At IGotAnOffer, we have helped more than 30,000 candidates prepare for their consulting interviews. Students who go through our full training programme are a happy bunch: more than 80% of them get an offer at McKinsey, BCG or Bain.
Developing fast and accurate maths skills is a big part of being successful at case interviews. In the following guide we've listed a number of free tools, formulas and tips you can use to become much faster at maths and radically improve your chances of getting an offer.
Part 1: Case maths apps and tools
Mental maths is a muscle. But if you are like us, you probably haven't exercised that muscle much since you left high school. As a consequence, your case interview preparation should include some maths training.
If you don't remember how to calculate basic additions, substractions, divisions and multiplications without a calculator that's what you should focus on first. Our McKinsey and BCG & Bain case interview programmes both include a refresher on the topic.
In addition, Khan Academy has also put together helpful resources. Here are the ones we recommend taking a look at if you need an in-depth arithmetic refresher:
Once you're feeling comfortable with the basics you'll need to regularly exercise your mental maths muscle in order to become as fast and accurate as possible. Our McKinsey and BCG & Bain case interview programmes include a calculation workbook PDF with maths drills. We recommend doing a few everyday so you get more and more comfortable over time.
In addition, you can also use the following resources. We haven't tested all of them but some of the candidates we work with have used them in the past and found them helpful.
- Preplounge's maths tool. This web tool is very helpful to practice additions, subtractions, multiplications, divisions and percentages. You can both sharpen your precise and estimation maths with it.
- Victor Cheng's maths tool. This tool is similar to the Preplounge one but the user experience is less smooth in our opinion.
- Magoosh's mental maths app (iOS + Android). If you want to practice your mental maths on the go this free mobile app is great. It lets you work on different types of calculations using mental maths flashcards. You can also track your progress as you study.
- Mental math cards challenge app (iOS). This mobile app let's you work on your mental maths in a similar way to the previous one. Don't let the old school graphics deter you from using it. The app itself is actually very good.
- Mental math games (Android). If you're an Android user this one is a good substitute to the mental math cards challenge one on iOS.
Part 2: Case maths formulas
The links we have listed above should go a long way in helping you bring your maths skills to a good level. In addition, you will also need to learn the formulas for the main business and finance concepts you will come across in your interviews.
We've put together a list of the important maths formulas for you with concepts that you should really master for your interviews and concepts which are optional in our experience.
2.1. Must-know maths formulas
Revenue = Volume x Price
Cost = Fixed costs + Variable costs
Profit = Revenue - Cost
Profit margin (aka Profitability) = Profit / Revenue
Return on investment (ROI) = Annual profit / Initial investment
Breakeven (aka Payback period) = Initial investment / Annual profit
If you have any questions about the formulas above you can ask them at the bottom of this page and our team will answer them. Alternatively, you can also read our in-depth articles about finance concepts for case interviews and for the McKinsey PST or watch this video where we explain these concepts in great detail.
2.2. Optional maths formulas
Having an in-depth knowledge of the business terms below and their corresponding formula is NOT required to get offers at McKinsey, BCG, Bain and other firms in our experience. But having a rough idea of what they are can be handy.
EBITDA = Earnings Before Interest Tax Depreciation and Amortisation
EBIDTA is essentially profits with interest, taxes, depreciation and amortization added back to it. It's useful to compare companies across industries as it takes out the accounting effects of debt and taxes which vary widely between say Facebook (little to no debt) and ExxonMobil (tons of debt to finance infrastructure projects). More here.
NPV = Net Present Value
Let's say you invest $1,000 in project A and $1,000 in project B. You expect to receive your initial investment + $500 from A in one week. And you expect to receive your initial investment + $500 from B in 5 years. Intuitively you probably feel that A is more valuable than B as you'll get the same amount of money but quicker. NPV aims to adjust future cashflows so different investments such as A and B can easily be compared. More here.
Return on equity = Profits / Shareholder equity
Return on equity (ROE) is a measure of financial performance similar to ROI. ROI is usually used for standalone projects while ROE is used for companies. More here.
