# Case interview math

## An overview of case math problems, why firms use them and how to prepare Kenton Kivestu, ex-Google, ex-BCG, Founder at RocketBlocks Published: October 14, 2020

Case interviews are chock-full of math. This makes sense since consultants do math everyday in their casework and need to be sharp analytically to be effective on the job.

In their interviews, consulting firms use case math to see if candidates are up to snuff in terms of their analytical abilities. In this article, we walk through why consulting firms use case math in their interviews, the essential case math skills and examples of how they can come up, and some tips for how to prepare.

## What case math problems are really testing (Top)

The obvious question for most candidates is "Why am I being tested on these mental math abilities? Won't I have Excel for that when I'm on the job?"

While it's true that you'll have plenty of analytics tools around you to do simple, and more often quite complex, calculations while on the job, that's not what case math is testing for.

Consulting firms use case math to test two things:

• A candidate's ability to quickly confirm or disprove a hypothesis
• A candidate's ability to prioritize analyses by quickly doing math to rule out areas which don't need further analysis

Workflow on a management consulting case is an iterative, hypothesis-driven process. Given a problem, consultants come up with a hypothesis for a solution and then do analysis to confirm or disprove that hypothesis.

For example, let's say the CEO of a large consumer-packaged-goods (CPG) player is looking to improve the profitability of an underperforming product line. The average margin on these products is \$100, and they sell about 500,000 units per year. She's set a target of \$5 million in profit improvement. The partner on the case's hypothesis is that increasing the profit margin through supplier negotiations will make that happen.

We know from previous experience with a similar client that supplier negotiations would at most improve margins by 2%. Faced with this hypothesis, a good consultant would be able to quickly disprove it, maybe even during the meeting in which it was proposed! A 2% margin increase would produce \$2 of marginal profit per unit, and if sales remain steady we would only see a \$1 million improvement in profitability.

This quick analysis is a powerful tool because it allows a team to quickly pivot to other aspects of the profitability problem. Maybe we can increase sales through promotional activity in addition to cutting costs through supplier negotiations. Whatever the solution ends up being, we know that supplier negotiations alone won't cut it.

Our quick numerical analysis drove the process forward and helped to prioritize our efforts towards potentially more high-yield solutions. This is exactly what consulting firms need from their consultants, and that's why they test for it in their interviews.

## The case math skills you need to master (Top)

Let's get into the nitty gritty of what type of math you'll see in your case interviews. For each skill, we'll walk through examples of how it may appear in a case interview.

### Division and multiplication

Big division and multiplication are staples of case interviews. They're an easy way to test a candidate's mettle - it's not everyday you have to multiply or divide two numbers in the millions!

Case problems will throw all sorts of multiplication and division problems at you. You'll get numbers with tons of zeros, odd numbers that can't be easily simplified, and everything in between. The key to solving these will be to use shortcuts, break-up messy numbers into easier to manage chunks, and stay organized.

Let's walk through two ways we can use shortcuts to make multiplication and division way easier.

#### Case Example: Dealing with a TON of zeros

Let's say our client wants to understand the average productivity of their employees on their manufacturing line. They have 50 workers and on a given day produce 100,000 widgets. How many widgets are produced per employee?

Figuring out how many times 50 goes into 100,000 isn't easy, but what if it doesn't have to be that complicated? Let's remove 3 zeros from 100,000, effectively dividing it by 1,000. 50 goes into 100 twice. Add those three zeros back, we get 2,000. So, (100,000 widgets) / (50 employees) = 2,000!

#### Case Example: Working with messy numbers

Our client is a massive, and I mean massive pizza joint in New York City. They have 200 pizza ovens that can each produce 125 pizzas per week. What's their total pizza making capacity?

To make this problem easier, we can break up the problem into two parts using the distributive law. Instead of 200 * 125, we can set up the problem as (200 * 100) + (200 * 25).

• (200 * 100) = 20,000
• (200 * 25) = 5,000
• Now, add it back together: 20,000 + 5,000 = 25,000.

So in total, our client can produce a whopping 25,000 pizzas each week.

### Percentages calculations

Working with percentages and proportions is all over business analysis, and the use of percentages is a key skill in tons of different case math problems (more on this later…).

Percentage problems aren't hard to conceptualize, they're just the multiplication of a proportion to a given metric. The trick is learning how to do them quickly, or how to structure more complicated questions so that you don't get lost in a sea of numbers.

Let's go through the two basic ways percentages calculations come up in case interviews:

1. Easy percentage calculations
2. Messy percentage calculations

#### Case Example: Easy percentages

Simple percentage questions can be quite easy.

