Case interview math: the ultimate guide
You cannot get around it. In 70% of consulting case interviews, candidates need to demonstrate the ability to handle numerical problems confidently. For top-tier firms such as McKinsey, BCG, and Bain this probability is close to 100%: the more prestigious, the more likely that you will have to handle quantitative problems during their cases. For example, candidates first have to analyze a problem mathematically before qualitatively investigating the particular reason for a change in the numbers and then providing a recommendation. In fact, many candidates fear this part of the interview and even fail it.
This does not need to happen. Fact-based decision-making and recommendations in case interviews are often based on numerical results. The math you need to do to arrive at a certain conclusion will never be too difficult in nature. What makes it more challenging is the fact that both planning the steps you need to take and then calculating happens in a stressful environment with the interviewer watching your every step along the way. Of course, calculators cannot be used.
Being able to solve easy numerical questions, do fast mental math, basic calculus (addition, subtraction, multiply, division, percentages, and fractions) and give reasonable estimates is usually enough to survive a typical interview problem. Advanced mathematical knowledge is not required. Some of the problems you have to solve may be tricky, include multiple steps and therefore require a certain degree of logic but the calculations itself should not be difficult.
For some candidates, this might mean to (re)-acquire skills not used since high school, for others it might imply dumbing down their approach and get used to basic math again. The latter part is especially true for engineers and other people with a quantitative background.
When approaching cases, always have a quantitative angle, even if the interviewer does not explicitly ask you for it. Many candidates are simply scared of digging into the mathematics of a case. Don’t be that person!
In the following section, we will show you what you need to know to keep you on track during the grueling case interview mental and pen-and-paper math.
Mechanics of case interview math
The challenge of case interview math usually consists of two components. First, you need to derive and lay out a path to get to the desired result. Second, you need to do the calculations to get to the actual numbers.
So, how should you quantitative problems during a McKinsey, BCG, or Bain case?
- Before you dig into the quantitative problem at hand, slow down: clarify the numbers you got with the interviewer or discuss what additional information you would need. Clarify the outcome
- Listen carefully and actively to what your interviewer tells you. Finding a solution should be a collaborative process.
- Set up your planned approach to the calculation. For more difficult tasks, you could ask for some time to get your head around the problem with one minute to be the upper boundary.
- Lead the interviewer through your approach. This way you’ll make sure that mistakes are spotted early.
- When the interviewer agrees with your approach, follow through with the calculations.
- Summarize the result of each step briefly.
- When you get a final result, DON’T STOP THERE. Quickly explain and interpret the numbers. Relate them numbers to the problem at hand. Remember why you set up the calculation in the first place. How does it tie in your planned analysis? How does it impact your hypothesis? Is it the final result or just some intermediate result? It is important to discuss the ‘so what’ of your analysis.
A word of caution for mental mathematicians. Write everything down. When you make a mistake while calculating mentally you will face a serious problem. You have no track record of your mistake and would have to re-do all calculations. It is also more difficult to spot your own mistakes in time, which could save your ass. Additionally, the interviewer is not able to help you or pick you up from the mistake. At least keep a record of your intermediate results for the interviewer to be able to follow and intervene and for you to quickly go back to find and solve a mistake.
Now that we have discussed the process, let’s discuss the skills needed that help you ace case math.
Case interview math tips and tricks
Tackle the problems aggressively
Interviewers want to see highly driven candidates. Show self-initiative. If you hesitate or make mistakes the interviewer will test if this was just an anomaly. They will give you even more calculations in the progress of the case whereas a candidate that proceeded flawlessly through a calculus process often gets short cuts for the next quantitative parts or whole results readily delivered by the interviewer.
In case you messed up one calculation, don’t mess up the next one! Don’t calculate everything in your head. Use pen and paper to structure the numbers. Choose the fastest and easiest approach to set up your calculations.
Keep calculations organized along the way. Ideally, you find what works for you in the first mock interviews and then apply it consistently (habit-forming).
Re-learn basic calculus
(Re-) learn basic calculus operations and practice until you can do it in your sleep. Many candidates struggle with the concept of being watched while doing these basic operations.
Therefore, the better your skill to compute quickly in a stressful environment, the bigger your quantitative muscle in the interview. Practice these calculations both mentally and with pen-and-paper under time-pressure and vigilant eyes of friends, family, and colleagues. It certainly helps to build resilience and stamina.
