's revenues within the first few years the company was then sold for $130 million what was the return on investment return on investment equals profit divided by investment cost or $30 million divided by $100 million giving 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 twoy key formulas you need to know output equals rate time time and utilization equals 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 time 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 10 types of charts and graphs you should be familiar with simple bar chart Compares a single variable a stacked bar chart Compares a single variable and shows the segments that make up each bar a 100% stacked bar chart shows the percentage each segment makes up in a bar enabling percentage comparisons across bars a pie chart shows the percentage each segment makes up of a total a merco or mecho chart is a more complex version of a pie chart and it shows the percentage each segment makes up of a total across two Dimensions waterfall chart this shows drivers of change between two comparison points a histogram shows frequency distributions a line graph shows changes in a variable over time a scatter plot shows the relationship between two variables and finally a bubble chart is a more complex version of a scatter plot that shows a relationship among many variables whenever you are given a charter graph the very first step you should do is to read the axes of the chart or graph out loud this will help you determine what information is being provided this is especially helpful when you are given several charts or graphs reading the axes of each Charter graph out loud will also help you see the relationship across different charts and graphs 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 a top down approach starts with a large number and then refines and breaks down that number until you get your answer a bottom up approach starts with a small number and then builds up and increases the number until you get your answer let's look at an 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 these 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 and then multiply all of these figures to determine the market size of tires for personal vehicles let's start with the United States population of 320 million we can estimate that the average household has 2.5 people therefore there are 320 million divided by 2.5 or 128 million households in the US assume that 75% of us households own a car that gives us 75% * 128 million households equals 96 million 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 times 96 million households equals 144 million cars each car has four tires so there are four time 144 million cars which equals 576 million tires for personal vehicles in the US tires are replaced approximately once every 6 years therefore in a given year 576 million tires ided by 6 equals 96 million tires that are sold if tires cost an average of $100 each then 96 million tires times $100 equals $9.6 billion the market size of tires for personal vehicles in the US is 9.6 billion let's move on to case interview math tips now that you know the case interview math fundamentals and the five types of case interview math questions let's move on to reviewing a few case interview math tips using these tips will help you differentiate yourself from other candidates tip number one develop a structure before doing math calculations 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 a confirmation on whether the approach makes sense once the interviewer approves of your structure the rest of the math is simple arithmetic tip number two 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 by 17 see if the interviewer will allow you to round so that you multiplying 200 by 17 you don't want to round too much since this may signal to the interviewer that you are are uncomfortable performing math calculations if the numbers are not easy in round however rounding occasionally can help simplify the calculations and reduce the likelihood that you make a calculation error tip number three use abbreviations for large numbers this tip is our top case interview math tip implementing this tip can instantly reduce your math Mistakes by 50% or more if you're working with numbers in the thousands millions billions or trillions use abbreviations rather than writing out all all of the zeros for example 10,000 can be expressed as 10K 200 million can be expressed as 200 m 30 billion can be expressed as 30b and 4 trillion 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 equal 1,000 1 m = 1,00 k or k * K 1 B = 1,000 m or k * m and 1 t = 1,00 b or k * b or M * m for example 14,000 * 5000 = 14 k * 5K 14 * 5 = 70 and K * k = m giving us 70 M 2000 * 14 M = 2K * 14 m 2 * 14 = 28 and K * m = b giving us 20 8 B 12 M * 6 M = 12 * 6 which is 72 and M * m = t giving us 72t tip number four use the rule of 72 to estimate growth 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 grb 9% then it would take approximately 72 / 9 or 8 years for the investment to double if a market is growing at 12% per year then it would take approximately 72 divid 12 which equals 6 years for the market size to double tip number five sense check numbers along the way accidentally missing zeros or adding extra zeros during your calculations is the most common math mistake after each step of a math calculation you you can do a quick sense check to see if your answer is in the right order of magnitude for example if you're multiplying 125 Million by 24 you should expect your answer to be in the billions because 100 million * 20 equal 2 billion and with that you have completed our brief overview on case interview math you are now fully equipped to answer any quantitative question in your upcoming case interviews you'll just need to practice and hone your math skills if you're looking for a quick step-by-step overview of how to solve the the entirety of a consultant case interview make sure to check out our video on learning case interviews in 30 minutes I've linked the video in the description below and last important message for you if you're looking for a step-by-step shortcut to learn case interviews quickly check out our comprehensive case interview course and books links are in the video description we teach the most robust effective case interview strategies in the least time-consuming way to save you hundreds of hours of prep time and help you land multiple consulting job offers
