Mental Math Practice: Getting the Most “Bang for your Buck”
In case interviews, you may be asked to undergo mental math gyrations from Minute 1. Consider the following excerpt from this sample case.
This deluge of information might come as a series of facts, sprinkled throughout the case – or it might arrive after a two-sentence prompt citing a client’s desire to reduce their R&D costs.
“[…] All of the R&D facilities and staff are concentrated in the Boston area. There are approximately 2,300 R&D employees and the total R&D costs are approximately $12B per annum. The client has traditionally received 80% of its total revenue ($24B) from drugs to treat heart disease, blood pressure, and liver disease; 80% of the patents on these [three] drugs will expire by 2015, and the company therefore expanded research into adjacent areas in 2010. Not getting satisfactory results, the company expanded the research portfolio in 2012 [emphasis mine].”
If you’re given all this information as a trainee, candidate, or interviewee, you’re likely to be experiencing hand cramps in the near future. Yet the interviewer will also expect that the cogs in your mind are turning, too – that is, that they are doing more than simply holding these bolded numbers in working memory and sending nerve impulses to your forearm to ensure you write everything down before you miss something. To put it bluntly, you should be trying, even as you jot down your notes, to “size” some important quantities, and wonder about things like:
- The total R&D cost per employee
- The Boston company’s total revenue
- The time remaining until the patents expire – as well as which patents will expire!
The degree of mental math fluency required to be successful as a consultant cannot be understated. In this article, I’d like to spend some time discussing a subset of mental math skills surrounding (no pun intended) the art of rounding numbers, and make suggestions for practice.
The Art and Necessity of “Rounding” for Mental Math
If you’re reading this article, you most likely know that the art of rounding will be critical to your mental math performance in case interview endeavors. Perhaps you’ve been taught it is necessary for reasons having to do with speed, efficiency, or capacity (what number of numbers you can keep in mind for a time, without forgetting). All three of these categories are valid!
Rounding is clearly useful. One motivating example can be found in the multiplication of two numbers — often, and fortunately for us, this can be observed even when a problem is easy enough to solve exactly. Take as an instance the following problem, from a sample profitability case: what is 12% of $12B? Anyone who still remembers their multiplication (or “times”) tables can see that, since 12 x 12 = 144, and one of the numbers is a percentage, we can simply move the hitherto invisible decimal point from behind the second “4” to rest between the “1” and the first “4,” giving an answer of $1.44B. Yet we could have rounded to estimate this one as 10% of $14B (i.e., $1.4B flat), or even 14% of $10B, and we’d have come up with an answer of $1.40B – only 4/100 of $1B (only – yes, only, because $40 million is a comparatively small amount) off!
By cleverly rounding one answer slightly up and the other down (alas, this “toy problem” probably represents the upper limit of how much rounding is safe to do; I say so intuitively now, but will make this intuition mathematically precise below!), one can sometimes lessen the work by virtue of dealing with numbers that are more familiar, or easier to multiply, than the original numbers in the problem.
What about the exact solutions?
That said, there always those dreaded moments, where interviewers asks candidates to provide “exact” answers; and any short-lived affair with your favorite search engine can turn up seemingly contradictory pieces of advice for how to handle such situations, or how to approach and emphasize rounding techniques in your case interview training (possibly, at a detriment to the honing of other skills).
For instance, even when rounding is strongly encouraged, how conservative should you aim to be? One might observe Felix, who had entered her training with good numeracy already, and wonder if her rounding of “90% of $14.4M” to “12.9” (instead of 12.96) was advisable, or if her rounding of “90% of 12.9” to “11.6” was necessary (or whether she could have, instead, just solved “90% of 13”) during her guided estimation of the number of clothing irons in Spain.
Especially if you’re good with numbers like Felix, or happen to be a lightning-fast human calculator like Dr. Arthur Benjamin, it might even seem moot to waste time with fancy rounding techniques in the middle of your calculations: might not it be better to compute exact answers, even if you report them to your interviewer in approximate terms?
I believe that the answers is a firm “no,” for there is a much more compelling reason to become fluent in rounding than is often discussed!
A Better Reason For Learning to “Round” in Mental Math
My reason is a simple one: many of those “exact” mental math techniques you can learn (and Professor Benjamin’s book teaches a lot of them) necessitate the use of rounding, in order to make them work! A perfect example is the technique of performing subtractions by addition of complements: 980 minus 39 can be transformed into “980 – 40 + 1,” where “980 – 40” can be a lot easier to visualize than the original problem, and– crucially – it can easily be corrected, via appending the final “+1” operation. Another, which I’ll display later, is a more subtle technique for multiplying numbers: one that requires the user to decide which roundnumber is closest to the original number being multiplied.
Beyond this reason, which deals with mental math “as such,” there are, of course, other reasons to practice rounding for a case interview. For instance, interviewers may try to simulate what an executive purports to care about in a given calculation, and the “bottom lines” in these discussions are often expressed as order of magnitude estimations (see comments on assessing “competing effects” on the basis of relative size in my recent article 3 Case-Interview Concepts I Learned From Physics). Indeed, in the video from which the “12% of $12B” example derives, the answer itself is even is rounded to $1.5B, since the interviewee cares only about demonstrating that other categories of costs besides labor (which makes up the aforementioned 12%), are two times this size in dollar value!
In the subsequent section, I’ll spend some time unraveling a few of the techniques that Art Benjamin espouses in his work. While I’ve never known him personally, his presentations of mental math skills were extremely inspiring for me, as a young physicist, the slam-dunk reason I have provided for “learning to round,” specifically, was one inference, or generalization, that I’d made by reading his Secrets of Mental Math, to which I also linked earlier. I will spend time on a few of his strategies that I’ve actually used in preparing for case interviews, and move on in the next sections to both a rule for rounding and some general advice on mental math in business.
