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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503 comments… add one
  • YASH CHOUDHARY Jul 17, 2020, 1:42 pm

    Let us assume that the mountain rested on the ground is of conical shape with below dimensions:-
    Height: 10 meters
    The base of diameter: 5 meters

    Total time is calculated as T: a+b+c
    a=Total time of loading the rubble on the truck.
    b=Total time taken by truck to move rubble from starting point to the destination for unloading the rubble, which itself is dependent on Volume of Truck and number of trips and by the truck.
    c=Total time taken for unloading the rubble from the truck to the destination spot.

    Also, we are assuming that:
    -The capacity of the truck is 4 metric tonnes.
    -The speed of the loaded truck is 50 meters/ sec.
    -The speed of the empty truck is 80 meters/ sec.

    Now, let us assume that we have 4 manpower at loading and unloading side with the same work efficiency. Each able to load about 0.5 cubics meter rubble/min. in the truck and unload 0.6 cubics meter/min. from the truck.

    {Calculating a}
    The volume of rubble=Volume of Cube=1/3*pi*r*r*h
    pi=3.14r=Dimater of base/2=5/2=2.5
    h=10
    Volume=1/3*3.14*2.5*2.5*10
    = approx.63 cubics meter

    Therefore,a=Total volume of Cube/Total rubble loaded by 4 workers per unit time

    As 4 workers are there, so total rubble loaded by 4 workers per unit time=4*0.5 cubic meters/min.= 2 cubic meters/min.

    So, a=63/2=32 min

    {Calculating c}
    c can be calculated in a similar way as a, but total rubble loaded by 4 workers per unit time will be different.

    so, c=Total volume of rubble/Total rubble unloaded by 4 workers per unit time.

    Now, Total volume of rubble is =63 cubics meter

    Total rubble unloaded by 4 workers per unit time
    =4*0.6cubics meter/min=2.4 cubics meter/min

    c=63/2.4
    =25 min

    {Calculating b}
    b= Total number of trips from starting to destination*Time took by truck from starting to Destination per trip
    +Total number of trips from destination to starting*Time took by truck from Destination to starting per trip

    Calculating each separately:

    *Total number of trips from starting to destination*=
    The total volume of Rubble/capacity of the truck=63/4=15.75

    So, at least 16 trips will be required to go from start to Destination point.

    *Total number of trips from destination to starting*=16-1=15 trips (We are assuming that truck started from the starting point)

    *Time took by truck from starting to Destination per trip*=The distance of haul/ Speed of loaded truck
    =10 miles/50 meters/sec

    Now 1 miles=1600 meters

    So, *Time took by truck from starting to Destination per trip*
    =10*1600/50 sec=320 sec=5.3 minutes

    *Time took by truck from Destination to starting per trip*=
    =The distance of haul/ Speed of empty truck
    =10 miles/80 meters/sec=10*1600/80 sec
    =200 sec
    =3.3 minutes

    So, b=16*5.3 + 15*3.3=84.8 + 49.5 minutes
    =134.3 minutes
    = Approx134 minutes

    So, our answer T=a+b+c
    =32+134+25 minutes
    =191 minutes
    =3 hrs 11 minutes

    • Vaibhav Jan 14, 2021, 12:55 pm

      Speed is too much

  • Est Apr 19, 2019, 5:53 am

    Mountain size : 1000high and 500 radius
    Mountain volume = between 1/3*pi*(r^2)*h (cone) and 1/2*pi*(r^2)*h (1/2 of cylinder)= 100,000,000
    Truck size : 5*5*8
    Truck volume= 200
    Truckloads = 100,000,000/200=50,000
    Time per ride= load (automatically) + drive+ unload (automatically) =20+20+20=60 min= 1hour
    Working hour: 8 hours per day
    Duration = 50,0000 loads x 1 hour/8 hours = 6,250 days=25 years (250 working days per year)

  • Ivan Apr 1, 2019, 11:03 am

    time = time required to move 1 mile * 10
    = weight/weight per hour
    =(assume the mountain is a triangle)
    =1/2*(100 mile * 100 mile) * density / weight per hour
    =(assume density is 20 kg/mile^3, weight per hour = 20 killograms)
    =2*10^4 hours

  • Asd Apr 1, 2019, 4:38 am

    What I did was I made assumptions about the mountain.
    The mountain contains only sand and rocks and its total weight of these elements is 10,000Kg. And our truck has a carrying capacity of 1,000Kg

    My frame work was:
    Loading time x number of rounds
    Unloading time x number of rounds
    Time from point A to point B x number of rounds
    Time from point B to point A x number of rounds

    Number of rounds is: Mountain weight/Carrying capacity=10
    Loading time x 10
    Unloading time x 10
    Time from point A to point B x 10
    Time from point B to point A x 10

    Loading time: I assumed that we have an advanced truck that can load 100 Kg per shovel per minute. 100×10=1000Kg in 10 minutes
    10 Minuts x 10 Rounds
    Unloading time x 10 Rounds
    Time from point A to point B x 10 Rounds
    Time from point B to point A x 10 Rounds

    Unloading time: unloading is much faster than loading so we’ll assume that 2 minutes is the time required.
    10 Minutes x 10 Rounds
    2 Minutes x 10 Rounds
    Time from point A to point B x 10 Rounds
    Time from point B to point A x 10 Rounds

    Time from point A to Point B: let’s assume that the average truck moves at 50 mph but since we are carrying 1000Kg of sand and rocks we’ll take 20% from its speed and say its 40 mph.
    10/40 x 60= 15 minutes
    10 Minutes x 10 Rounds
    2 Minutes x 10 Rounds
    15 Minutes x 10 Rounds
    Time from point B to point A x 10 Rounds

    Time from point B to point A: Here we’re not carrying any loads so we’ll stick with the average truck speed.
    10/50 x 60= 12 Minutes
    10 Minutes x 10 Rounds
    2 Minutes x 10 Rounds
    15 Minutes x 10 Rounds
    12 Minutes x 10 Rounds

    Our formula is:
    (10×10)+(3×10)+(15×10)+(12×10)= 6.5 hours or 6 hours and 30 minutes.

    now I know this is somewhat unrealistic but for simplicity sake I tried to simplify each step.

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