Become comfortable thinking quantitatively
Get a feeling for numbers, especially percentages (e.g. be able to instantly estimate percentages in your head, also % of %) and magnitudes. This helps you to interpret results and put them into context and spot more obvious mistakes you made in your calculation. Be able to interpret your results and demonstrate good business judgment (e.g. if the numbers indicate that the goal of the company seems to be out of reach). Make approximations and estimations quickly and correctly. See the implications of your calculations and conclude correctly (ask the question ‘so what?’)
Relate numbers to each other
Quickly relate numbers and outcomes with each other. You can make a habit out of using equations to describe particular relationships. It both helps your thinking and shows that you are structured in your approach. A brief example:
To improve the over-utilization of train tracks draw up the equation: utilization = demand/ capacity. From this equation, you can instantly see that you need to decrease demand and increase the capacity to improve the utilization situation. Obvious but very effective, such an approach also demonstrates composure and logical thinking.
Sanity check everything
Try to spot your own mistakes before the interviewer does, be vigilant and sanity-check your approach to the problem and outcome of each calculation. Use your judgment to spot calculation and estimation results that seem out of line in the first place (e.g. 18.3% vs. 183%). Maybe you have done a mistake in the calculations or your assumption were off. In this case, act quickly to re-think and give reasons why your numbers might be off.
Keep track of units
Don’t lose track of your units. Is it kg, tons or USD? Set up the calculation before actually doing them, and already prepare (either mentally or preferably on paper) a space for the end result including the correct unit. Keep the units for your intermediate results organized and labeled.
Be efficient and use shortcuts
Be efficient in your calculations. Most answers rely on multiple assumptions and reasonable estimates anyway, therefore not providing a 100% level of precision. Use shortcuts!
Most of the time, close-to-correct answers are expected. If you come up with a population number use 80mn instead of 82.5mn (state it beforehand that you will trim the fat a bit; if the interviewer agrees, proceed with your calculation). Similarly, if you get 42.65 as an intermediate result say that in the following calculations this will be rounded to 40. Round numbers! Of course, ask the interviewer before you do so. 99% of the time, they will agree. The numbers won’t be 100% accurate either way and are not expected to be. Make plausible shortcuts in your approach and calculation to reach plausible numbers.
Similar to the above point, try to simplify calculations as much as possible, thereby reducing the number of steps you have to calculate and minimizing the chances of mistakes.
Watch the 0s
You would not believe how many candidates fall into this trap.
Watch out for 0s that you have trimmed or left somewhere along the way to facilitate calculations. One particular way that we find quite useful is to make use of the powers of 10 (e.g. 1.4bn / 70mn = 1.4 * 109 / 7 * 107). You can trim the power of tens and do the simple division. Once you reach a conclusion you are able to immediately see the magnitude of your number (in this case: 0.2 * 102 = 20).
How to prepare for case interview mental and pen-and-paper math
There are several things you could do to get up to speed with mental and pen-and-paper math that should suffice for McKinsey, BCG, or Bain interviews. The trick is to be confident in your ability to efficiently do simple math and resilient enough to external pressures in the process.
Get number affine by working with numbers you encounter in your daily life, be it the bar tap or the receipt at the grocery store or figures and data you find in the news (especially business news). Put them into context; create relative numbers and percentages. Try to calculate some simple business cases (e.g. waiting at the doctor: how much profit does he make a month, a year, etc.). The opportunities are unlimited. Additionally, get some apps for Android or iOS that train your mental math abilities. There are some really fun and entertaining apps out there.
Do all this in a stressful environment. You want to build stamina and resilience to outside influence and stress. Use the apps in the crowded and noisy subway; calculate in mock interviews, in front of friends and family or simply with a time limit (some apps include this function).
Make sure to be able to recite the 1×1 in your sleep. From there you mostly just add some 0s in case interview calculations.
Learn simple calculus shortcuts and see the examples below as starting points:
Build groups of numbers that add up to ten or multiples of ten.
7+3+12+8+5+5 = 40
(10)+(20)+(10) = 40
Learn quick subtraction by finding out what makes it to ten.
- Reverse the subtraction (5-2 = 3)
- Find what makes it to 10 (3+7 = 10)
- Add 1 to the digit on the left of the number you are subtracting
Convert percentages into divisions, e.g. 33% of 500 = 500*1/3 = 500 / 3 = 167
Split numbers into tenths, e.g. 60% of 200 = 10% of 200*6 = 120
Use the table below to learn the division table by heart:
Cut the 0s, but be careful to add them again in the end, e.g. change 34x36mn to 34×36 = 1224, then add six 0s –> 1,244,000,000
Break apart multiplications, e.g. 18×5 = 10*5 + 8*5
Exchange percentages and simplify the calculation, e.g. 60×13% = 0.6×13 or 6x 1.3 = 7.8
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