Mental “Mathemagics”: Tips for Playing with Numbers
Benjamin is the expert, but I (one of many) have profited tremendously from adopting some of his practical tips for negotiating mental arithmetic. So, I’d like to show you some of the tips and tricks for case interview math that I’ve internalized, using examples from his book.
I’ll break these into two categories: the first will be “General Go-To’s,” and the second a couple applications of “Specific Strategies” for multiplying non-round numbers. Since division is, in essence, multiplication by a reciprocal (at least in everyday arithmetic!), both approaches can also be used to engage quotients, ratios, and fractions – you’ll see why after General Go-To #2!
After reviewing these mental math tips, I’ll revisit the topic of trying to approximate an answer by rounding the numbers in a multiplication problem – the multiplicand and multiplier – in opposite ways. In particular, I’ll try to show how you can be confident that their product will be within the ballpark, given how much you’re rounding the individual numbers that are being multiplied. This latter subsection will be more technical – feel free to skip to the final section!
Go-To #1: How Many Zeros?
Even as a trained scientist, my sticking point in a “back-of-the-envelope” calculation can often be the order of magnitude assessment. To parallel what I have stated here, is the ratio of 700 billion to 46 million best expressed in millions? Hundred-thousands? I’d say ten-thousands (unless I had mistakenly divided by 460 million and come up with an answer in the thousands!):
700,000,000,000 divided by 46,000,000 is approximately 15,217
The usually touted trick is to express the answer in some scientific notation, or separate an appropriate number of zeros by converting the first few integers to decimal coefficients — as many as necessary to make the problem convenient, and multiple choices that work for a given problem. Yet there’s another, perhaps simpler, strategy you can invoke for your sanity checks… “count” the number of digits you should expect the product of your multiplication (or quotient for any division) to have!
How can this be done? Consider the equation below:
321 x 29 = 9309.
The first number, called the multiplicand, has three digits; the second digit, or multiplier, has just two, and their answer – the product – has exactly four. In general, products that do not require you to carry over additional digit will contain a number of digits that’s just one less than that of all the digits being multiplied (the sum of the numbers of digits in the multiplicand and multiplier, minus one). What about problems that do require you to make a new column?
999 x 99
and convince yourself that, even when the answer does “spill over,” you will only need to add a singleadditional digit – so when multiplying a number with X digits against a number containing Y digits, the answer itself have, at its most, X+Y digits. To be precise, this happens whenever the very first, “leading” digits of each number themselves have a product that is equal to or greater than 10, and occasionally when this product is between 5 and 9. You might impress your friends (and, hopefully, your interviewer!) if you find time to internalize this knowledge, since you can tell immediately that your answer will have the smaller possible number of digits, X+Y-1, if that product is 4 or less! Of course, something similar holds for division: dividing a number that has X digits by another number with Y digits yields an answer that has either X–Y (or X–Y+1) digits.
Go-To #2: Common Decimal-to-Fraction Conversions
Who knows how to express ½ as a decimal number? The answer, naturally, is 0.5.
Who knows how to express “5 out of 6” in decimal? You’re correct if you said “.8333…”!
Who knows how to express the ratio 1:7 as a decimal? Nice job – 0.143 is a fine answer! You’re doing quite well, so how about 2:7? At this point, the easiest thing to do might be simply be to “double” your previous answer. Consider that certain fractions are just worth memorizing.
The argument here should be self-evident: it’s not that certain fractions appear with an eerily large frequency in consulting (although, you might enjoy taking a look at Benford’s Law if you’re that suspicious), but that having sufficiently large repertoires of numerator-denominator pairs can help you break other problems into smaller, more efficiently processed ones.
I’m going to go out on a limb and say that you’ll shine (at the very least among nerdiest companions, if not an interviewing partner!) if you make a concerted effort to commit various fractions to memory, just like you would a multiplication table. I’ve listed some of the fun and useful ones below, but this doesn’t mean that these are the only (or, most) important ones!
- When the denominator is “5,” multiply by 2 and divide by 10 (or, you can reverse it!)
- 1/5 = 2/10 = 0.2 (literally, “two tenths”… etc.)
- 2/5 = 4/10 = 0.4
- 8/5 = 16/10 = “1 & 6 tenths” = 1.6 (simplifies improper fraction, no division!)
- When the denominator is “6,” multiply by 2 and divide by 10 (or, you can reverse it!)
- 1/6 = 0.1666…. (a repeating fraction, sometimes approximated as “0.1667”)
- 2/6 = 0.3333…. = 1/3
- 3/6 = 0.5 = 1/2
- 4/6 = 0.6666…. = 2/3 (see 2/6)
- 5/6 = 0.8333…. (the only other one that’s new!)
- When the denominator is “7,” we see a new kind of “repeating number” pattern:
- 1/7 = 0.142857142857142857….
- 2/7 = 0.285714285714285714….
- 3/7 = 0.428571428571428571….
- 5/7 = 0.714285714285714285….
To know where to start the repeating pattern, multiply numerator by “14”:
- 1*14 = 14 142857….
- 2*14 = 28 285714….
- 3*14 = 42 42857….
- 4*14 = 56 57142….
- … etc.
- When the denominator is “8,” you can pretend it’s “4” and cut in half… or…
- 1/8 = 0.125
- 2/8 = 0.25 = ¼
- 3/8 = 0.375
- 4/8 = 0.5 = 1/2
- 5/8 = 0.625
- 6/8 = 0.75 = ¾
- 7/8 = 0.875
- When the denominator is “9,” decimal conversions are particularly easy:
- 1/9 = 0.111…. (dividing by 9 is similar to multiplying by 11 – already useful!)
