Percentage Math Practice

One of the basic skills a candidate requires is good mental math abilities. Your interviewer will test this skill either directly by asking a math question such as "What is the square root of 759?" or in an indirect manner while solving a business case problem.Interviewers can differentiate between hard and soft mistakes. Calculation errors are considered hard mistakes because there is no partial answer in math. Therefore, you must improve or refresh your mental math skills before interviews. Remember, it is alright to use scratch paper for most calculations, but it is expected that you are able to mentally perform easy calculations. Try the following tool to improve your mental math skills.*box-open*Train your brain! Improve your math skills using our Mental Math tool!*box-close*Impress the interviewer with outstanding mental math skillsPossessing excellent mental math skills is crucial during a case interview because you will need these skills as a consultant. In general, there are two main reasons why consultants should possess excellent math skills:As a consultant, you will come across several occasions in which you will need mental math skills. Estimation skills can save a lot of time and, in some cases, help figure out unnecessary analysis that can waste a lot of time if a quick estimation is not done (see 80/20 or pareto principle). Without excellent math skills, you could end up spending the entire night using Excel.You can easily impress your client by demonstrating to them that you are a potential math genius. Clients often pay a lot of money to you (or your employer) and expect real wisdom and cleverness. Quick math solutions can be an effective way to convince clients of your recommendations.Math or "quant" skills account for 20-25% of overall performance evaluationQuant's skills are only one of the skills assessed in case interviews. As a candidate, you are evaluated on your overall combination of skills and your perceived potential. Generally, a candidate’s quantitative assessment consists of 20-25% of the overall interview evaluation weight. The goal of Management Consulting firms is to find the set of candidates with what they perceive is the best set of skills overall.Most firms have a “minimum bar” for quant skills for their candidates, which means you are unlikely to receive an offer if you do not exceed this bar—unless you are otherwise a highly exceptional candidate (such as being a Rhodes Scholar). At some firms, dramatically exceeding this “bar” does not strengthen your candidacy—that is, the quantitative evaluation is binary: you either pass the quant assessment or you don’t. At other firms, such as McKinsey, if you can demonstrate very strong quantitative skills, it will differentiate you from other candidates. The quant problems from these firms can be more complex, which allows candidates with strong quant skills to fully demonstrate their ability. The corollary is that, for some very complex quant problems, you don’t need to correctly solve all the components to receive an offer.The relative importance of quantitative skills in the case interview varies by firm, office, and individual interviewer. As an example, if you are interviewing with a consulting firm partner who has a Ph.D. in a very quantitative field like Applied Mathematics or Physics, they may ask more challenging quant questions and place a greater emphasis on quant skills in their interviews. McKinsey, in particular, places more emphasis on quant skills than other firms, and, as mentioned previously, demonstrating exceptional quant skills can significantly strengthen your candidacy at McKinsey.Given the large weighting of 20-25% that quant skills have in consulting interviews - and the quant “minimum bar” at most consulting firms - you can certainly sink your chances of receiving an offer if you perform poorly in the quant portion of the interview.*box-open*For extra practice, check out our Quizzes on Logical Reasoning and Math Assignments for Case Interviews!*box-close*SummaryA quant's skills are needed to be an effective Management Consultant; hence, they are assessed in Management Consulting interviews. Quant's skills are one of the skills consulting firms consider in their interviewsAt most firms, if you do not exceed a “minimum bar” for quant skills, it is unlikely you will receive an offerYou cannot use calculators or spreadsheets in Management Consulting interviews, and the assessment of your quant skills usually contributes about 20-25% of the overall interview evaluation weight.

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.