‘Quants Interview Series’ – Part 1

If you’ve ever faced an interview question like:

“How many real Christmas trees are sold each year?”“Estimate the revenue of a coffee chain in your city.”<br>“How many customers would buy this product?”<br>

…and your first thought was:

“Wait… I don’t have enough data to answer this.”

Congrats! You’re thinking correctly.

Because these questions are not really math questions.

They are assumption questions disguised as math.

In this post, we’ll break down what assumptions are, why interviewers care about them, how you can create them confidently, and a list of common assumptions you can memorize for interviews.

1) What are assumptions?

An assumption is a piece of information you accept as true without proof.

That sounds a little scary, right?

But in interview estimation problems, assumptions are normal because the interviewer is intentionally not giving you all the information.

Why assumptions matter in Fermi problems

Fermi problems often involve estimating:

• Revenue
• Customer demand
• Market size
• Production capacity

And your answer becomes much more credible when you don’t just throw out a number like:

“Umm… maybe 10 million?”

Instead, you build the answer using intermediate building blocks.

That’s what makes your estimate believable.

2) Why interviewers love “building block” thinking

Let’s take a classic example:

Example: Estimating demand for real Christmas trees

Suppose the interviewer asks:

“Estimate how many real Christmas trees are purchased each year.”

A strong candidate won’t guess randomly.

They’ll break it down like this:

• How many people celebrate Christmas?
• What % of those households buy a tree?
• Among tree buyers, what % buy real vs artificial trees?

Even if your final number is off, the interviewer is looking for:

“Good structure.”<br>“Logical reasoning.”<br>“Realistic assumptions.”<br>“Clear communication.”

That’s what they want.

3) “Isn’t it ridiculous if I can’t ask or research assumptions?”

Honestly? It feels ridiculous at first.

But it’s done on purpose.

Estimation questions are meant to simulate the real world.

In real life, decision makers often have to act when:

• information is incomplete
• data is missing
• time is limited
• research is not possible

Professionals don’t freeze.

They estimate, decide, and move forward.

And that’s exactly what interviewers want to test.

What the interviewer is actually testing

They’re not grading you on whether you said:

“It’s 31.6 million trees.”

They’re grading you on:

• Judgment → Are your assumptions reasonable?
• Structure → Did you break it down properly?
• Communication → Can you explain clearly?
• Math fluency → Can you calculate confidently?

4) The secret skill: sound judgment

One of the biggest differences between weak and strong candidates is judgment.

Strong candidates propose assumptions that feel realistic because they are:

• well-read
• intellectually curious
• exposed to real-world numbers

5) How do I come up with assumptions?

There are three main ways to generate assumptions in interviews:

1. Research critical assumptions before the interview

2. Ask the interviewer

3. Come up with your own

Let’s go through each one.

Method 1: Research critical assumptions before the interview

This is the best method because it reduces stress in the interview.

You don’t need to memorize a textbook of statistics.

You just need a few high-impact numbers like:

• population of your country and major cities
• household size
• income levels
• GDP scale
• common penetration rates

These numbers come up again and again.

Bonus: Research role-specific metrics

If the role is specialized, learn the numbers used in that field.

Example: Digital marketing roles

It helps to know:

• CPM (cost per thousand impressions)
• CPC (cost per click)
• CPA (cost per acquisition)
• Click-through rate (CTR)
• Conversion rate

Even if you don’t know exact values, knowing reasonable ranges makes your assumptions sound confident and grounded.

Method 2: Ask the interviewer

A job interview is a dialogue, not a one-way exam.

You can ask things like:

• “Should I assume the U.S. population is ~335 million?”
• “Can I assume the target audience is urban adults only?”
• “Is it okay if I assume 2.5 people per household?”

When interviewers will help

In consulting-style interviews, interviewers often expect you to ask.They may confirm your number or give a hint.<br>

When interviewers may not help

Some interviewers want you to generate assumptions yourself.

Still: it doesn’t hurt to ask

Even if they refuse, you didn’t lose anything.

A great line is:

“I’ll make an assumption here—please stop me if you’d like me to use a different number.”

That shows confidence and flexibility.

Method 3: Come up with your own assumptions

This is the default scenario in many interviews.

Your goal is to make assumptions that are:

• realistic
• internally consistent
• easy to calculate

A simple framework for creating assumptions

When you don’t know a number, build it using:

• benchmarks (population, households, income)
• percentages (adoption rate, penetration rate)
• frequency logic (daily/weekly/yearly usage)
• capacity constraints (how much one person/store can do)

Most estimation problems follow this structure:

• Start with a big number (population)
• Filter down (target segment %)
• Multiply by frequency
• Multiply by price/value

This is how you turn guessing into reasoning.

6) Can I round numbers up or down?

Yes — and you should.

Interviewers expect you to calculate by hand:

• no calculator
• no computer
• no Google

Rounding makes mental math manageable.

But rounding must be “judicious”

Good rounding makes you faster without making you look careless.

Sensible rounding examples

• 331M → 330M
• 8.1B → 8B

Sloppy rounding example

• 142M → 100M

That kind of rounding can signal poor attention to detail.

Rule of thumb for rounding

Try to round in a way that changes the number by less than ~5–10%.

If you must round aggressively, say so:

“I’m rounding aggressively to keep the math simple; this will be a rough estimate.”

That keeps you safe.

Common Assumptions to Know and Memorize

These numbers are updated to reflect more modern estimates.They are meant to be <br>fast defaults, not perfect.