If cost of goods sold (COGS) is 10% of a Company A's revenue and they did \$160 million in revenues this year, what is the exact level of COGS?

Calculation: 10% of \$160 million can be calculated as \$160/10, which is \$16 million.

Let's say an analyst from Company A approaches us and tells us COGS are actually 15% of revenue. We can break up the calculation into two parts. Instead of directly calculating 15% of \$160 million, we can calculate 10% and 5% of \$160 million and add them together.

Calculation: We know 10% of \$160 million is \$16 million, and 5% of \$160 million is half of that. So in total, COGS is \$24 million.

#### Case Example: Messy percentages

Okay, let's make it a bit more complicated. As a result of a cost cutting initiative, Company A has reduced COGS to just 13% of revenue. Revenues have remained stable at \$160 million. What is the exact level of COGS?

We can use the same technique from before. We'll split up the percentages into easy to manage chunks. Instead of a direct calculation of 13%, we can set it up as (\$160 million * 10%) + (\$160 million * 3%).

We know 10% of \$160 million is \$16 million

To further split up our 3% figure, we can set it up as 3*(1% * \$160 million).

1% of \$160 million is \$1.6 million. Multiplied by three, that's \$4.8 million

In sum, we get COGS = (\$16 million) + (\$4.8 million) = \$20.8 million.

### Breakeven analysis

Breakeven analysis asks an interviewee to determine the amount of sales necessary to recoup a large upfront investment or cost - the breakeven point for a certain product or service. To put it simply, breakevens ask "How many units (or services) do I need to sell to make up for my upfront costs?"

Solving these problems follow a pretty standard format. Determine the marginal profit per unit or sale, and divide your initial investment by that metric. So the formula is:

(Investment) / (Unit revenue - unit cost) = Units required to "break even"

Quick tip: Breakevens often involve big division type problems. Mastering that skill will help a lot when dealing with breakeven calculations.

#### Case Example: Launching a snazzy new tech product

Tech products have high R&D costs, and a critical goal for technology companies is to recoup that initial investment within a reasonable timeframe after launch. For our product, we are given the following information:

1. Our client expects to spend \$1 million in development
2. Each unit costs \$100 to produce, and it's sold for \$300

So, how many units would we need to sell per year to recoup the initial investment?

So, how many units would we need to sell per year to recoup the initial investment?

Let's apply our formula

• Investment = \$1 million
• Unit revenue - Unit costs = \$200
• Calculation: (\$1 million) / (\$200) = 5,000 units

### Growth estimations

Growth estimates are a staple in business analysis. Companies are always thinking about and forecasting the future, and to do this they apply estimated growth rates to current metrics to inform where a business is going and how that may affect their strategy.

The simplest growth estimation problems will be one-period estimations. For example, if a business is currently doing \$1 million in sales and growth is expected to growth over the next year by 20%, sales in the next year will be (\$1 million) * (1 + 20%) = \$1.2 million.

More complicated growth estimations will have multiple periods, and really tough problems will have varying growth rates. Let's walk through an example of each type below.

#### Case Example: Determining future revenues from fixed, multiperiod growth

Let's say that our client is currently doing \$10 million per year in revenues, and revenue has historically been growing at a rate of 5% year-over-year. Their investors have asked the CEO to prepare a report on how revenues will grow over the next 2 years. We have been told we can assume growth rates will stay the same.

To determine this, we can use the formula for compound growth:

(Present Value) * (1 + growth rate) ^ (number of periods).

For this problem, the formula would be: (\$10 million) * (1 + 5%) ^ (2). (1.05)^2 is equal to 1.1025, and (\$10 million) * (1.1025) = \$11.025 million.

Alternatively, if you don't want to deal with exponents, you could calculate this in a stepwise fashion.

• Year 1: (\$10 million) * (1.05) = \$10.5 million
• Year 2: (\$10.5 million) * (1.05) = \$11.025 million

#### Case Example: Determining future costs from variable, multiperiod growth

In this case, imagine we were working with the same client, but they now want to know what their revenues will be in 4 years. Importantly, they expect growth to be 10% in years 3 and 4, up from 5% in the first two years.

To solve this problem, we can break up the growth estimates into two steps while using the compound growth formula. In the first step, we'll apply the 5% growth rate to the original revenue figure and project 2 years of revenue growth. Then we'll take the result and do the same calculation with the 10% growth rate for the final two years.

• Step 1: (\$10 million) * (1.05)^2 = \$11.025 million
• Step 2: (\$11.025 million) * (1.10)^2 = ~ \$13.34 million

Again, if you don't want to deal with the exponents or the numbers are more complicated, you can do the calculations in each step in a stepwise fashion.