Return on assets = Profits / Total assets
Return on assets (ROA) is an alternative measure to ROE and a good indicator of how profitable a company is compared to its total assets. More here.
Part 3: Fast maths tips and tricks
The standard long divisions and multiplications approaches are great because they're generic and you can use them for any calculation. But they are also extremely slow. In our experience, you can become MUCH faster at maths by using non-standard approaches we've listed below.
All these approaches have ONE thing in common: they aim at rearranging and simplifying calculations to find the EASIEST path to the result. Let's step through each of them one by one.
3.1. Rounding numbers
The first step towards becoming faster is to round numbers whenever you can. 365 days becomes 350. The US population of 326m becomes 300m. Etc. You get the idea.
The tricky thing about rounding numbers is that if you round them too much you risk a) distorting the final result / finding, and b) your interviewer telling you to round the numbers less.
Rounding numbers is more of an art than a science, but in our experience the following two tips tend to work well:
- We usually recommend to not round numbers by more than +/- 10%. This is a rough rule of thumb but gives good results based on conversations with past candidates.
- You also need to alternate between rounding up and rounding down so the effects cancel out. For instance, if you're calculating A x B, we would recommend rounding A UP, and rounding B DOWN so the roundings compensate each other.
Note you won't always be able to round numbers. In addition, even after you round numbers the calculations could still be difficult. So let's go through a few tips that can help in these situations.
3.2. Handling large numbers
Large numbers are difficult to deal with because of all the 0s. To be faster you need to use notations that enable you to get rid of these annoying 0s. We recommend you use labels and the scientific notation if you aren't already doing so.
Labels (k, m, b)
Use labels for thousand (k), million (m) and billion (b). You'll write numbers faster and it will force you to simplify calculations. Let's use 20,000 x 6,000,000 as an example.
- No labels: 20,000 x 60,000,000 = ... ???
- Labels: 20k x 6m = 120k x m = 120b
This approach also works for divisions. Let's try 480,000,000,000 divided by 240,000,000.
- No labels: 480,000,000,000 / 240,000,000 = ... ???
- Labes: 480b / 240m = 480k / 240 = 2k
When you can't use labels, the scientific notation is a good alternative. If you're not sure what this is, you're really missing out. But fortunately Khan Academy has put together a good primer on the topic here.
- Multiplication example: 600 x 500 = 6 x 5 x 102 X 102 = 30 x 104 = 300,000 = 300k
- Division example: (720,000 / 1,200) / 30 = (72 / (12 x 3)) x (104 / (102 x 10)) = (72 / 36) x (10) = 20
When you're comfortable with labels and the scientific notation you can even start mixing them:
- 200k x 600k = 2 x 6 x 104 x m = 2 x 6 x 10 x b = 120b
To be fast at maths, you need to avoid writing down long divisions and multiplications as they take a LOT of time. In our experience, doing multiple easy calculations is faster and leads to less errors than doing one big long calculation.
A great way to achieve this is to factor and expand expressions to create simpler calculations. If you're not sure what the basics of factoring and expanding are, you can use Khan Academy again here and here. Let's start with factoring.
Simple numbers: 5, 15, 25, 50, 75, etc.
In case interviews and tests like the McKinsey PST or BCG Potential Test some numbers come up very frequently and it's useful to know shortcuts to handle them. Here are some of these numbers: 5, 15, 25, 50, 75, etc. These numbers are frequent but not particularly easy to deal with.
For instance, consider 36 x 25. It's not obvious what the result is. And a lot of people would need to write down the multiplication on paper to find the answer. However there's a MUCH faster way based on the fact that 25 = 100 / 4. Here's the fast way to get to the answer:
- 36 x 25 = (36 / 4) x 100 = 9 x 100 = 900
- 68 x 25 = (68 / 4) x 100 = 17 x 100 = 1,700
- 2,600 / 25 = (2,600 / 100) x 4 = 26 x 4 = 104
- 1,625 / 25 = (1,625 / 100) x 4 = 16.25 x 4 = 65
- 2.5 = 10 / 4
- 5 = 10 / 2
- 7.5 = 10 x 3 / 4
- 15 = 10 x 3 / 2
- 25 = 100 / 4
- 50 = 100 / 2
- 75 = 100 x 3 / 4
Once you're comfortable using this approach you can also mix it with the scientific notation on numbers such as 0.75, 0.5, 0.25, etc.
