Redrock Study percentage questions

Correlation and causality are two important terms in statistics that are often confused with each other. In this text, the difference between the two is explained with understandable examples and related to the business consulting industry. Keep in mind that math is an important aspect of the case interview.The Principle of CorrelationCorrelation describes the degree of relationship between two variables. When two variables correlate, it means that changes in one variable are accompanied by changes in the other variable. A positive correlation occurs when the values of both variables increase or decrease together. An example of a positive correlation is the relationship between the number of hours spent preparing for an exam and the score achieved. As a rule, the more time one invests, the better the exam score.A negative correlation occurs when the values behave in opposite ways, that is, when the value of one variable increases, the value of the other variable decreases, and vice versa. As an example, there may be a negative correlation between the number of hours one spends watching TV and physical fitness. The more time one spends in front of the TV, the less time one has for physical activities, which can lead to lower fitness.However, it is important to understand that correlation alone is not sufficient to conclude causality.The Principle of CausalityCausality means that a change in one variable is a direct cause of a change in another variable. In other words, one variable causes a change in another variable. An example often used to explain the difference between correlation and causality is the relationship between the number of ice cream sales and the number of sunburns. There is a positive correlation between these two variables because more ice cream is sold on sunny days and more people also get sunburned. However, it would be wrong to assume that ice cream consumption causes sunburns. In reality, the common cause of both is sun exposure. On sunny days, people buy more ice cream and are also at higher risk of sunburn. This example illustrates that correlation does not automatically indicate causation.Correlation and Causality in Management ConsultingIn consulting, understanding correlation and causality can play a critical role. When consultants analyze data to help companies or organizations, it is important that they draw the right conclusions and not make false assumptions. Usually it is important to decide which factors should be qualitative and/or quantitative analyzed.Let's say a company finds that there is a strong correlation between the use of a particular marketing strategy and sales growth. The company might be tempted to assume that the marketing strategy is directly responsible for sales growth and therefore continue to invest heavily in that strategy. However, it could be that the correlation only indicates that both variables are influenced by another, as yet unknown, cause. To confirm causality, further research would be required, such as experiments or bigger control groups to test the direct impact of marketing strategy on sales. Possible causes could also be pricing, a change in the product portfolio or the changing size of the market.Overall, it is important to understand that correlation and causality are different concepts. Correlation simply describes the relationship between two variables, while causality implies a cause-and-effect relationship. Just because two variables correlate does not automatically mean that one variable is the cause of the change in the other variable. Further investigation and evidence is needed to conclude causality.

Nearly all the math required in a case interview is arithmetic: addition, subtraction, multiplication, and division. You will likely need to perform these calculations with percentages, decimals or fractions, and calculate a percentage value (which requires division). Most candidates understand these math concepts; the challenge for many people is performing multiple calculations with large numbers quickly and without calculators or spreadsheets, in a high-pressure environment with a six-figure salary and your future career on the line.If practiced, the maths part of the case interview can be completed competently and without much stress. Demonstrating good maths skills can be your differentiator in your interview so it is worth spending the time to practice. Being quick is not only a major plus for your performance but it also helps keep a natural flow to your interview, helping you build rapport with your interviewer.Good mental maths skills are particularly useful in market sizing/guesstimate questions and consulting case studies.