- 2/9 = 0.222…. (see the pattern?)
- 3/9 = 0.333…. (= 1/3)
- 8/9 = 0.888…. (which is very close to “0.89,” not repeating)
- 9/9 = 1 (try “0.999”…. it’s indistinguishable from 1!)
- When the denominator is “11,” decimal conversions are also particularly easy:
- 1/11 = 0.090909…. (a two-digit, repeating pattern)
- 2/11 = 0.181818…. (see a pattern forming already?)
- 3/11 = 0.272727….
- 9/11= 0.818181….
- 10/11 = 0.909090…. (again, the relationships of 9 & 11 can be super useful!)
- Other fraction-simplifications may be useful for ambitious candidates:
- 1/12 = 0.08333…. (nothing new, since this is 0.5/6, or 5/6 divided by 10)
- 1/13 = 0.076923076923…. (another long, repeating decimal)
- 1/14 = 0.07142857142…. (also nothing new, if you think of 0.5/7)
- 1/15 = 0.0666…. (here, think of 2/30 = 0.2/3 = “0.666…. divided by 10”)
- 1/16 = 0.0625 (similarly, think of .5/8)
To see more like these, visit http://www.asiteaboutnothing.net/p_speed-arithmetic.html#new.
Specific Strategy #1: Break multiplication into steps, rounding in the most convenient direction
Consider the problem of multiplying the numbers 41 and 17. At first glance, this might seem “messy” … but rounding can help! Since “41” can be broken into “40 + 1, ” imagine that you just want to do the (slightly simpler) problem of 40 times 17. Really, this is only “ten times the quantity ‘4 times 17,’” and you can do the latter any way you’d like. Then, just add 17!
It looks like “rounding down” the multiplicand, 41, helped quite a bit. What about the other way – rounding up? Benjamin offers the following example:
53 x 89 = (53 x 90) – 53 = 4770 – 53 = 4,717
In this case, he rounded one of the numbers up and then subtracted the very amount by which this operation caused him to overestimate his answer. He calls this heuristic, by which he adaptively decides whether to round up or down when breaking down multiplication problems, finding the “math of least resistance.” One major alternative, of course, would be to break the multiplication problems into factors that you know how to memorize by heart. For me, finding the closest round factors works quite well – and once you’re practiced, can even be scaled-up.
Try doing it this way:
53 x 89 = (53 x 90) – 53 = (50 x 90) + 3 x 90 – 53 = (5 x 9) x 100 + (3 x 9) x 10 -53
Specific Strategy #2: Break multiplication into steps, rounding in the most convenient direction
Consider the problem of multiple the numbers 26 and 23: they are both close to 20, but rounding down the second (23 20) is probably the easiest way to find their product. If I said you that you had to round the first number, which way would you go?
While 26 is technically closer to 30, rounding down (26 20) still seems quite appealing to me. If we leave 23 as it is for the moment, this might look something like
26 x 23 = (20 x 23) + (6 x 23) = (2 x 23) x 10 + (6 x 23),
and the benefits would include operating with small numbers and only additions, which I find to be more intuitive than subtractions – which, admittedly I often do by adding the complement. If we had instead rounded up, we might have written
26 x 23 = (30 x 23) – (4 x 23) = (2 x 23) x 10 – (4 x 23),
and that’s still not too bad… but here we are, on paper, when the name of this game is mental math. There must be an even easier way to do this multiplication – right?
Try this. Recall that both numbers are near to the number 20; the first is 6 units away on the number line; the second is 3 units away. Now, let’s fantasize about pooling all those units of distance into a single number: 26 + 3 (or 23 + 6) = 29. What is the product of that so convenient, nearby number – 20 – and this new number, 29? We can easily verify that
20 x 29 = 580,
exactly 6 x 3 = 18 units away from the true answer of “26 x 23” = 598! Coincidence? I think not!
This approach to multiplication (dubbed the “close-together method” by Dr. B.) works because, if you take out the product of the digits in the “ones” column, 6 x 3 = 18, what’s left is the set of subproblems
(20 x 20) + (20 x 3) + (20 x 6) = (20 x 20) + (20 x 9) = 20 x (20 + 9) = 20 x 29,
and is precisely the first sub-product we computed. This method is powerful, and generalizable.
I’ll walk through another problem to reinforce how the close-together method operates. The name of the tactic refers to the fact that a multiplicand and its multiplier must be near each other. Let’s multiply 107 and 111.
Step 1. Figure out how far each number is from their common neighbor, 100.
This can easily be done in your head, at least for this particular case!
Step 2. Perform the addition 107 + 11 (or 111 + 7) to find arrive at the sum, 118.
Step 3. Multiply 7 x 11 (or 11 x 7) to find the product, 77, and remember it well!
Step 4. Multiply the “neighbor” number, 100 with the “ancillary” number, 118.
This should really be straightforward – separate into two subproblems.
In this case, it’s especially easy – one needs only to append two zeros!
100 x 118 = 11,800
Step 5. Recall the product of the “ones” digits, and add it to the running total:
11,800 + 77 = 11,877
Apart from the “rounding” that gave us our “nearby” number of 100, this was all exact! With practice, it’s also lightning-quick, compared to its pencil-and-paper equivalent. Yet again, in case interviews, rounding is often the name of the game. How can we make that fit back in?
How Much Rounding Can I Get Away With in My Mental Math?
