Case Interview Example: Estimation Question

This article provides the answer to this estimation question. The process and rational I use to answer this specific estimation question can be used as a template to practice answering other estimation questions as you prepare for case interviews.

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

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 does it take 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.

So 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. Lets 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 cubit 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. Lets 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. So 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 cubit 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 fairly 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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18 comments… add one
  • Erica X Feb 29, 2012, 3:15 am

    Hi, I’m a bit confused by how you got the volume of a cone here to be 125,000,000,000 cubic ft:

    “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.
    So 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.
    So then underneath my original formula, I would write the following:
    125,000,000,000 cubic ft / volume of a truck * time per truck trip = total time to move mountain”

    You say that the volume of a cone would be a little more than *half* the volume of a cube, but isn’t the 125,000,000,000 cubic feet you end up using for the equation the volume of a cube? Not a cone? Wouldn’t you have to half it and end up with 62,500,000,000 cubic feet?

    In other words – 5,000 x 5,000 x 5,000 would be the full volume of a cube. But you’d want only about half of that to get the volume of a cone with the same base and height, according to your original estimated formula.

    Please explain to me if I’m missing some fundamental simple step here. Thanks!

    • Victor Cheng Sep 27, 2012, 5:57 pm

      Erica – Your assessment is correct. The error was on my end and has since been revised in the post.

      -Victor

      • Zhu Feb 18, 2014, 8:57 am

        1280 cubic feet is around 36.2 cubic meter. Consider a density of rocks of 2.5 t/cubic meter, it is actually 90 t!

        I do not think it is a normal truck, it is a very big truck…

  • Muhammad Jun 13, 2012, 2:01 am

    I second Erica.

    That would render the estimation to be 25 mil hours,
    which is relatively the same with my estimation (30 mil hours) on the estimation page

  • steph Jan 23, 2013, 10:17 am

    assumptions:
    a truck can hold 100 kilos of soil and an average mountain has 10,000 kilos of soil
    a truck will travel the same rate (time and distance) with or without the load from point A to point B. Let us assume that it takes 10hrs for the truck to travel

    we would like to find out how many hrs it would take
    Thus our equation should be
    10,000 / 100 = number of trips
    number of trips x (HRStotravelpoint1-2 (x2))
    i multiplied it with x2 because it goes back and forth thus doubling the time

    This is my answer.. i wonder if this is okay? i did easy numerical assumptions to prevent myself making a mental math blunder….

  • DB Feb 9, 2014, 1:18 pm

    I used personal experience to base my initial assumptions. Being from Colorado, our mountains are basically two miles high. We live at 6000 feet and the mountains are upwards of 14,000 feet. So I rounded off to two miles high which really skewed the rest of my results and I ended up with 70,000 years.

  • Alexey Nikityuk Apr 22, 2014, 3:50 am

    Victor, I am a bit confused why the time for loading and unloading cargo was not included in the estimation.

    In my assumptions I took 10 mins for loading, and 8 mins for unloading (here I include the fact that as mor cargo will be delivered to destination as higher the new mountain will be and as longer the distance will became)

    Also you assume 60m/h speed which is almost 100 km/h. Is this a reasonable assumption for a packed truck? I assumed 30 m\h in my solution

    In the end I got 70.000 years…

    Could you please clarify?

  • Paul Jun 27, 2014, 10:31 pm

    A basic assumption here is that the majority of the time is spent in moving the mountain. However if its a mountain, it has to be broken down, and then rebuilt. I am assuming that is what is meant by “move” or “relocate” the mountain. By my estimates the time required to break down and reassemble the mountain one truck load at a time, is much larger than the time required to moved the same volume by truck. SO I dont agree with this number.

    The solution provided by Victor was my first guess as well, but on further thought that is what I came to the conclusion. But I may be wrong and looking too much into the idea of “moving” the mountain.

  • Anu Jul 31, 2014, 1:45 pm

    How good an idea would it be to include loading, off-loading and refueling time into consideration ,while solving this case?

    • jermaine Jul 6, 2015, 3:01 pm

      I was think that the whole time. So obviously you would need to figure out the approximate tank size and then also the rate of pour for that tank and how many trips needed before gas is needed. But if you do that you also have to figure out the distance thst each tire can go in normal condition before wear and tear. If you ads gas then you HAVE to add every intangible in as well.

  • Hoa Aug 26, 2014, 2:09 pm

    Hi Victor,

    How long should a candidate take to answer an estimation question? I ran through one with a partner this morning and rushed through the analysis because it wasn’t a case. Should I regard estimation questions as similar to a case interview? Is it okay to take 15 minutes for an estimation question or should I aim for 5-10 minutes?

    Thanks!
    Hoa

    • Victor Cheng Aug 26, 2014, 4:52 pm

      Hoa,

      It depends on if the estimation question takes place within a case. If so, 5-10 min is ideal. If the estimation question is given instead of a case and it’s an elaborate one, then 15 min might be okay. That said, 15min feels long to me. I would take another look at your answer and see if you could have simplified your approach. 5-10min is much safer. The key isn’t to rush, the key is to use a simpler approach that isn’t as time consuming.

      Victor

  • Akshay Oct 6, 2014, 6:37 am

    how much time one shud take so as to answer “etimation question” on an avergae?

  • jim Jan 22, 2015, 4:00 pm

    Did I miss the time estimation of using the excavator to fill the truck….making the assumption of emptying/dumping negligible?

  • Polymath Sep 8, 2015, 9:40 am

    The formula for the volume of a cone is something that you learn in 10th grade math, if you are being considered for a job that requires quantitative skills it will count seriously against you that your estimate of “half the volume of the cube it’s contained in” was so far off. A cone is 1/3 of the cylinder it is contained in, and the cylinder is pi/4 of the box it is contained in, so you should have pi/12 instead of 1/2 which means you are off by a factor of pi/6 already. You very nearly doubled your final estimate simply because you could neither remember the formula nor visualize a cone clearly enough to realize that “1/2 the cube” was a gross overestimate.

  • Aditya Dec 19, 2015, 1:17 am

    1 truck
    Length = 10 yards, Breadth = 10 yards, Height = 5 yards
    Volume carried by 1 truck = 500 cubic yards

    Shape of mountain = Conical
    Height of average mountain = 100 yards
    Radius of mountain base = 210 yards
    Volume of soil in mountain = 22,000 cubic yards

    Total truckloads needed to move mountain = 22000/500 = 44

    Moving 1 truckload of mountain 10 miles involves = Loading + Transporting + Unloading + Coming back

    Assuming time taken to load and unload are the same = 2 hrs
    Transporting full truckload 10 miles = 30 mins
    Coming back empty = 20 mins

    Total time per trip = 2 + 0.5 + 2 + 0.33 = 4.83 hrs
    Total trips = 44 x 4.83 = 215 (approximately)
    Total time = 9 days (approximately)

  • Amit Jul 25, 2018, 12:43 pm

    In the above explanation as we have to relocate a mountain so obviously we are in a mountain terrain hence shouldn’t we consider that truck would not operate at its full speed of 60 mph ?

  • Ramsey Chahine Apr 9, 2019, 2:50 pm

    You guys are missing the point, you can get into all the details you want while considering this information. Its just the simple thought process that Victor is trying to explain, not details about rocky roads etcc

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