5 Shortcuts to Ace the Case Interview Math
Updated: Jan 2, 2021
Consulting case math is not hard. You won’t have to calculate any integrals, or even solve multivariable equations, yet, quick and accurate mental math with pen and paper (you won’t be able to use a calculator) is critical for your interview success. While, great mental math won’t secure your offer, multiple math mistakes in an interview for a top-tier firm will guarantee you a rejection. This is because interviewers use quantitative questions, as a proxy for your mental agility, analytical skills and attention to detail.
The good news is that anyone can reach a “hygiene” level in mental math. While, someone who just completed a degree in math is likely to start from a better position, learning the basics, practising (a lot!) and applying a few simple techniques, will go a long way in helping you impress the interviewer. During interview coaching, we saw many clients go from 0 to 60 in their math skills over a span of a month. As another piece of evidence, McKinsey is full of consultants with liberal arts degrees, including history, geography, and literature among others.
First things first, brush up on the basic arithmetic operations. You will need to get comfortable with addition, subtraction, multiplication, division, percentages, fractions and averages, in addition to simple algebra – turning a word problem into an equation and solving for x. If you need a reminder for how to do long division, check out our QuantMaster course on Udemy (20% off for ConsultingCasePro visitors). Practice accuracy first, before starting to work on speeding up the calculations.
Luckily, there are a few shortcuts that can help you simplify calculations, which in turns increases your speed and reduces the chances of an error. In this article, we will dive into the 5 most useful techniques that will help you ace the case interview math, not to mention impress your date, by calculating the tip in your head.
1. Rounding: that’s the simplest technique and the “lowest hanging fruit”. It is generally acceptable to round numbers by +/-10%, but confirm with the interviewer (in 90% of the cases, the interviewer won't object). This way 52 weeks in a year, become 50 and 365 days, turn into 350. If you are rounding multiple numbers, alternate the direction of rounding (i.e., if you round one number up, round the other down) to stay closer to the exact answer
Example: a restaurant generates an average revenue of $23,550 per week, what is their annual revenue? Exact answer: 23,550*52 = $1,224,600; with rounding: 25,000*50 = $1,250,000. The rounded answer is within 2.2% from the exact one, which is an acceptable margin of error, and the calculation was much easier.
2. Re-ordering: for a sequence of calculations with the same operator (e.g., multiplication) the order of execution doesn’t matter. This way you can start from the easiest calculations (generally, the smaller numbers), before moving to the more complex ones. Don’t forget though that multiplication and division take preference over addition and subtraction, therefore, 2+3*4 ≠ (2+3)*4. Was that news for you? Go back and brush up on the basics!
Example: a plane has 350 seats, with an average occupancy of 80%. If 25% of passengers travel for business, how many business travellers are there on an average flight? Compare calculating in order: 350*80%*25% = 280*25% = 70 to starting from multiplying the percentages one by the other first: 80%*25%*350 = 20%*350 = 70
3. Factoring and Decomposition: in multiplication and division, in particular, it might be beneficial to break down the numbers into easier to calculate components. This way 847 becomes 800 + 40 + 7 and 125 becomes 25 * 5. Don’t forget the rules of expanding double brackets: (800 + 40 + 7)*(100 + 25) = 800 *100 + 40*100 + 7*100 + 800*25 + 40*25 +7*25.
Example: a company produced 11,500 devices last year that they sold for $125 each. What were their total revenues last year? 11,500*125 = (10,000 + 1,000 + 500)*125 = 1,250,000 + 125,000 + 500*125 = 1,375,000 + 5*125*100 = 1,375,000 + 625*100 = 1,437,500
4. Getting rid of zeroes: the majority of clients of top tier consultancies, are large multi-national corporations, which count their revenues, costs and investments in millions or even billions. As a result, a lot of the data you will encounter in case interviews will include multiple zeroes. Instead of doing the calculations with all the zeroes in place, it is a good practice to replace them with the appropriate labels to simplify the calculation and avoid the (frequent) mistake of forgetting or adding an extra zero. This way 1,000 becomes 1K, 1,000,000 becomes 1M and 1,000,000,000 becomes 1B. Keep in mind also how to handle operations between labels, e.g., K*K = M, M*K = B, B/M = K, M/M = 1
Example: assuming that the top 10% of the US population earn $120,000 per year on average, what are the total annual earnings of this group? 330,000,000*10%*120,000 = 330M*10%*120K = 33M*120K = (30+3)*120M*K = 3,600+360B = 3,960B = 3.96T
5. Borrowing: it is much easier to calculate with round numbers that end with a “0”. But what can you do if you only have “ugly” numbers? During the calculation, “borrow” a factor to create a round number. Just don’t forget to “return it” in the end. This way 457 + 123 becomes: 450 + 125 + 7 - 2 = 575 + 5 = 580, and 45 * 16 = 45*2/2*16 = 90*8 = 720.
Example: our client considers an investment of $1.25m in a new production line, which will save them $150k a year. What is the payback period on this investment? 1,250k / 150k = 1,250k / 150k * 2 / 2 = 1,250 / (300 / 2) = (4 + 50 / 300) * 2 = 8 1/3 years. Note that the division of the denominator by 2, is the same as the multiplication of the numerator by 2.
In your next case interview practice, spend 5-10 seconds thinking about how you can make the calculation simpler with shortcuts, before jumping in. These will make the quantitative section of your interview a bit less stressful, and remember practice makes perfect!
Are you familiar with other useful shortcuts? Let us know in the comments.
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