If we had rounded the numbers in the above problem both to the nearest hundred, that is, we had solved instead the problem 100 x 100, we’d have been about 15.8% below the actual answer. If we rounded the first number down to 100, and the second up to 120, we’d have got to within 1% of the answer – amazing! But could we have known this would happen in advance, before invoking the close-together method? How can you predict the effect of liberal rounding?
To answer this curiosity (and, finally, lay to rest the woes of your most skeptical friends about the dangers of severe rounding errors), we’ll switch from an arithmetical approach to an algebra-based formalism. The idea will be to figure out how far we’re off, exactly, even if we do not know in advance the amount of the rounding we will do; we will write out the equation that describes a multiplication in which an arbitrary degree of rounding is applied to each number.
Algebra Interlude [Optional]
Let the first number, the multiplicand, be represented by A; the variable for the second number will be B. Rounding either number means adding or subtracting a small amount, which can itself be expressed as a fraction of the number being rounded. In mathematical “language,” we can call the “rounded” version of the first number “A + (a x A),” where the lower-case letter a is the ratio of the number of units by which A has been increased or reduced to the value of A itself. For now, say “a” is a positive number – A is being rounded up. In other words, if you add a certain number of units to “Big A,” then divide by the value of A, you get “little a.” Rounding down works the same, but in that case (since we are subtracting from A) we assign a negative sign to the front of “little a” – this works out identically, though, because subtraction is really the same thing as adding a negative number. We’ll do all of this for B as well, so we’d write:
A x B [ A + (a x A) ] x [ B + (b x B) ],
that is, the right-hand side is the new multiplication problem that forms for rounded numbers.
What happens when we rewrite the multiplication using the laws of algebra, such as the Distributive Law? We see that the final answer to the new multiplication is given by
(A x B) x [ 1 + a + b + (a x b) ].
Here, “little a” and “little b” are both fractions, and we know that multiplying a fractions against any number gives a smaller number. This means that multiplying two fractions gives an even smaller fraction; since these fractions describe a “rounding” process, and rounding usually means adjusting by only a small number of units anyway, we can justifiably assume that (a x b) term doesn’t affect our answer enough to be included in deciding about “too much” rounding. Thus, we can assume that, by rounding, we changed our multiplication problem like this:
A x B (A x B) x [ 1 + (a + b) ].
Let’s do one more simplification step. Distributing again, we see that our new answer is
(A x B) + (A x B) x (a + b).
Said another way (i.e., English), the “rounded” answer is approximately (A x B) x (a + b) “more” than the original answer. Since what we really we want to know is not how much the rounded will deviate by – a million dollars might matter if you only make 2 million dollars per year, but will be entirely negligible if you’re looking at a 50-billion dollar proposition – but rather the % error in the product of A and B after all the rounding and multiplication has been carried out, we will make another fraction. We divide that “error” amount (A x B) x (a + b) by the actual value of the product (A x B) to find the important quantity, which turns out to be (a + b).
Mental Math Sanity Checks: Sizing a Rounding Error
Whether or not you read the last section, the upshot of the algebra is that it’s actually relatively simple to calculate the “percent error” you can expect in the product of two rounded numbers (as compared with the value of the “exact” product). The way we can do this is define two numbers – “little a” and “little b” – as the fractional number of units, respectively, by which the multiplicand and multiplier are rounded. So, if I round the first number in the multiplication up by 5 units, I divide this 5 by the original value of the first number to get “little a.” Both “little a” and “little b” should also keep track of the “sign,” or direction of rounding – so, if I round the second number in the multiplication down by 7 units, I divide -7 by that number to get “little b.”
Now, either “little a,” or “little b” (or both) can be positive or negative. So let’s “plug in” some values and see if this approach works. Before, we multiplied 26 x 23 to get 598. If instead we had done 25 x 25 = 625, we would have landed about 4.5% too high. Here, “little a” is equal to -1/26, and “little b” would have a value of 2/23; adding them together, we see (a + b) = 4.8%.
Not too shabby! Let’s check this on our other close-together multiplication, 107 x 111. If we’d treated this as 100 x 110, our error would be about 1.8% – we could have anticipated this by computing (a + b) = (-7/107 – 1/111) = (-0.065 + -0.009) = 7.44%! What if we’d gone instead with 105 x 105? Then we’d have had (a + b) = (-2/107 – 6/111), still only about 7% error. Note that, in last two these examples, we didn’t even try to round in opposite directions!
Mental Math Implications for Case Interviews
Even without conducting these numerical tests, you can see immediately that it pays to round in opposite directions, especially when you’re rounding by comparable amounts: (a + b) is closest zero when “little a” and “little b” nearly cancel one another. Of course, all of this logic also applied to division: the important quantity for a quotient is 1 – (1 + a) / (1 + b), which is the percent error for A divided by B, when both A and B are rounded as described above.
Maybe even more importantly, it is also now quite simple to picture how much rounding is “too much.” If one’s interviewer is comfortable with a 20% error in an approximate product, a rule of thumb could be to make sure that one does not allow (a + b) to exceed 20. You might be rounding each number up by 10%, or one up by 30% and the other by down by 10%, but that is your threshold. It’s actually not so strict when you look at it like that; and not having to guess in advance what your overall error will be for a pairwise product helps to eliminate some anxiety.
Check yourself on this! Watch this video on YouTube – but first, try to think about how you would move to approximate the answer to 819,593 x 7,669 (Mbbmath’s solution begins at the 55-second mark). Then, figure out the value of “little a” by counting how far from “819,593” you’ll get on the number line when rounding that multiplicand, and dividing (yes, roughly!) that figure itself by 819,593. Do the same for 7,669 to find the ratio we’ve been calling “little b,” and finally perform the sum (a + b). At the 1:54 timestamp, he flashes slightly different values than I would have calculated with my algebraic formalism; I get (-0.024 + 0.043) = 1.9% – cozily within the 20% range, and exceedingly close to the rigorously hand-calculated error value of 1.8%!
