Estimation Question

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.

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501 comments… add one
  • Dennis Nov 11, 2010, 10:38 am

    I’ve used the following assumptions:

    -mountain volume:
    ->Assume cone shape: 1/3*heigth*base area = 1/3*h*pi*r^2
    ->heigth: 2km
    ->width: 4km

    -transport means: I assumed transport by trucks (vs. a big lifting device, cartwheels or a ‘star trek beamer’ thingy… etc)
    ->truckload volume: lxbxh = 5x3x1.5 = 23m^3 = 23*10-9km^3
    ->truck speed: 15km/hr (unpaved road)
    -># trucks: 100
    ->trucks operate at max efficiency (without timeloss due to having to wait in line)
    ->operating hours: 24/7 for 300 days/year
    ->time/truckload: pick-up: 30 min; transport: 60min (10miles/(15km/hr)); delivery: 30min; Total: 120min = 2hr
    ->refueling: happens within 30mins pick-up and delivery

    Which gave me the following results:

    So, based on these assumptions you’ll find a mountain volume of 33km^3
    Per session 100 trucks transport 23*10-9km^3 each = 23*10-7km^3
    Therefore 33/23*10-7 ~ 1.5*10^7 = 15 M sessions are required
    In total 15M sessions*2hr = 30 M hrs are required
    Leading to 30 M hrs/ (300*24 hrs/year) = 30M/7200 ~4000 years

    What does that mean: I’d build a tunnel rather than moving the mountain

  • Reem Nov 11, 2010, 10:31 am

    I don’t know if am right, but i will say it will take minutes, i will draw the mountain and i will right down it’s volume then i will just move it using my mouse on a scal drawing 1 mile or two or whatever!!! if it’s not right i still got a point 🙂

    • Frank Apr 14, 2017, 12:22 am

      Is this the quiz answer page to the mountain question?
      My answer to it is—–9 minutes and 1 second.
      Am I correct?

  • ahmet Nov 11, 2010, 10:20 am

    what I left out, though it is implicated: once we know how much mass we have and how much force we need to move it, its simple to map that to number of workers / machines and thus time needed. but again, i think the important points are:
    – what does “average” mean?
    – what does movement / relocation mean?
    the simplest relocation is making a change on the map (there are precedents for this in history, ie. border negotiation in cyprus as a recent example)

  • ahmet Nov 11, 2010, 10:14 am

    My first assumption would be that we are thinking about mountains on the earth’s surface, excluding mountains on the ocean floor and mountains on the moon or other planets.

    Next, we need to think about what qualifies as a mountain and at which point an elevation is just a hill. In addition, one has to define how a mountain is differentiated from another mountain, that is, at which point is it a mountain range, where are the limits / borders of a mountain? This becomes especially important to calculate an average. Another question is whether the distinction between mountains are colloquial or geological. There are mountains which are geologically separate but are regarded as “one” mountain.

    Also, one has to regard what “average” means. Whether the assumption is a global, national or local average defines the mass we will be dealing with. In defining a national or local average, one has to take into account mountains which are laid out on borders and are only partly within a region.

    In regarding the average mountain, it is important to take the surface vegetation (trees, bushes, etc) and wildlife (rabbits etc) into account as well. While some mountains are not very habitable, others are teeming with life. Further aspects in calculating the mass are water (rivers, lakes, underground reservoirs) and cavities (caverns, mines etc).

    Taking all these factors into account, one can calculate a well defined average. The act of calculating the movement by 10 miles is then a simple formula of distance times force necessary to mobilize the mass.

    However, there could be asked further questions complicating the issue, such as: why does the mountain have to be moved and does it have to be moved intact? If we are talking about the parts, then detonation and drilling would be viable means (in which case, we would be thinking of practical methods instead of abstract calculation of force). If we are talking about geolocation, one could also simply move the points of reference, thus changing the coordination system. This would be “moving the map” instead of moving the mountain.

  • M Nov 11, 2010, 10:09 am

    Seeing as we are not given any information to relate to, this makes it very easy to be as abstract as possible in my opinion, so i’d start relative to Earth’s rotation. We are all moving, but in relation to what (deeeeep I know!)? I think that’s for the person who is answering the question to decide! Anyway, this is just how i’d think about doing it!

