Case Math and Computations
When I first started practicing for case interviews, I was told to “walk the interviewer through my math.” And I would dutifully do so: 350 multiplied by 8, so that’s 0, carry 4, 24 plus 4 is 28, so that’s 2,800. (Ta-da!)
That’s not so painful, but try something more complex. How about $472M divided by 19 months? Go do that calculation by hand and talk aloud as you do, and record yourself. Then play it back. How do you like the sound of you “walking through” your math?
It wasn’t until I started finding strangers on the internet to swap cases with for practice that I realized this mandate isn’t gospel, it’s just a suggestion. It is so boring to listen to someone do their calculations. Everyone is in misery: you, for sweating it while you do your calculations, and your interviewer, for having to pretend to follow along. No matter how fast, slow, brilliant, accurate or plain off-track you are, no one’s coming out of this energized and cheerful. With all the good vibes sucked out of the room, how do you think your interviewer will think back on how you made them feel?
So pro-tip #1: don’t walk anyone through your math, not literally.
Instead, walk through how you plan to do your math. Set up the formula you’re going to use in words, explain what it’s going to get you and why that’s important. Then ask for permission to do that math, SHUT YOUR MOUTH, and do that math. (Ahhh, silence, what a lovely reprieve!) When you have your answers, look up, make eye contact, smile, and then proceed to state your findings in an organized and composed manner.
Set up the formula in words? Like this:
I’d like to find out the absolute number of patients in each of these three segments. Because I have the total number of patients, and the percentage for each segment, I’m going to multiply the percentages to the total number of patients, which should give me the absolute number of patients in each segment. Does that sound reasonable to you?
I recommend throwing in a quick “collaborative” sense-check to your interviewer when you walk through your formula to get their buy-in before you spend the time on calculations. You don’t want to motor-mouth through this part, only to find later that your interviewer has been politely waiting to tell you that you misread the column title and therefore have multiplied the wrong values together entirely. Tell them what you’re looking to obtain from your calculation. Tell them how you’re going to do it: which values you’re going to use, and how you’re going to manipulate them. You can even repeat what the result is, and further add why it’s important or what you’ll do with that result next. Then ask what they think - so they can tell you yay, nay, or some additional information right now before you lose precious time on unnecessary mathematics.
Once you have your numbers, share them in an organized manner and have an opinion. If you don’t know enough to have an opinion, tell me what you need in order to form an opinion. Here’s what I mean:
…so what I’ve found is that the total market size for this new product is $30M in the US, $22M in the UK, and $12M in Canada, taking into consideration the current currency exchange.
[Those are your numerical answers, clearly labeled.]
Since our client is the only player in the market for the foreseeable future, this means they could capture an additional $64M in revenue by launching this product.
[Bringing this back to the main objective!]
While $64M sounds like a lot of money, I would like to make sure it’s a worthwhile opportunity for our client - and that they don’t have a product that could capture $2B in revenue instead waiting in the wings that this launch might detract from. It’d be helpful to contextualize how much $64M would mean to our client. Do we know anything about their current annual revenue?
[This whole comparison bit, leading up to the question, is you trying to form an opinion on if $64M is good or bad - but if you don’t know, you move into finding out what would allow you to make that judgment. Hence asking for more data.]
Here comes pro-tip #2: never present a number without contextualizing it.
Don’t just tell me the answer is X. You have to give me the “so what” to help me really understand it and know what to do next. Because that's the actual reason we’re doing this: I, your client, want to know what action to take! So you calculated that there is a $64M opportunity in this market. Is that good? Is that bad? Should I go into this market? Should I pass because it’s not worth the effort? All these questions - and you thought you just gave me “the answer!”
You can’t judge a number without some context. In order to have an opinion, you have to know what to compare it to. Don’t have anything to compare it to? Don’t panic! A lot of times in a case interview you aren’t given anything to compare your “calculated answer” to - and that’s okay. You can still describe what information would be useful to have for you to evaluate the answer and form an opinion. Other times, your calculations may give you a couple of answers to compare to each other. For example, there are three segments in the market and your client wants to prioritize one because they don’t have the resources to pursue opportunities in all the segments. In addition to coming up with a final answer on which segment, you also need to justify why you recommend that segment. (Otherwise how do I know you didn’t pretend to do the math and just mentally threw a dart instead?)
And that’s pretty much it. The rest is actually on your math skills. Don’t make silly mistakes - like be off by a factor of 10 (my signature move). You can drill calculations any time of the day - just grab pen and paper and make up random numbers to add, multiply, divide into each other, etc. Case math doesn’t have to be mental math, although if you’re confident in your abilities it can look quite impressive. For the rest of us mortals, erring on the side of being safe, pen and paper works just fine.
