I recently received a question from a reader about case interview math and a 3^{rd} round McKinsey interview.

The reader asked what kind of cases and skills they should be practicing to improve their math skills for an interviewer-led case interview. I realized that this is most likely a problem many readers have so am including my response below which includes my tips for how to best prepare for the math portion of a case interview.

Two tips I highly recommend using during your interview and to practice prior to your interview are:

1. Slow Down and Make Sure Your Math is Correct

2. Test the “Reasonableness” of a number

**Slow Down and Make Sure Your Math is Correct**

My suggestion is write out the formula in words before doing the computations.

As in (over simplified):

Line 1) Profit = Revenue – Cost

Line 2) Profit = $10 – $5

Line 3) $10 – $5 = $5

Most candidates immediately jump to Line #3, but I generally recommend taking the extra effort to write Line #1… mostly to prevent yourself from being confused.

For example, when you spot check the following line:

Line 3) $10 + $5 = $15

It’s much harder to notice the error, because the actual computation you did was correct. $10 + $5 does after all = $15.

But, if you write the extra line, it is much easier to spot the error. Here’s an example:

Line 1) Profit = Revenue – Cost

Line 2) Profit = $10 + $5

Line 3) Profit = $15

By writing out the extra step especially Line #1, you can do a reasonable test very quickly.

Which brings me to my 2^{nd} tip of **Test the “Reasonableness” of a number**

You would look at the computation, but then you’d conclude that Profits *exceed* Revenue… wait a minute, that’s not possible in the real world.

That would in theory prompt you to re-check your work, and as you compare Line #2 to Line #1, you could easily notice the change in the – sign to a + sign.

For example, if the market size is $100 Million, and competitor #1 has 17.9% market share, I’m expecting the competitor’s sales to be around $20 million.

If there’s a competitor #2 that’s a little smaller than this, I’m guess the number should be in the mid-teen millions… say $13M – $19M range.

If I do a calculation where I compute revenues from competitor #2 as $500,000, I say, “Wait… that doesn’t seem reasonable, given the other data.”

This is known as “triangulation” — you try multiple approaches to calculate a number, or a reasonable range for a number. If you calculate a number three different ways, and you got the same approximate answer, then most working consultants would deem that number “reasonable.”

The “reasonable” test is usually applied to making estimates, as in estimate the market size of X product. We might use three different methodologies to estimate and see if they all give us the same answer.

For the kinds of computations, you’ll see in a McKinsey interviewer-led case, you can use a similar approach.

One of the techniques used in determining reasonableness is “bracketing.” Bracketing is a way of confidently determining a number that you’re certain is *above* the actual number… and one that is definitely *below*.

It is the process of establishing a high / low range for a particular number.

For example, let’s say you’re working on a case where you’re asked to compute the impact of changes in the client’s pricing strategy and how it would impact profits.

If you lower the price, you would expect the new profit to be *lower* than the profit with the previous price.

If you lower the price by 10% on a product which originally had a 22% profit margin (defined as profit / price), you should ideally notice two things while doing this calculation slowly:

Observation #1: By dropping the price, you’re actually dropping the profit margin by around 50% or so.

Observation #2: The # units sold would have to approximately double for profits to stay the same.

If the scenario is: “client drops prices by 10% and volume increases by 250%,” before you even do the actual math calculation, you would expect that overall profits would be higher than under the original scenario.

Here’s the secret to doing this type of reasonableness checking well:

You need to *think holistically* during the case… *not just do what you’re told*.

Observations 1 & 2 above are things you should ideally notice if you’re “thinking” during the case. But most likely in such an example, the interviewer will not explicitly ask you about #1 and #2.

If I were to venture a guess, I would say most candidate’s mindset going into the interview is to “solve the case,” when in many ways it should be to “serve the client.”

The former involves answering the explicit interview questions asked. The latter involves being thoughtfully analytical and bigger picture oriented *while* answering the explicit questions.

In addition, when you synthesize a case, you want to answer the explicit questions asked, and then mention the other issues you notice that if time permitted would be interesting to examine further.

Incidentally, the reason McKinsey cares so much about this skill is because it potentially risks embarrassing the firm with a client.

Analysts and Associates *do *make mistakes from time to time. The key is that with a McKinsey analyst and associate, it’s expected that you would be able to notice and catch your own mistakes before a client ever sees your work.

For practice, I suggest going through sample cases, with the intent of doing a reasonableness check for each case. By getting accustomed to this process, you’ll be better able to do it in an interview.

In addition, if an interviewer ever says, “is this number reasonable?” or “how would you determine if this number is reasonable?”…*never* say you have no idea.

If you must, *stall* for time and figure out some way to cross-validate your number. Seventy percent of the time, this comment from an interviewer is a hint that you made a math mistake. Thirty percent of the time they want to test your confidence level in your answer, and to observe how you determine whether that confidence is justified. And in both cases, they want you to do a reasonableness check.

For more math practice tools and resources I recommend checking out my math practice resource page.