We could also use a simple trick to get a "good enough" estimate of our answer. Instead of figuring out a complex exponent, we can add the compound growth rates together and multiply our original value by that sum. Let's see this in practice:

• Step 1: We have two periods of 5% growth, so we'll multiply (\$10 million) * (1.10) = \$11 million
• Step 2: In this case we have two periods of 10% growth, so we'll multiply (\$11 million) * (1.20) = \$13.2 million

Notice that while our answer is not exactly correct, it's within 1% of our answer and is certainly close enough for the purpose of a case interview! Plus, you can do this sort of math way faster. It's a win-win situation. More on this trick later...

### Market size calculations

Market math problems are an extension of percentage problems applied to a company's market share or a total market size. They'll come in all shapes and sizes, but always ask you a version of the following question:

If X% of a market is \$X, how big is the total market?

Market math problems come in two forms: ones with "easy" numbers, and ones with "messy" numbers. Let's walk through an example of each.

NOTE: For more in-depth market sizing estimate drills, see our overview of on how to approach market sizing estimates.

Quick Tip: Mastering percentages will make these problems a breeze!

#### Case Example: Market math with "easy" percentages

We know that our client has captured 10% of the market, and currently does \$10 million in revenues. What is the full market size?

Mathematically, the formula is: (Revenues) / (Marketshare). In this case, (\$10 million) / (0.1) = \$100 million total market size.

A far easier way to do this is to recognize that 10% goes into 100% ten times. So, we can multiply our client's revenues by 10 to get the market size. So (\$10 million) * (10) = \$100 million. This shortcut can be applied to all sorts of "easy" percentages.

• For 5%, the multiplier is 20
• For 20%, the multiplier is 5
• For 33%, the multiplier is 3

You get the idea!

#### Case Example: Market math with "messy" percentages

Now let's imagine that our client has a market share of 17%, and revenues are still \$10 million. What's the total market size?

There's no easy multiplier that we can use here, at least at first... The trick with these messier problems is to make a rough estimation by rounding the market share to an "easy" number. Interviewers usually don't expect an exact answer, and as long as you don't round to aggressively you should be in the clear!

In this case, we could round 17% to 20%. Then, we can use a shortcut to multiply \$10 million by 5 to get a total market size of \$50 million. Since we rounded up, you can say that the total market size is just north of \$50 million - which we know since we rounded the divisor up.

## Tips for how to prepare (Top)

### Isolate core skills and master them

We just went over the skills necessary to rock your case math. Just like you did in school, you need to study and master them.

A tool like RocketBlocks makes this process easy. We have tons of content that walks you through the different types of math encountered in a case interview and comprehensive strategies for approaching and solving these problems.

💡 Shameless plug: Our consulting interview prep can help build your skills

### Learn mental math short cuts

In an interview, you're expected to be able to apply these math skills quickly in the context of the case. But as we just went over, not all of the math is super simple and there can be plenty of room for error. Learning math shortcuts will make your case math far more efficient and accurate. You'll make less errors, move more quickly through a case, and have more time to apply the results of your analyses to the problem at hand.

Here's are two examples of these types of shortcuts:

The Rule of 72: The time it takes a metric to double given a certain growth rate can be roughly determined by dividing 72 by that growth rate. So given a 10% year-over-year growth rate for revenue, you can say that will take roughly 72/10 = 7.2 years for revenue to double.

Estimating Compound Growth Rates: Instead of figuring out a complex exponent, you can quickly estimate a compound growth by adding the component growth rates together. So if you are estimating 3-year compound growth of a metric at 5% growth year-over-year, a good estimate would be to apply a 15% growth rate to your metric. This can be very helpful when you don't need an exact answer for the problem at hand.

Note: This technique doesn't work well for super high growth rates or a ton of periods.

## Conclusion: Practice makes perfect (Top)

The key to getting good at case math is to do A LOT of practice problems. The more the better. As you do more you'll get faster, see more of the different types of problems, and get used to applying core case math skills in different contexts.

One way to practice is to do lots of mock cases. This will present relevant case math but the downside is you might spend 1hr on a case and only do math for 5 minutes! To excel at the math component, you should do targeted practice on math specifically.

A more targeted way to prepare for case math is to use a tool like RocketBlocks to isolate the skills you're weakest on and gain access to an almost unlimited number of problems. RocketBlocks helps you practice case math in two ways:

1. Practice cut and dry case math problems using our Math drills
2. Practice case math in the context of applied data analysis in our Charts and Data drills

Bottom line: to get good at case math you have to do a lot of example problems.

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