**Case Interview Example - Estimation Question and Answer**

I was asked the following management consulting estimation question by a McKinsey interviewer many years ago:

**"Estimate how long it would take to move or relocate an average size mountain 10 miles using an average size truck"**

Below you will see my answer to this estimation question and the process and rational I use to answer this specific question can be used as a template to practice answering other estimation questions as you prepare for case interviews.

The first thing to realize in an estimation question is that an acceptable answer MUST mention a specific number.

This question was how much time it takes to move an average mountain 1 mile (or something along those lines).

If the answer does not include a specific unit of time like X hours, Y days, Z years, ** then the answer is not acceptable**.

By the way, I use the word "acceptable answer" instead of "correct answer" very deliberately. The interviewer's evaluation in this type of question is in assessing the approach you took, not necessarily the specific answer you gave.

The next thing to the answer must include is that explicit assumptions must be made.

It is not possible to answer this question without making some assumptions. They key is to EXPLAIN to the interview that you are going to make some assumptions. Once you do and once you make a specific assumption, explain your rationale behind that assumption.

For example, when I was given this question. I knew that I needed to estimate the cubic volume of the mountain. And since the mountain loosely resembles a cone, I knew there was a geometric formula to calculate the volume of a cone--except I did not recall the specific formula off the top off my head.

So my interviewer suggested that I estimate the formula of a cone, which in turn I would use to estimate the volume of an average size mountain, which would then be part of a calculation to estimate the average time it would take to re-locate it.

Notice the estimate that is nested within the estimate here. This is very common. Most important thing is to not get mixed up and confused by your own work.

I find it is useful to just write out the formula that will produce the estimate FIRST, THEN go about making reasonable assumptions.

For the move the mountain case, the formula I wrote up on the white board during my interview was:

volume of mountain / volume of a truck * time per truck trip = total time to move a mountain

I would literally write that on the board. That is the amount of time it would take 1 truck to move an average size mountain 10 miles (the 1 truck is an assumption as well)

Then I went about estimating each of those 3 factors.

Assume the average size mountain is 1 mile tall, 1 mile wide, and the shape of a cone. That's approximately 5,000 ft in height and base.

I forge the formula to calculate the volume of a cone, but if I eye ball it, it is probably a little more volume than half of a cube of similar size height and base.

The volume of a cube that's 5,000 ft tall, 5,000 ft wide, and 5,000 ft deep is 125,000,000,000 cubic ft.

Since I'm trying to estimate a CONE, and not a CUBE, I'd then take 125,000,000,000 x 50% (my approximate guess as to how much smaller a cone is vs a cube of approximately the same height, and width and length at the base.

With some slight rounding, that gets us 60,000,000,000.

Then underneath my original formula, I would write the following:

60,000,000,000 cubic ft / volume of a truck * time per truck trip = total time to move mountain

Next, I would move on to estimate the volume of a truck.

The carrying capacity of a cargo truck is the width x length x heightof the cargo container.

I said, well I know those big trucks are a little wider than my car, but not by much since they still must be able to fit into a lane on the freeway. My car sits 3 people across, assuming 2 ft in shoulder width per person, that's 6 ft of interior space. Let’s add on a little more and assume those big trucks are around 8 ft in width.

I know they are about double the length of most passenger sedans. And lets see if I were to lie down in the driver's seat to take a nap, I cover most of the interior cabin space. And the hood and trunk of the car combined are about the same length as the interior cabin. I'm a little under 6ft tall, so that makes my car around 12 ft long. If I double that, I get the length of one of those trucks to be 24 ft long. I subtract out say 4 ft for the driver compartment, and that leaves me about 20 ft in length for the cargo area.

Last time I looked, I saw a worker standing in the back of one of the cargo areas, and the cargo area was taller than the person. I figure the cargo container is about 8 ft tall. And since most freeway bridges have signs that say "height 13 ft" and I know those trucks can go under those bridges, assuming an 8ft cargo section and a 4ft for the tires and chassis under the cargo area, that gives me 12 ft...which does seem to triangulate with the height of those underpasses. So I'll say the cargo section is approximately 8 ft tall.