Population: United States (Updated)

• United States: ~335M
• New York City: ~8.3M
• Los Angeles (city): ~3.8M
• Chicago: ~2.7M
• San Francisco: ~0.8M
• Seattle: ~0.75M

Interview rounding tip:NYC ≈ 8M, SF ≈ 1M, Seattle ≈ 0.7M is totally fine.<br>

Population: Outside the United States

• World: ~8.1B
• Europe: ~0.74B
• Asia: ~4.8B
• South America: ~0.44B
• Africa: ~1.5B
• China: ~1.41B
• India: ~1.43B
• Japan: ~123M
• United Kingdom: ~68M

Other Useful Assumptions for the U.S.

• Life expectancy: ~77 years
• People per household: ~2.5
• Median household income: ~$80K
• GDP: ~$28T
• GDP growth rate (long-run): ~2%
• Corporate tax rate (federal): ~21%
• Smartphone penetration: ~85%
• Percent with Bachelor’s degree (25+): ~38%
• Percent married adults: ~50%
• Percent under age 18: ~22%
• Percent over age 65: ~17%

Final Takeaway

Assumptions aren’t a weakness in interview math questions.

They are the entire point.

If you can:

• break big problems into drivers
• choose reasonable assumptions
• round intelligently
• calculate cleanly
• explain your logic clearly

…you’ll do well in estimation interviews even without perfect data.

Let’s take an example: How many BMW dealerships are in United States?

This is a classic “Fermi Problem” or market sizing question often asked in quantitative finance and consulting interviews.

The Estimation Framework (Step-by-Step)

We will calculate this using the formula:

$$text{Total Dealerships} = frac{text{Total Annual BMW Sales}}{text{Annual Sales per Dealership}}$$

Step 1: Estimate the Number of US Households

• US Population: $approx 340 text{ million}$ (Round numbers make mental math easier).
• People per Household: Assume $approx 2.5$ people per household.
• Total Households:

$$frac{340 text{ million}}{2.5} = 136 text{ million households}$$

(Let’s round to 140 million for simplicity).

Step 2: Estimate Car Ownership & New Car Sales

We need to determine how many new cars are sold per year, as dealerships primarily survive on new inventory turnover (service and used cars are secondary for this specific sizing).

• Car Ownership Rate: Most US households have a car, but we care about annual turnover.
• Average Car Life: A car lasts about 10–15 years.
• Turnover Rate: Therefore, roughly $1/15$ or $1/14$ of households buy a new car every year.
  • Let’s assume there are roughly 15–17 million new cars sold per year in the US (a known industry standard, but you can derive it: $140M text{ households} times 10% text{ buying rate} approx 14M$).
  • Let’s use 15 million new cars per year.

Step 3: Estimate the “Premium” Segment

BMW is a luxury/premium brand.

• Premium Market Share: The car market is dominated by economy brands (Toyota, Ford, Honda). Premium brands (BMW, Mercedes, Audi, Lexus, Tesla) usually make up about 10–15% of the market.
• Premium Car Sales:

$$15 text{ million} times 10% = 1.5 text{ million premium cars/year}$$

Step 4: Estimate BMW’s Market Share

BMW is a top-tier player in the premium space, competing closely with Mercedes and Lexus.

1. Market Share: If there are ~5 major players (Mercedes, BMW, Audi, Lexus, Tesla), BMW likely holds a significant chunk, perhaps 20% of the premium market.
2. Total BMW Sales per Year:

$$1.5 text{ million} times 20% = 300,000 text{ BMWs sold/year}$$

Step 5: Estimate Dealership Capacity (The Supply Side)

Now we need to figure out how many dealerships are required to sell 300,000 cars.

• Daily Sales: A dealership is open ~300 days a year. How many cars does a typical dealership sell per day?
  • Small dealers might sell 1–2 per day.
  • Large dealers (like those in LA or NY) might sell 10+ per day.
  • Average: Let’s assume an average dealer sells 3 cars per day (roughly 20 per week).
• Annual Sales per Dealership:

$$3 text{ cars/day} times 300 text{ days} = 900 text{ cars/year}$$

(Let’s round this to 1,000 cars/year to make the division easy).

Step 6: Final Calculation

$$text{Total Dealerships} = frac{text{Total BMW Sales}}{text{Sales per Dealership}}$$

$$text{Total Dealerships} = frac{300,000}{1,000} = 300 text{ Dealerships}$$

Sanity Check (The “Geography” Method)

As a quant, you should always cross-verify your result with a second independent method to check for major errors.

• Major Cities (Hubs): There are roughly 50 major metropolitan areas in the US.
  • Assume 4–5 dealers per major metro area (North, South, East, West, Central).
  • $50 times 5 = 250 text{ dealers}$.
• Mid-Sized Cities: There are roughly 100 mid-sized cities (places like Dayton, OH or Fresno, CA).
  • Assume 1 dealer per mid-sized city.
  • $100 times 1 = 100 text{ dealers}$.
• Rural Areas: BMW is a luxury brand; rural density is near zero.
• Total Check: $250 + 100 = 350 text{ Dealerships}$.

Both methods (300 and 350) converge on a similar range.

The “Actual” Answer

In an interview, you stop at the estimate. However, for your own context:

• Estimated Range: 300–350
• Actual Number (2025/2026 data): There are approximately 348 to 369 BMW passenger car dealerships in the United States.

and more about whether you realized that population $to$ market share $to$ unit economics is the correct logic flow.