BONUS Advice: Why can Mental Math seem so Difficult within the Context of a Case?
Mental math “hacks” are great, but can you pull off such wizardry during a case?
One might imagine three reasons that performing feats of mental math becomes more intimidating than it otherwise might be, in the specific setting of consulting case interviews:
- The candidate is ill-prepared (simply not well-practiced enough at negotiating numbers)
- Nerves impair clear thinking (including inhibitions, like “tip-of-the-tongue” phenomena)
- The context is too unfamiliar
While the first two might seem more obvious, I believe the third requires special attention. I would like to spend the remainder of this article not on generic advice, or even further methods that I’ve enjoyed learning from FirmsConsulting’s training programs, but an honest-to-goodness unpacking of that last point – truly, context is key.
Mental Math With (Almost) No Numbers
Let’s switch, for just a moment, to a different kind of “mental math” question – one that tests logic, rather than focusing purely on numeracy skills.
Pretend you’re sitting at a table with friends out at a restaurant or social club. On the table there are four ordinary playing cards, with one minor modification: on the reverse side of each card has been printed a letter of the alphabet. You are told: “if there is a vowel on one side of the card, then there is an even number on the other side.” Glancing briefly down, you note the following arrangement.
A P 2 3
The youngest (and most precocious) of your friends asks you which cards would need to be flipped in order to discover whether the “even number-consonant” rule had been violated. Did you figure it out yet? She claims that you need to turn over two of the cards.
The answer, of course, is that you must check the numerical value of the card imprinted with “A,” as well as the letter that’s the printed on the back of #2. As straightforward as it may seem once you’ve arrived at the solution, it’s admittedly unfair of me, the author, to ask such a question without a warm-up. If you subscribe to “better late than never,” try an easier version.
Social Problems are Simpler
While you’re sitting at the same table, your server changes shift, and a new server comes to introduce himself (and take another round of drink orders). He has not a clue who was drinking what earlier; for all he knows, you and your friends have swapped drinks, or have been sharing. This is of concern for him, since your mathematically-inclined friend is clearly underage, and he cannot tell from afar what she is drinking. From his eyes, your table looks something like this:
Underaged College-aged Drinking Coke Drinking beer
Who does he have to check on? Clearly, he needs to make sure your underaged friend isn’t drinking something alcoholic, and by rights he should also ask for your beer-drinking friend’s ID (you’re probably the one drinking soda because you have a case interview tomorrow, for which I sincerely hope this last-minute read comes in handy). Not so hard, is it, this problem? Yet, you might have noticed that the structure of the question is identical to that of the 4-cards problem. The only thing that’s different is the context, and somehow this matters tremendously. Humans are often much happier solving problems that look like what’s already in their wheelhouse.
So, while a person might (or might not!) be comfortable calculating the tip at a restaurant, or “guess-timating” how much you’ll pay at the gas tank for filling up your mechanical steed, it’s not guaranteed that you’ll have similar speed, precision, or overall comfort – even on problems with identical structure – in the middle of a case, even if you’re fluent with numbers and nerves don’t significantly affect your performance.
Implications for Mental Math Practice
What’s the takeaway? You need to practice the way you’ll play – once you’ve gotten down the basics, this means training in-context, on business-specific mental math problems. I’ll finish this “round” (okay, pun intended this time) of discussion by naming a few “themed” examples.
First, you must be an expert at converting percentages to “raw” values. What I mean by this is that, given one number and a percentage figure, you must be able to rapidly, and accurately, perform a mental multiplication problem to tell me what the value of interest is. This is needed in the reverse directions as well – converting ratios or proportions to percentages, or solving for some original amount given a share and its percentage – but problem of breaking down figures into smaller components appears often, in many business contexts. Furthermore, it is easier to compare the relative sizes of business segments or effects using numbers than percentages!
Some examples of “use cases” are:
- Computing the profit margin for a given product line in a company
- Breaking a market into segments based on the share of individual players
- Figuring out required upfront costs to attain a certain return in investment (ROI)
- Segmenting costs into categories like “labor, equipment, land, and inventory,” etc.
- Calculating “utilization,” as when assessing time available to actually treat patients
- Estimate the number of infections, to see if it exceeds 5% of all people in a country
Another example from Benjamin goes like this. Your bank offers you a certain % savings rate for your new account: how long will it take your money to double? It is not terribly difficult to derive an approximatealgorithm that works, and that algorithm is known as the “Rule of 70.”
There are harder (and slower) ways to compute the time it takes a principal investment to grow, but usually these involve inverting an exponential expression, using logarithms. I don’t know about you, but I can’t fathom trying to round and multiple against “logs” in my head, so I would just think:
70 divided by ( interest rate) = # of years to double.
So, a 3% interest rate should take about 23.33 years (it’s actually about 23.45 – off by a month and a half, only!) and a 7% rate should take about 10 years (10.25 – also about 3 months off) according the Rule of 70. Of course, if the interest rate is given not annually but monthly, it still works – but the answers would be 23 and 10 months (off by 3 days or 1 week, respectively). Get this: the same idea also applies if you want the money to triple – that’s just the Rule of 110!
What about a harder problem? Benjamin offers us the following amortization example*. Pretend you borrow money to buy a house – say, $200,000 – and the bank charges 6% interest annually, “compounded monthly.” In plain language, this means that they plan to charge you
(6% divided by 12 months/year) = 0.5% interest,
each month, on whatever amount of money you haven’t yet paid back. If the bank gives you 30 years to pay off the loan (and you plan to use all the time, rather than trying to pay it off faster), can you figure out the amount of money you’ll need to pay the bank each month?
