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”
17 years.
We need to estimate three things: volume of all the rocks, debris, etc that make up the mountain, the volume of material that an average truck can contain, and the time it takes to load, drive 10 miles from the mountain to the dumping site and back, and to unload.
1. Volume of the mountain: an average mountain has a base and a peak, roughly the same geometry as a cone. let’s assume the average height is 1,000m (i’m from Vietnam and the highest mountain here is over 3,000m. so take into account high and short mountains, even hills, 1,000m seems reasonable) and the radius 100m. so volume is 100^2*1000*Pi (3.14, let’s make it 3) =30 million cube meters
2. Volume of an average truck: i’m not talking US-sized trucks which are bigger than the ones I usually see at construction sites that are dotting the map of Hanoi. the ones I see are about 20-25 meters long, 3 meters tall and 8-10 meters wide. For length, i subtract the 5 meters at the front used for cabins where drivers sit, so the length of the containing area is 20 meters. 3 meters tall, take out 1 meter (for tires). that leaves 2 meters. Volume of an average truck, then, is 20*2*10=400 cubic meters
3. So we will need to use this same truck: 30 million/400 = 75000 times over to move this mountain. Now let’s estimate the time. The truck has to load, drives 10 miles with rocks on it to the dumping site, unload, then drives 10 miles back empty before loading again. I call that 1 cycle. How much time is in a cycle?
4. Assume there’s only 1 crane and it is continuously scooping rocks into the truck. From scooping to dumping and back, it’s about 5 minutes. How many scoops? Well, the part of the crane that does the scooping (excuse my lack of vocab) is similar to an upside down cone, with the radius at the base and its height roughly 2meters (i’m 173 cms tall and from the look of it, I can easily fit in that scoop). Therefore, each scoop is about 2^2*2* pi = 24 cubic meters. A truck’s volume is 400 (calculated above) => 400/24 = 15 scoops * 5 minutes each = 75 minutes to load that truck.
5. The distance is 10 miles. I assume the driver travels at 60mph (quite fast for a truck load of rocks, but time is money), so 10 miles to and from will take 20 minutes. Unloading takes another 10. So from loading to unloading, it’s 75 20 10=105 minutes. Take into account of minor malfunction or disturbance on the road, let’s call it 2 hours. We have to use the truck 75,000 times over -> 150,000 hours/24/365 = 17 years.
Estimate how long it would take to move or relocate an average size mountain 10 miles using an average size truck.
1. The avg dump truck bed is 8′ x 3.5′ x 20′ which equals a volume of 560 cubic ft. It is never filled exactly to the rim, so let’s use 500 CF for estimating.
2. The mountain is 5000′ high and 5000′ across at the base and roughly cone shaped for a volume of 32724923475 CF
3. The truck travels good roads at highway speeds and loading/unloading is done in the same time each load- Estimated time per round trip: 5 min load/get on the road/ 10 min travel time at 60mph/ 5 min unload/start back/ 10 min travel time back to reload= 30 min trip time.
4. Taking these numbers: 32724923475CF/500CF/trip= 65449847 trips to move the volume of the mountain
5. 65449847 trips x 30min(.5 hr)/trip = 32724923.5 hours
6. 24 hours/day, so the number of days working 24 hrs/day would be: 1363538.5 approx. and with 365 days in a year, it would take approx. 3735.7 years.
7. IF we estimate this based on a workday of 10 hrs/day x 5 days/week, the number of years goes even higher: 12587 years rounded up- a job for your family forever!
“Estimate how long it would take to move or relocate an average size mountain 10 miles using an average size truck”
Assuming that the truck being used to move the mountain can carry about 50 cubic feet of material in a given trip and moving at around 50 miles per hour makes each roundtrip approximately 40 minutes an additional 20 minutes for loading/unloading and the and the mountain itself is around 200000 cubic feet large then it would take approximately 40000 trips for the truck to fully move the contents of the mountain. 40000*1 hour for each roundtrip equals 40000 hours total to move the mountain. 40000/24 = 1666 days to move the mountain
340 years
60 days of continuous work
39 days of continuous work
Suppose an average sized mountain weighs 2000 ton, and an average sized truck can carry 5 ton. Also suppose it takes one day to move a part of the mountain onto the truck (a process involving shoveling), but once the part is on the truck it should take a negligible amount of time to transport 10 miles. So 2000/5*1=400 days
8 years
We need to calculate the amount of time required to relocate a mountain (of avg size) 10 miles away using an avg truck.
Let’s assume that our goal is to calculate and express the time in number of years.
Having said that,
T: time.
T: (numers of required trips) x (speed of each trip).
Speed of each trip: how long does it takes to an avg truck to cover, once it is full charged, a distance of 10 miles?
Let’s assume the avg speed to be: 60 miles per hour.
It means that to cover 10 miles, our avg truck will take app. 10 minutes.
Now, focus on the number of required trips (nrt): this depends upon 2 variables,. indeed:
NTP: (AVG Mountain Size)/(AVG truck charge capacity).
So, we need to calculate the avg mountain size.
Let’s check what we know:
First, let’s assume that to be classified as a mountain, one peak should not be lower than 1.000 meters.
Meanwhile, the highest peak in the world is app. 8.000 meters.
We can asssume that on average, a mountain tend to be around 2.000 meters.
We need to calculate the lenght and how large it is, on average.
Let’s assume as a proxy the avg population of each mountain village in europe…we finally arrive to consider:
base: 5 km, h: 2km, l: 1km, avg volume: 10 km^3.
Now, focus on a truck: in europe there are different standard.
We can assume an average as one of those to drive which you do not need a special driver license.
According to existing rules, we know that this size, in volume is:
lenght: 4 mts. high: 2 mts. large: 1 mts.
So we have a 8 cubic meters of volume (chargeable).
Now, we need to calculate the ratio between the avg mountain size and the average volume of charge:
number of trips required: 1.250.000 (avg mountain volume/avg chargeable volume per avg truck).
So:
Duration: 1.250.000 trip x 10 mins/trip: 12.500.000 mins.
1h: 60 mins. So: 208.333 hours.
1 day: 24 hours, so: 8.680 days.
Number of working days per year: 220.
Number of years required to relocate the avg mountain with an avg truck: 39.45.
Are u sure u want to be stuck in this bizi?
Thanks for your precious help!
Luca
Assumptions:
1. Avg Mountain size = 1000 tons
2. Avg Truck size = 20 tons
3. Sp approximately 1000/20 visits are needed to move the complete mountain i.e. 50 visits.
Now we calculate time for each visit.
Load time to move the gravel to truck (Vol = 20 tons).
Assume 0.5 ton is lifted and put in truck in 1 minutes. So for 20 tons you will need 40 minutes
For simplicity assume avg truck speed on the road from start to end is 40 miles / hour. To cover 10 miles one way the truck will need 15 minutes. Assuming unloading requires 1/4th time of loading, the unloading time = 10 minutes. (0.25 x 40 minutes). Assume returning speed is same and hence duration is same. So for one visit the time required for the truck is 40 15 10 15 = 80 minutes.
Since we said that 50 visits will be needed, the total time needed to move the mountain is 80 x (time for 1 visit) = 80 x 50 = 4000 minutes.
If we assume a working day is 8 hours (and lunch is 1 hour) there are 7 productive hours in a day.Hence, in a day truck can make 4 complete visits. To completely move the mountain the truck will need approximately 50/4 i.e. 12.5 days which is 12 days and 6 hours.
Further to be more accurate, the mathematical time needed will be 4000 /420 = 9.5 days. So it should take anywhere between 9.5 and 12.5 days if we assume that the workers are 100% productive during those 7 hours.