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.
504 thoughts on “Estimation Question”
I approximate the volume of a mountain to that of a piramid. Which is 1/3 base multiplies the height.
I guesstimate that the average mountain is about 1000 metres high and has a base of 1000 metres x 1000 metres.
Total volume of the piramid is: 300 millios of cubir metres
Average track: I guesstimate 10 metres long, 5 metres wide and 3 metres hight. Volume: 150 cubic metres.
300million m*3 / 150 m*3= 2 million trucks needed or 2 million trips of the same truck.
I guesstimate that average speed of a truck transporting heavy stuff is 80km/ h. 10 miles are 16 km, that means the track takes 20 min to go and 20 min to come back, total 40 min.
40 minutes x 2 million= 80 millions of minutes
60 min x 24 hours= 1440 min. I round it to 1500.
1500 min x 7 days a week= 10500 min a week
In one month0 42000 min, I round it to 40000.
80Millions of minutes / 40000 = 2000 months= 166 years
Assuming average mountain height is 2000 ft and for ease of calculation is one shaped like a rectangle of dimensions 2000 ft x 2000 ft x 1000 ft, the volume is 2000 ft*2000 ft* 1000 ft = 4e9 ft^3.
Assuming the truck is an average size passenger truck such as a Tundra, etc. equivalent, the cargo area is only filled to the top of the tailgate, and the cargo dimensions are 10 ft x 5 ft x 1 ft, the volume of 1 load is 10 ft x 5 ft x 1 ft = 50 ft^3.
Therefore, the number of loads would be 4e9 ft^3/50 ft^3 =
8e7.
Assume the truck can travel 60 mph each way (both to and from the mountain) and load/unloading time is negligible. 1 trip to take a load and return to the mountain would be 20 miles. For the last load, the truck would only travel 10 miles. Nonetheless, transporting each load would take 1/3 of an hour.
Therefore, 8e7 loads* 1/3 hr per load = 26,666,666.6 hrs With the last load only traveling one way, subtract 10 minutes, or 1/6 of an hour, giving 26,666,666.5 hrs.
it depends on speed of truck
Assuming the average mtn volume was equivalent to 100k average trucks and each unloading roundtrip took 1 hr
it would take 100k round trips or 100k hours (assume 25hr/day, 4000 days or 12 years)
First we need to determine volume of the average mountain.
Lets assume that average mountain is 3 km tall and has a 20km diameter. and lets assume it is a pyramid rather than a conus for easier calculations.
So the volume will be 3*10*10/2*4=600 cubicle kilometeres
Lets assume that we are efficent and are extracting truck-size blocks from the mountain at the time when our track makes its journey so it does not consume additional time.
So now lets calculate how much time we need to transfer the mountain.
Let average truck volume be 10 cubicle meters.
So we need 600 bn/10 = 60 bn trucks to move the mountain.
let truck speed be 60 miles per hour and 5 minutes to load and unload and we get 0.5 hour for each trip.
Now 60 bn*0.5-3 bn hours. lets round it to 2.4 bn and divide by 24 to have days. we got 100 mm days now we divide it by 365 and we get around 250 000 years
60 Years.
250 yrs
30days
In order to solve for how long it would take to move an average size mountain 10 miles using an average size truck, I would have to define an average sized mountain, based on my skiing experiences I can classify large mountains as 4,000 ft vertical based on Kicking Horse Resort and small “mountains” as 500 ft based on Blue Mountain Resort. I am assuming that there are equal amount large as small mountains based on the random fact that if the earth was shrunken down to the size of a cue ball it would actually be smoother than a cue ball – establishing relativistically the distrubution of large and small mountains to average out, resulting in the height of an average mountain as 2,250 ft. Assuming the mountain is conical, and that the city of Macchu Piccu was built near the top of a mountain and assuming that one could walk across Macchu Piccu in 4 hours at 3mph the diametre at the top of the mountain would be 12 miles. Therefore the diametre at the bottom, assuming a slope of 1:1 as the moutain wouldn’t be too tough to climb, nor too easy, would result in a diametre of 34,230 ft, a radius of 17,000 ft. The volume of the moutain is 1/3 * r(squared) * pie * height, assuming pie and 1/3 cancel each other out r2 equals 289M sqft multiplied by the height yields 650B cuft of mountain to move. To determine an average truck, I know that average trucks in Canada are limited to a length of 53′ height 12′ , width 8′. Assuming the cab length takes 4ft for engine, 4ft for bench, 4ft for sleeping/storage, and 1 ft for gap, that leaves ~40ft for storage, and assuming of the 12ft height, the wheels go up 2ft that leaves 10′ so the volume of an average truckload would be 40x10x8 = 3,200 cuft. Therefore moving the mountain would take 200M loads. To move a mountain 10 miles a truck has to drive 10 miles there and back, assuming on a highway at 65mph, therefore a truck can travel 20 miles in 0.3 hrs x 200M loads means it would take 60M hrs or 60,000 years.
Volume of mountain (60 m radius, 80 m height) = 1/3*22/7*60*60*80=300,000 cubic meter
Truck volume = 2*4*6=48 cubic meter
On an average it takes the truck to travel up and down 10 miles is 1 hour
Loading and unloading 2 hours with a crane and JCB
Therefore in a working day of 9 hours 3 trips are possible
Therefore 300000/(48*3)=2100 days = 5.7 years