First of all, you know you’ll need to make (30 years x 12 months/year) = 360 payments … this you can (hopefully) calculate exactly, without too much trouble! If there were no interest, it would simply be a matter of paying
$200,000 divided by 360 “monthly” payments in all,
which would amount to (20,000 divided by 36) after knocking off the extra zeros; best of all, we can also divide by the dividend and divisor by 4 to reframe the problem as (5,000 divided by 9), and then divide both again, by 5, to reveal a much cleaner-looking problem, (1000 divided 5/9). If you’ve done your due diligence to memorize the pattern for fractions containing the number “9” as the denominator, you’ll know that 5/9 = 0.555…. moving the decimal says this is $555.
What about the interest? In the very first month, it will be ($200,000 x 0.005) = $1000, and it will never be higher since the multiplicand above ($200,000) goes down every time you make a payment, while the rate stays the same! Thus, as a ballpark estimate, you might say a monthly payment for this case will amount to $1,555 (or less). If we had rounded the original division problem to (20,000 divided by 40) for expedience, we’d have estimated it at $1500!
VERDICT: Mental Math Practice Should Begin with Rounding
No matter what you do to rehearse your mental math skills, I’d suggest beginning with a firm grounding (I know, enough with the puns) in establishing your rounding capabilities. Arthur Benjamin himself performs addition and subtraction from left to right (instead of the commonly taught right-to-left procedure, which is predominantly a pencil-and-paper algorithm) in order to preserve information about the largest orders of magnitude in a problem above all else; indeed, this is the very essence of what rounding does for you, and that’s how it allows you to prioritize.
Feel free to take the most common mental math “advice” I hear in consulting interview-prep communities: avoid ever hesitating, or grabbing your phone-calculator, to solve arithmetic problems that come your way. “They say” that you should be actively practicing with every bill, receipt, and baking recipe you encounter – and while there’s nothing wrong with this, I do see two counterpoints as particularly persuasive. First, I know a lot of MBB consultants – juniors as well as a few partners – and somehow, all of them seem to have managed to ensure that their inner human calculators don’t dominate their livelihoods… while none of them have lost basic numeracy abilities. Second, without the right context your preparation will never be as robust.
Let us know in the comments what you would like to see next – more examples of mental math breakdowns? More mental math strategies, from FirmsConsulting and beyond? Or more about learning theory, and how to train your skills in the environment, social, and psychology context that will help you transfer your latent abilities into case interview success? Until next time!
This article was contributed by Joe, a valued member of the FIRMSconsulting community.
Case Interview Math (Mental Math) Tools, Formulas and Tips
Consulting case interview mental math practice is a must as part of one’s overall consulting case interview preparation. All management consulting firms, and certainly McKinsey, BCG and Bain, expect candidates to be very comfortable with quantitative data, statistics, and the ability to make decisions and client recommendations based on data.
Management consultants at firms like McKinsey, BCG, Bain, Deloitte spend a lot of time working with numbers, charts, calculations, financial models in excel and other math work, often mental math work. So any consulting case interview mental math test, and there are really multiple mental math tests scattered throughout the consulting case interview process is something you have to be well prepared for.
This does not mean that you need to have a math degree to have the right level of consulting case interview mental math skills. But you do need to know what is expected of you and you do need to practice mental math a lot.
How is consulting case interview mental math different from academic math?
Because management consulting is all about solving difficult problems, usually under extreme pressure, the case interviewer is expecting a candidate to approach math problems in a specific way. In academic settings the most important element of solving math problems is accuracy. Accuracy is also very important for case interview math but management consultants usually work under extreme time pressure. And so answers are often required to be close enough to guide towards the “right” recommendation, versus being 100% accurate.
For example, imagine you are asked to calculate the market size for baby diapers for sensitive skin in Singapore. If this was a problem within an academic setting you would be expected to give an accurate answer correct to the decimal point. In consulting case interview settings you will have to make many educated estimations to arrive at, hopefully, a close enough answer. And then you will be expected to do what we call a sanity check to ensure that your answer actually makes sense.
Let’s take a look at an example from a real McKinsey engagement, mentioned by one of our trainers, Kevin P. Coyne. In case you don’t know, Kevin is a former McKinsey worldwide strategy practice co-leader and he leads The Consulting Offer II, which you can access if you join our Premium membership or FIRMSconsulting Insider level.
In this example, Kevin mentioned serving a large bank and during initial interviews with employees of the bank, Kevin’s team noticed that 100% of the profit for that bank was coming from one business unit. That does not mean that all other business units were operating at a loss. But combined all other business units of that bank had zero profit. So the bank was dependent on this one unit to generate all their profits.
Kevin’s team further uncovered that a lot of clients that the unit served were really old. To give a more accurate answer on how bad the situation was Kevin’s team selected only 1 letter in the alphabet and studied the age of all the clients whose name started from that letter, let’s say it was letter B.
This exercise uncovered that within the next 5 years that bank would lose something like half of its clients. And it does not mean the bank will have those clients for 5 years and then they will disappear. No, the clients will start dying now and within 5 years the client base will be about half smaller than now.
And younger people were not interested in that type of service. Doing the same analyses for all clients within the unit would be cost-prohibitive and will take significantly longer, and the limited analysis conducted was more than enough to understand that the bank was in serious trouble and drastic action was required.
This is a great example of how math in real consulting settings is often focused on getting close enough/good enough answers fast and cheap. And as a great management consultant, you will need to have strong enough business judgment to know what is good enough and what is required and to never waste the client’s money and other resources on unnecessary analyses.