​Case numbers are “round” with few significant digitsThe good news is that most of the numbers you encounter in case interviews (what we call “Case Numbers”) are “round” numbers and have only a few significant digits (i.e. only a few digits are non-zero), but may be in the Millions or Billions. For example, the number 200 Thousand, which is 200,000 in long-form, has a single-digit that is not zero (“2”), and so has one significant digit. The number 2.5 Million, which is 2,500,000 in long-form, has two non-zero digits (“2” and “5”) and therefore has two significant digits. Case numbers frequently have only one or two significant digits, less frequently three significant digits, and in rare cases four or more significant digits. The term "Case Calculation" refers to a calculation required in a case interview, and may provide specific Case Numbers or refer to a general calculation without specifying the numbers involved, such as calculating Revenue, given Price, and Quantity. Examples​Here are some examples of the types of questions you would get in a case interview:​Q. A hairbrush manufacturer has fixed costs of £3,000,000 and they sell 175,000 units a year, what is the fixed cost per unit?​A. £3,000,000 ÷ 175,000 = £17.1​ Q. A global software company sells products in Germany, France, and the US. The German market had revenues of £2.35bn last year, France had sales 23% higher than Germany, what were their total revenues?​A. £2.35bn x 1.23 = £2.89bn Methodologies​Maths skills take practice but before you begin your practice it is best to choose calculation methods that you are most comfortable with. The method that you were taught when you were younger may not always be the method you want to use now.​Below we have found some methods that we think are good for speed and accuracy in pressure situations along with any useful tips or shortcuts that we are aware of. Addition and SubtractionMost of the Case Numbers you need to add and subtract will have only a few significant digits. If they have more than one significant digit, the last or “trailing” significant digits will often be “5” or “25,” which makes them easier to add/subtract. Example 1: Add 250 Million, 300 Million, and 150 Million.Answer: 700 MillionIn this additional example, the leading digits (prior to the Million) are: 250, 300, and 150. These numbers have either one significant digit (300) or two significant digits, where the last significant digit is a “5,” which makes addition easier.When you need to add Case Numbers with three significant digits, the last two significant digits are often “25” or “75,” which are also easy to calculate with. Example 2: Add 225 Million, 375 Million, and 200 Million.Answer: 800 MillionYou may need to add some Case Numbers with three significant digits that don’t end in “25” or “75,” but they most likely end in “5.” For example, you might need to add numbers like 115 and 165. Since these numbers both end in “5”, they are relatively easy to add without a calculator, and the answer is 280.It is highly unlikely you will need to add or subtract a series of numbers with three or more significant digits, where all the digits are effectively random, such as 147, 368, and 434 (where the last significant digits are not “25” or “75”).However, if you have to do so, we recommend the regrouping method.The regrouping method aligns the two numbers on top of each other by their units and then you sum each unit column to find the total number. It reduces the complexity of the equation down to single-digit additions and can be used for decimal places too. The long subtraction method we prefer is built on the same principle as the long addition and so you don't need to learn a new method but rather apply it differently. The two numbers are again aligned on top of each other and instead of summing each column, you subtract the two numbers.*box-open*For extra practice, check out our Quizzes on Logical Reasoning and Math Assignments for Case Interviews!*box-close*Multiplication, Division, and PercentagesAs you can see from the prior examples, candidates need to perform multiplication and division calculations using numbers in the thousands, millions, or even billions (but with a limited number of significant digits). In case interviews, candidates often need to perform multiplication and/or division with percentages, decimals, and fractions. For example, you might need to multiply a number by a percentage (e.g., calculate 25% of $500 Million), or divide two numbers and express the result as a percentage (e.g. what percentage of $80 Million does $16 Million represent?). Similar operations using fractions instead of percentages are also frequently required.Most of the Case Numbers you need to multiply or divide will be round numbers with only a few significant digits. For instance, the number of units sold for each product could be 5 Million, 10 Million, and 12.5 Million. In a case interview, you are unlikely to receive a similar problem where the number of units sold is something like 9,618,493, which has many significant digits that appear random. The percentage values you need to multiply/divide within case interviews will also usually have only a few significant digits. A typical Case Calculation would be calculating 20% or 25% of another number. It is unlikely you would need to calculate 23.7% of a value in a Case Calculation.Example 3: Calculate $120 Million times 250.Answer: $30 Billion*box-open*Don't know how to derive that answer? Check out our next article for practical tips & tricks*box-close*MultiplicationLong multiplication methods have the most variance of the calculations but most follow the simple principle of breaking the large numbers into their component parts e.g. 728 is 700, 20, and 8. To keep things as similar as possible across the different calculations we again choose to work vertically with the numbers aligned by units.DivisionDivision gets most difficult when dealing with decimals but the method we prefer allows you to continue into decimals seamlessly, this is particularly helpful when dealing with small amounts of money e.g. $3850.45, the 45 cents make the calculation more difficult.