The volume of the cargo area of an earth moving truck is:

8 ft wide x 20 ft long x 8 ft tall = 1,280 cubic feet

For sake of simplicity, I'm going to round that down to 1,250 cubic feet and plug this number back into my original formula which now reads as follows:

60,000,000,000 cubic foot mountain / 1,250 cubic foot truck capacity * time for truck trip = total time to move a mountain

The only factor missing in our estimate is figuring out the round-trip time for a trip to move 10 miles, drop its load, and return the 10 miles. Let’s figure out the travel time first. Assume the truck travels on the freeway at 60 miles per hour.

For it to travel 10 miles, it does so in 1/6 and hour or 10 minutes. The drive time is 10 minutes to the new location, and 10 minutes returning to the old mountain for a total of 20 minutes. Assume that the off-loading process has been designed to be pretty quick. The load is just "dropped" and then repositioned while the truck is on its return trip (as opposed to being scooped out of the truck, one scoop at time which seems more time consuming).

That means each round trip takes 30 minutes or 0.5 hours.

Let's go back to our formula again and update it.

60,000,000,000 cubic ft mountain / 1,250 cubic foot track capacity * 0.5 hours per truck trip = total time to move a mountain

Let me do the math now. For the first 2 components of the formula, that works out to about 50,000,000 (50 million truck loads).

50 million truck loads x 0.5 hours, thats 25 million hours to move a mountain.

If we assume a typical day has 25 hours (to make our math a little simpler), that's 1 million days to move the mountain using only 1 truck. That works out to a bit under 3,000 years

That is the logic I just presented is a pretty good one that would most likely pass most estimation question interviews.

You will notice that for every little component I explain WHY I felt that was a reasonable assumption.

There is a big difference between making a wild assumption vs. a reasonable one. Your goal is to make as reasonable assumption as you can come up with. When you make such an assumption, it is very important you explain WHY you made the assumption you did.

The math is not that complicated (it's math we all learned before high school) BUT communicating what you are doing is just as important.

It is also important that you do not make a math mistake. I wrote out this example quickly and hopefully I did not make a math mistake.

If I did make a math mistake, I would full expect to get rejected even if I got the logic and assumptions largely right.

That's just the way it works. Practice your mental math. You DO use it a lot not just in interviews but with clients as well.

How did we do? :p

Why would you want to do that?

Based on the wording of the question I’ve assumed that there’s one truck.

First I split the problem into # loads and time per load.

# trips:

*I assumed that the average mountain can be approximated as a cone of radius 2000m (went metric as I’m from Australia) and an average height of 2000m.

This gives a volume of (1/3)x3.14×2000^3, which is approximately 8 billion cubic metres.

*I made assumptions about the average truck size and decided to go with 4m long, 2m high and 1.5m wide, giving a volume of 12 cubic metres (marked this assumptions to test later).

Dividing the volume of the mountain by the truck size gives the required number of trips. 8billion/12 is approximately 700 million trips.

Time per load:

*I assumed that the average speed limit in the area would be 60m/h (rural)

Therefore travel time would be 20miles (return)/60 = 0.33hrs of driving.

*I assumed that there would be a crew at each site to unload and pack the matter, and that they had the appropriate machinery.

Therefore I assumed roughly 20 mins delay at each site. I approximated this to 0.66hrs, making total time per load 1 hr.

This gives a total time of 700million hours.

*I assumed 24/7 operating hours.

This figure should be divided by # hours in a day and # days in a week –

I divided by 7, then by 25 (up from 24 for simplicity) to give 4 million weeks.

Dividing by 50 (down from52 for simplicity) yields 80,000 years.

Sense check – I would check the following assumptions:

*average mountain size

*average truck capacity

*equipment and crew available at the sites (would affect answer dramatically – and if they can only afford one truck, they likely won’t be staffing crew and lots of equipment at each site for 80,000 years)

*hours of operation

Overall, the number is too high for any feasible plan; but then again, nobody would do this with one truck.