What do you mean by case interview math?
Strengthening your mental math and written math skills is one of the most important elements of preparing for case interviews.
As part of a case interview process, your mental and written math skills will be tested in multiple ways. If you are strong in academic math you are in a good place. However, the style of math used during case interviews is quite different vs. math problems in the academic context, as we discussed above, and takes time to get comfortable with.
Some examples of what case interview math test can include:
Case interview math test can include word problems. Word problems used as part of case interviews are similar to the type of word-based problems you practiced for as part of your GMAT preparation or preparation for other standardized tests. And such a case interview math test may or may not include a business-based context.
Case interview math can be tested during a full case. In fact, full cases almost always test math along with other skills. For example, coming back to the example above, you may be asked to estimate the market size for diapers for sensitive skin babies in Singapore as part of a full case of your client considering entering the Singapore market. As part of the case, you may also be asked to work with many graphs and charts, which we refer to as data cases. We cover data cases extensively in The Consulting Offer, our flagship program where we help real candidates prepare for interviews with McKinsey, BCG, Bain, Deloitte, etc. You can track candidates’ preparation at various stages, all the way from networking, editing resume and preparing for standardized tests to getting an offer and deciding if they should accept an offer. You can track Ritika joining McKinsey Chicago, Jen joining Bain Boston, Assel joining McKinsey Europe after 5 years out of the workforce and with no prior work at MBB (never before been done), Sanjeev joining BCG, Alice joining McKinsey NYC and much more.
Mental math is also tested as part of case interview math tests. In fact, it is tested a lot as part of the case interview process. You will be required to do math in your head and very fast. This is often one of the most difficult components of a case interview for candidates. The Consulting Offer will help you prepare.
Standard math such as multiplication, division, fractions, percentages, and other concepts are routinely tested. Case interview math tests are usually baked into a case and math is just a component to finding a solution within a specific business context.
In all examples of case interview math above, speed and relative accuracy matter. And the use of calculators is not allowed. So it is crucial to practice and be ready to handle case interview math tests fast, accurately, and without a calculator.
Consulting case interview math formulas
Revenue = Volume x Price
Cost = Fixed cost + Variable cost
Profit = Revenue – Cost
Profit margin / Profitability = Profit / Revenue
Return on Investment (ROI) = Annual profit / Initial investment
Breakeven / Payback Period = Initial investment / Annual profit
EBITDA = Earnings Before Interest Tax Depreciation and Amortization. EBIDTA is essentially profits with interest, taxes, depreciation and amortization added back to it. It’s useful when comparing companies across various industries.
Fast mental math: rounding numbers
What will help you become faster in doing mental math during consulting case interviews is rounding numbers. For example, ~82 million population of Germany becomes 80 million, ~46.7 million population becomes 45 million. The key to rounding numbers is to round them carefully, in a way that does not distort too much the final answer. A good guideline to follow is not to round by more than 10%. It is also helpful to round both up and down as you are working through the case, so the effects, to some degree, cancel each other out. At the end also make sure you check if your answer actually makes sense.
Fast mental math: dealing with large numbers
The key to dealing with large numbers, like 200 million, for example, is to remove zeros and then add them back later. Use labels (m,k,b) to help you keep track. So if you have 200 million, it becomes 200 m to help you remember that it is millions. 200,000 will be 200k. 10 billion will become 10 b. The key to achieving fast mental case interview math is to simplify. For example, 5 x 30 million becomes 5 x 3 = 15 with 7 zeros.
Fast mental math: break down numbers into smaller parts
When dealing with case interview math, another trick that will help you work through the problem faster is breaking down numbers into smaller parts. For example, 14 x 6 = (10 x 6) + (4 x 6) = 84.
Fast mental math: subtracting from numbers with 1 followed by zeros
This is another trick for faster case interview math. Again, simplify. 1000-536 becomes 999-536+1 = 464.
Fast mental math: group numbers into multiple of 10 (addition)
Another trick for fast case interview math is to group numbers into multiple of 10 (for addition). 3+7 + 4 + 6 +13 +7 +21 becomes 10 + 10 + 20 + 21 = 61.
Other tips to achieve fast case interview math (mostly mental math) during a consulting case interview
Here are a few tips to keep in mind to help you perform better during a consulting case interview when it comes to case interview math (and mostly mental math).
- At the beginning of the case ask your interviewer if it is ok to round numbers. Most of the time they will say yes and it will make math calculations much easier and faster.
- Do not rush. If you make a mistake it will take you even longer to fix it. This is if you even catch your mistake. You may also catch your mistake by the time when the interviewer will not give you an opportunity to fix it. And case interview math mistakes can be very embarrassing and lead to a completely wrong recommendation. Of course, there is a lot of time pressure during consulting case interviews so do not take any longer than you need. You need to find a good balance. This comes with a lot of practice. We provide a lot of opportunities for you to practice case interview math. Some full cases are provided below and you will find more on our YouTube channel. And, of course, you can unlock access to all candidates and seasons of The Consulting Offer when you become Premium member (more details below).
- Do not be afraid to write things down when you feel you need it.
- Keep your writing organized. Let say you are estimating how many cars will be purchased in Germany in 2020. As you are putting down numbers for each element of your equation keep it neat and organized so you don’t get confused and it will also help you avoid silly mistakes.
- Do not state your answer to an interviewer as a question. Be confident in your answer.
- As part of your preparation refresh key math topics like ratios, fractions, percentages, averages, and probability. Khan Academy is a great place to refresh your math skills. And you will have more than enough opportunities to practice fast mental case interview math as you go through various candidates and seasons within The Consulting Offer (part of Premium membership).