Here’s another few tricks to try if you’re nervous about doing calculations under the scrutinous gaze of a judging stranger:
Round up or down sensibly if you can:
Sometimes you get really wonky and specific numbers. Instead of trying to choke your way through it, ask your interviewer (casually) if they’d be okay if you round up or down to [insert an easier-to-work-with whole number]. Most times you’ll be given permission to give a rough estimate by using round numbers. Occasionally you’ll be asked to stick to the specific number, in which case, at least you tried.Be familiar with your multiplications and fraction-and-decimal conversions:
What’s 5400 divisible by, apart from 2, 5 and 10? 6 or 9 - and therefore also 3. If that felt rusty, set yourself some math drills. Trust me, it’s a real shame to choke and lose the offer because of “a silly math mistake” - and that’s my honest voice of experience there. Remember: you don’t rise to the occasion, you fall to the highest level of your preparation.
By fraction-and-decimal conversions, I mean being able to look at 0.11111 (recurring) and know off the top of your head that is equivalent of 1/9. You probably know 0.33333 is ⅓ and 0.66666 is ⅔ but the others are a little trickier. If you want to be extra prepared to impress, revising a few of these wouldn’t hurt. There are charts online you can search around and memorize if you like.Try using Ks, Ms and Bs as shorthand:
I was horrendous at tracking all the zeros in big numbers: ten thousands, hundred thousands, millions, ten millions, urgh. The first problem was that I couldn’t recognize a million at a glance. (Hint: six zeros behind the single, non-zero number.) I chalk this up to a flawed Chinese-English translation error that got stuck in my head as a child: I always thought the unit for “ten thousand” in Chinese meant “million” - and yes, this meant I somehow counted like, nine thousand, nine hundred and ninety-nine…one million! The second problem is just that, when I diligently write out all the zeros and then attempt to do division or multiplication, I somehow inevitably miss some zeros or add some extra. And of course the third problem was that writing out all the damn zeros took a damn lot of time.
The best piece of advice I got from an MBA student in the year above me who generously did a handful of cases with me was to convert to using the K (thousand), M (million) and B (billion) system. One K for every three zeros. So one thousand became, on my scratch pad, instead of 1000, 1K. Ten thousand, 10K. Two Ks for an M, and ignoring what the abbreviation could be here, three Ks for a B.
How was this helpful? It was time-saving, for starters. It also kept me a lot more organized. When I was given a number like 87M and I had to divide it by 43K, I could quickly sketch out 87KK / 43K and immediately cross off matching Ks top and bottom. (If this is no revelation to you, congratulations - even the likes of me get hired to be a consultant, so that should prove to you math skills aren’t that rigorously evaluated.)If you’re given the chance to estimate numbers, pick wisely - and know a few pieces of “general statistics” for those estimates:
I don’t recommend trying to find out and memorize answers to things like “how many pigs there are in China” or “how many diapers does the US go through in a year” - those questions are definitely about seeing how you approach the problem and structure your thinking. For those questions, you can go bottom-up or top-down.
Bottom-up means you think of the smallest relevant unit and then try to scale it up. For example, one baby may go through 6 diapers in a 24-hour period, and will probably live in diapers until they’re 1.5-years-old (so round that up to 2), so now you need to find out how many newborns, 1- and 2-year-olds are in the US at any given time.
Top-down means you take the broadest view possible and “drill down” to the details. For example, we can assume the US has a steady population of about 320M people at any given time. We can also assume a fairly even age distribution and an average life expectancy of 80 years. So that means at any given time, we’re looking at 4M people within the same age group. Add in the aforementioned estimates about babies and their diaper-wearing habits, you have 4M babies for each age group, so that’s 4M diaper-wearing babies at any given time…
I chose this diaper question because it helpfully illustrates what kinds of “general statistics” it’d be helpful to know. World population, US population, relevant local country population, etc. - all useful to know. You probably can get away with estimating life expectancy at 70 or 80, but if you want to get precise and research that, be my guest. Notice how broad some of my assumptions are - even if I know they’re not quite true. As long as you call them out, you’re fine to proceed. Is the age distribution really even? No, we have aging populations in the majority of countries right now, but there’s no point in getting caught up in that nuance. Similarly, US population doesn’t have to be a super precise number; a reasonable ballpark is probably anything within 100M. Why pick 320M instead of 340M or 300M? Because I already had in mind I was going to propose a life expectancy of 80 years and 320M is much more divisible by 80 than either 300 or 340 is. Yep, that’s how “quick and dirty” your estimates can be - you won’t be seen as lazy for not doing the math even though it’s not that difficult - all even numbers, etc. Keep it simple - remember, your interviewer has likely had to sit through dozens of people answering the same question, they’re just as keen as you are to get this over and done with.