Factoring the numerator / denominator
For divisions, if there are no simple numbers (e.g. 5, 25, 50, etc.), the next best thing you can do is to try to factor the numerator and / or denominator to simplify the calculations. Here are a few examples:
- Factoring the numerator: 300 / 4 = 3 x 100 / 4 = 3 x 25 = 75
- Factoring the denominator: 432 / 12 = (432 / 4) / 3 = 108 / 3 = 36
- Looking for common factors: 90 / 42 = 6 x 15 / 6 x 7 = 15 / 7
Another easy way to avoid writing down long divisions and multiplications is to expand calculations into simple expressions.
Expanding with additions
Expanding with additions is intuitive to most people. The idea is to break down one of the terms into two simpler numbers (e.g. 5; 10; 25; etc.) so the calculations become easier. Here are a couple of examples:
- Multiplication: 68 x 35 = 68 x (10 + 25) = 680 + 68 x 100 / 4 = 680 + 1,700 = 2,380
- Division: 705 / 15 = (600 + 105) / 15 = (15 x 40) / 15 + 105 / 15 = 40 + 7 = 47
Notice that when expanding 35 we've carefully chosen to expand to 25 so that we could use the helpful tip we learned in the factoring section. You should keep that in mind when expanding expressions.
Expanding with subtractions
Expanding with subtractions is less intuitive to most people. But it's actually extremely effective, especially if one of the terms you are dealing with ends with a high digit like 7, 8 or 9. Here are a couple of examples:
- Multiplication: 68 x 35 = (70 - 2) x 35 = 70 x 35 - 70 = 70 x 100 / 4 + 700 - 70 = 1,750 + 630 = 2,380
- Division: 570 / 30 = (600 - 30) / 30 = 20 - 1= 19
3.5. Growth rates
Finally, you will also often have to deal with growth rates in case interviews. These can lead to extremely time-consuming calculations so it's important that you learn how to deal with them efficiently.
Multiply growth rates together
Let's imagine your client's revenue is $100m. You estimate it will grow by 20% next year and 10% the year after that. In that situation, the revenues in two years will be equal to:
- Revenue in two years = $100m x (1 + 20%) x (1 + 10%) = $100m x 1.2 x 1.1 = $100m x (1.2 + 0.12) = $100m x 1.32 = $132m
Growing at 20% for one year followed by 10% for another year therefore corresponds to growing by 32% overall. To find the compound growth you simply need to multiply them together and subtract one: (1.1 x 1.2) - 1= 1.32 - 1 = 0.32 = 32%. This is the quickest way to calculate compound growth rates precisely.
Note that this approach also works perfectly with negative growth rates. Let's imagine for instance that sales grow by 20% next year, and then decrease by 20% the following year. Here's the corresponding compound growth rate:
- Compound growth rate = (1.2 x 0.8) - 1 = 0.96 - 1 = -0.04 = -4%
Note how growing by 20% and then shrinking by 20% is not equal to flat growth (0%). This is an important result to keep in mind.
Estimate compound growth rates
Multiplying growth rates is a really efficient approach when calculating compound growth over a short period of time (e.g. 2 or 3 years). But let's imagine you want to calculate the effect of 7% growth over five years. The precise calculation you would need to do is:
- Precise growth rate: 1.07 x 1.07 x 1.07 x 1.07 x 1.07 - 1 = ... ???
- Estimate growth rate = Growth rate x Number of years
In our example:
- Estimate growth rate: 7% x 5 years = 35%
In reality if you do the precise calculation (1.075 - 1) you will find that the actual growth rate is 40%. The estimation method therefore gives a result that's actually quite close. In case interviews your interviewer will always be happy with you taking that shortcut as doing the precise calculation takes too much time.
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Any questions about case interview maths?
If you have any questions about case interview maths, do not hesitate to ask them below and we will be more than happy to answer them. All questions are good questions, so go ahead!
The IGotAnOffer team