Practice consulting case interview math / mental math with full cases
As you work through the cases remember to focus on all elements of good case performance, not just math. People usually underestimate how important other elements of case interview preparation are, including FIT. And only realize after being rejected that the elements they ignored during preparation were the reason for the rejection. Learn from the mistakes of others. Take all elements of case interview preparation seriously.
BUSINESS CASE EXAMPLE #1: MCKINSEY, BAIN, BCG ACQUISITION CASE
This case is a McKinsey style case, of medium level difficulty. It should take you 15-20 minutes to solve this case.
The question is given upfront, at 2:02. The part in black is the part the interviewer would share with you and a part in grey is the part interviewer may share as the case progresses. The interviewer wants to see if the interviewee understands the case and asks the right questions.
The case question is quite explicit but even so we will show you how you can adjust the case and make the case more explicit.
Everything rests on the key question. If anything is not part of the key question, ignore it. Even though lots of information is provided, take time to understand and set up the case.
Always show why information is needed, and show progress so the interviewer is they are willing to provide more information. It is a barter. And always use the case information provided and the appropriate language to push the case forward.
BUSINESS CASE EXAMPLE #2: COMPREHENSIVE MARKET ENTRY CASE
We did this recording a few months after we completed the training with Rafik (TCO I). This is one of the most complex market entry cases we had to put together. It has elements of operations, elements of pricing, elements of costing and, obviously, elements of market entry. And it is probably the most difficult market entry case we can do because most market entry cases that most interviewers focus on have a strong market attractiveness element, market profitability element. But very few people actually look at the operational issues of entering the market. And it does not matter who you are interviewing with: Bain, BCG or McKinsey. The bulk of the focus usually goes towards analyzing the market worthiness but not a lot on the operational issues. So we decided, in this case, to flip it around and give this case a strong operational theme.
BUSINESS CASE EXAMPLE #3: PEPSI’S LOS ANGELES BOTTLING PLANT
Operations cases can be tackled in two ways: strategy and operations and within operations from productivity and the supply chain side. This case uses the supply chain side.
This case is candidate-led. As we mentioned above, candidate-led cases are much harder than interviewer-led cases. That is why we at FIRMSconsutling place so much more emphasis on teaching you how to lead cases vs. relying on the interviewer to lead. This will be considered an operations case. Pay attention to a very insightful brainstorming at 14:50 which includes at least one idea you most likely would not come up with if you were solving this case before watching this video.
What else can I do to improve my case interview math?
Mental math is a muscle. But most of us do not exercise it enough once we leave school. So your case interview preparation needs to include math training.
First refresh your knowledge and ability to calculate basic multiplications, divisions, additions, and subtractions, without a calculator. The Consulting Offer program (a part of Premium membership) includes ongoing opportunities to practice this. We also have many cases available for free on the FIRMSconsulting YouTube channel to get you started.
And there are other tools you can use for case interview math prep.
Khan Academy has some resources that you may find helpful. Here are some helpful links:
You will need to regularly practice to get comfortable with mental and written math. Case interview math tests require you to do all math calculations fast and accurately. We recommend working through a few sessions of The Consulting Offer a day to ensure multiple opportunities to practice math and other skills you need to give yourself the highest chance to get an offer from firms like McKinsey, BCG, Bain, Deloitte, etc.
Go through a few sessions every day and you will start feeling more comfortable over time not just with case interview math but with your resume, networking, estimations, brainstorming, answering FIT questions in a way that answers what the interviewer is REALLY asking you.
You will also develop or strengthen the ability to lead and handle difficult cases, and the ability to develop your own framework uniquely tailored to solve a particular case, and much more. View it as an investment in skills that will serve you for the rest of your life vs. just searching for tips and tricks to get an offer from McKinsey, BCG, Deloitte, et al.
Additionally, some candidates found the following tools helpful as supplemental materials along with The Consulting Offer. We have not tested those tools but are sharing them in case you would like to explore them.
Mental math games (Android). This one is similar to the mental math cards challenge app on iOS (below).
Mental math cards challenge app (iOS). This mobile app is a good choice if you are an iOS user.
Magoosh’s Mental Math Practice – Arithmetic Flashcards (iOS + Android). And here is another free math app that uses flashcards. And it allows you to track your progress as you study.
Preplounge: mental-math (registration required)
Case Interview: calculations (registration required)
How FIRMSconsulting can help me?
You will need to get comfortable doing calculations fast and accurately. And this comes with a lot of practice. If you will be using The Consulting Offer to prepare for your consulting case interviews you will have what seems to be never-ending opportunities to practice mental and written math as part of the full cases and as part of particular questions such as estimations, etc.
Management consulting jobs are very competitive, and working with FIRMSconsulting can mean the difference between getting an offer, or multiple offers, from your target firms and barely getting an offer from the company you hoped you never would need to settle for. And the latter example is something I, unfortunately, observed many of my MBA classmates settled for.
When it comes to case interview math The Consulting Offer program, all 5 seasons of it and counting, with various candidates, includes everything you need to master not just case interview math, but all key aspects of consulting case interviews.
Don’t miss out by investing your time with general math drills when you can practice real-world case interview math examples while being taught by former consulting partners.
If you want the most comprehensive guidance for consulting case interviews math, and other aspects of case interview preparation, so you go to your interviews confidently, become a Premium or FC Insider level member now. And if you still have questions contact FIRMSconsulting ([email protected]) to find out why candidates even from top schools like Harvard, Stanford, and MIT choose us when they need consulting case interview preparation help, and stay with us for years and years once they get coveted jobs at McKinsey, BCG, Bain, Deloitte, etc.
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