​This method moves away from the vertically aligned method but we find it easiest and most versatile for long division.PercentagesRelativity is an important concept in consulting case interviews as a number on its own does not demonstrate the value of the numbers in comparison to the other number(s) in the question. Being able to convert to percentages is an important skill and will often be required in case interviews.​There are a number of simple ways to find percentages but if you are stuck try finding 10%, 5%, and 1% first, using a combination of these amounts you will be able to find other percentages fairly quickly e.g. 37% = (10% x 3)+5%+(1% x 2). Compounded GrowthCandidates also need to understand compound percentage growth and how to make approximations with compound growth. It is a very common Case Calculation to be given a firm’s revenue (or another metric), and the associated Compound Annual Growth Rate (CAGR), and be asked to estimate that value at some point in the future. Net Present Value (NPV)Another common Case Calculation involves assigning a financial value to monetary payments that will occur in the future. Management Consultants usually determine the current value of future payments using a financial method called Net Present Value or NPV. Hence, candidates need to be able to calculate the NPV under a variety of scenarios. NPV Example 1: How much would your company be willing to pay for another company that generates $20 Million in profit annually, if your firm requires an annual Return on Investment of 10%?Answer: $200 Million NPV Example 2: A real estate development firm is evaluating a project that involves buying a parcel of land and building condominiums on that parcel. The company forecasts they can sell the condominiums for a total of $250 Million six years in the future. What is the maximum the real estate company would be willing to spend now to buy the land and develop the condominiums, if all the associated costs for the project would be incurred today, and they require a 12% annual return on invested capital? Answer: $125 Million*box-open*Don't know how to derive that answer? Check out our next article for practical tips & tricks*box-close* More Useful TipsTackling questions like this can be daunting. To learn how to approach questions like this, have a look at our 5 top tips:1. Don’t worry about getting the EXACT answerIt is important to remember that the interviewer is unlikely to be directly testing your maths skills, they just want to see how you approach the question so finding the answer to 5 decimal places is not important. They will usually be happy with an approximate answer, especially if it makes your calculations quicker. 2. Round the numberIf you have the choice, make the numbers easy for yourself. As an example, it is helpful to take the UK population as 60 million rather than its true value of 66 million. They are not always looking for perfect answers, just a good thought process and rough figure. 3. Shorten long numbersIf you have the opportunity to write down the numbers, shorten a thousand to ‘k’, million to ‘m’ and billion to ‘b’. This will help you write faster and keep the numbers smaller as too many 0’s can be confusing. 4. Verbalize your reasoning to the interviewerExplain out loud the calculation you are about to make. The interviewer may agree it is the right direction, they may give you part of the answer, or they could course correct and push you to a different part of the problem. It is important to give the interviewer the opportunity to help you as much as possible. 5. Make reasonable assumptionsOften you will be asked questions in which you are required to come up with the numbers yourself. These questions do not test accuracy but test your logic skills and common sense. As long as you are clear about the assumption you are making e.g. Every person living in the city gets their haircut once a month, then the interviewer can see your logic and challenge it if they think they need to. Helpful ResourcesMake sure to not only read through this article but to actually practice! There are a number of resources out there to help you with your maths skills and to improve your performance:PrepLounge Mental Math Tool - interactive and user-friendly tool to train your case interview math with respect to all basic operations (e.g. addition, multiplication, percentage). You can even compare your performance to the overall PrepLounge community!The Khan Academy - the website and app provides videos and practice for all your maths needs. They have exercises for all different levels and on an extensive range of topics.​BBC Skillswise - BBC has curated lessons to help adults gain skills for the workplace. Their maths skills section provides a good overview of the maths skills you would need.