If it is an average mountain of shit, it would take about 10 minutes of driving time.

Firstly, how big is the average size mountain:

The tallest being apprx.9000 ft and the shortest being apprx. 1000ft. Thus, average size is about 5000ft.

What is the weight of an average mountain (This is where I’m utterly confused but I’ll give it a go):

I will take a common measurement; namely the BMI for humans, is weight (Kg)/height (m) squared. The BMI for a very an average person is about 25, if you are to make a pyramid from the person, you’ll need 5 of him (I sound ridiculous!) i.e middle maintaining the height, and 4 projecting from the central midpoint. This could equate to a BMI (remember the height is maintained) of about 125.

Assuming that the [weight:height squared] of a mountain is about 125, the weight of a 5000 foot mountain would be 3125million Kg which is about 3,125,000 tonnes.

Weight of truck: A medium sized truck would weigh apprx. 5 tons and the weight of the driver is negligible.

Average speed of a truck: On a straight stretch of road with no traffic apprx. 50 miles/hr. And the speed decreases as the weight increases by a constant (5 tons – 50miles/hr and 10tons 25miles/hr).

Therefore to carry 3,130,000 tons the average speed would be 5/3,125,000 = 1/625,000 miles per hour.

To travel 10 miles = 10 x 625,000 = 6,250,000 hours (Note I haven’t taken into account the time it will take to start the mountain moving, to overcome inertia).

Therefore:

it depends on how long it would take to dismantle the mountain, piece by piece. you would need a team of workers with various tools (picks, axes, etc) to essentially “break up” the mountain into smaller rocks. then, the rocks can be placed into the truck and the truck will transport them the 10 miles down the road. so, the rate limiting step isn’t the truck transport, it’s getting the mountain broken down into smaller, manageable pieces. this is actually a good allegory for knowing how to tackle seemingly insurmountable problems: just break the problem/task down into bite-size pieces, and soon, it will be accomplished/overcome.

We can divide this task into two parts.

a) number of rounds

b) time required in each round

Total time required = a x b

To answer this I need to know what is the average size of mountain and what is the carrying capacity of an average size truck. This would help us know how many rounds of a truck are required. (We will divide total size by capacity of the truck)

Then we want to know how long each round takes and to calculate that we have to know average speed of a truck which with distance can give us average time required in one trip.

1. Estimate the total volume of the mountain. Suppose its radius is 3 km and its height is 1km. The total volume will be 1/3*pai3^2*1,which approximately equals 20 km^3

2. Estimate the total amount of substance that an average truck can carry for one trip. suppose the size of truck is 3*2*3 (m), so the total amount that the truck can carry for one trip approximately equals 20 m^3.

3. Estimate how many trips the truck will have to travel in order to dig out the entire mountain. 20 km^3/20m^3 equals 1000,000,000

4. Estimate how long does it take for a round trip. Suppose it takes the truck half hour to load and it takes the truck 10 minutes to drive 10 miles and it takes another half hour for the truck to unload and then 10 minutes to drive back. So it takes approximated 1 hr 20 minutes for the truck to do a round trip

5. Calculate the time. 1,000,000,000 times (1+1/3) approximately equals 1,300,000,000, which approximately equals 50,000,000 days, which approximately equals 150,000 year.

The time depends on many factors as follows:

1. The average speed of truck. Higher the speed, lesser would be the time required to cover 10 miles.

2. The number of rounds required. Its very unlikely than an average size mountain can be shifted in one attempt. It may be required to break it into pieces and reassemble after relocation. Larger the volume ratio of mountain to truck, more trips will be required and hence more time.

3. The time required to dismantle and reassemble the mountain also contributes to the total time in question.

Well, this is tricky one, main hit is that question was given by guy who hold PhD in Astrophysics. Thus, if we put the origin in to the Sun. As far as I remember the orbital speed of the Earth averages about 30 km/s. thus to cover 10 miles (which is 16 km) we need about 0.5 second