    In this case, an ‘average mountain’ is therefore irrelevant. It could be a flea, human or Everest. It doesn’t matter. All objects on earth move with the earth and move at the same speed relative to the earth’s rotational spin. The fact a mountain is stationary makes it that bit easier. If it were a dynamic object like a car then we would have to assume all sorts of things to

    Let’s assume the mountain is at the equator. I’d assume the distance across the earth’s equator is 22,000 miles. It takes 24 hours to complete a full cycle. For the purposes of ease in this question (im simply demonstrating my method not arithmetic ability here!), lets say 24,000 miles:

    This means every 60 mins the earth and any object on it’s equator would travel 1000 miles. So, every 10 miles, it would take 0.6 mins or 36 seconds.

    This method cuts A LOT of corners but the object (“average sized mountain” in this case) still moves 10 miles in 36 seconds relative to the Earth’s rotations. One could also use the Earth’s rotation round the Sun but im not the astrophysicist so i’ve stuck to what I roughly know. There are INFINITE answers to this question, therefore.

  • Bobby Nov 11, 2010, 10:08 am

    I will assume that a mountain is really just a very large pyramid. This means that it has a square base.

    I will assume that an average sized mountain is about 10,000 feet tall.

    I will assume that the slope/incline of the mountain is easily climbable and is about 30 degrees. Since Sin(30) = 1/2 and cos (30) = (sq rt 3)/2, then the length of one of the sides of the base of the mountain/pyramid is 20,000 x (sq rt 3) feet.

    The volume of a pyramid is the (area of the base x height) / 3. Thus the volume of this mountain is 20,000 x 20,000 x 10,000 x 3/ 3 = 4,000,000,000,000 cubic feet.

    Now suppose that the mountain/pyramid is in sections of 1 cubic foot.

    If we think of a single worker going to the top of the mountain, removing one of the cubic sections and putting it in his wheelbarrow and then transporting it a mile to the site of the new mountain and putting it there, we quickly realize that the man will travel approximately the same distance every trip. He would have to go to the top of the mountain to take the first stone, and he would then be able to place that stone at the bottom of the other mountain.

    It is 20,000 up the side of the mountain, 20,000 feet to the bottom and then about 5000 feet (just under a mile) to the new site. This is a total of 45,000 feet. Using 5000 feet as our approximation, this is about 9 miles. We will then assume that the average worker can walk about 1.5 miles an hour while pushing this wheelbarrow (normal walking speed is about 3) which means that it will take him 12 hours to take one cubic foot from one mountain to the other. The time that it takes him to hoist and then dumb his stone we will assume is negligible.

    Therefore, putting this all together, it will take 4,000,000,000,000 x 12 man hours or 2,000,000,000,000 man days to move this mountain.

  • Bin Nov 11, 2010, 9:59 am

    How can we define move? Can we consider the sun as the original point? In case so, no man is needed, and we could actually find somewhere as the original point, so that the mountain can be moving 1 mile in a second…

  • Medic Nov 11, 2010, 9:39 am

    * typed in a hurry
    One man can complete all three tasks for 3.6 m3 in 100 hours (Not metric tons)

  • Medic Nov 11, 2010, 9:37 am

    OMG … This is tough! Long winded and probabley incorrect

    I would make a few assumptions to make my life easy:

    The average mountain is barren and consists of only rocks and soil and perhaps minor planr structures … grass and no trees. There will be no animals or ice.
    It is single peaked with peripheries. So it is cone shaped with extra peaks not as high as the core peak. There are two extra peaks, each comprising 10% of the volume of the core peak.
    We can restructure the mountain using the base as the top without any complications
    We have 100 men working

    The core peak has a square base of 30 * 30m area and a height of 1000 m. Hence the volume would be 1/3 *900*1000m = 300,000m3
    other peaks will account 60,000 hence a total volume of 360,000m3

    three tasks involved in moving the mountain
    picking up the soil
    Transporting
    offloading at new destination

    1 man can move 3.6 metric tons in 100 hours
    Hence one man can move 360000 in 36000 hours

    Assuming we have a work force of 100 – it will take 360 hours to move that mountain!

    Assume a working day is 6 hours, it will take 100 men working 60 days!

    And then I get fired.

  • d Nov 11, 2010, 9:27 am

    what ressources do we have?
    What is an average mountain?
    What obstacles are there within the ten mile distance?
    Machine that can pick up certain voulume of mountain, number of those volumes in the mountain, time to take them, move them to the other location (depends on the height, take average)

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