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”
an average size mountain: H=1,5 km, R=2,6 km (tilt angle approx 30).
Volume of the Mountain (cone) = Pi*R*R*H/3 = 3,14*1,5*1,5*2,6/3 = 10,6 km3 = 1 191 985 437 300 m3
Truck volume = 40 m3, working hours 24h * 7 days a week
Truck speed (incl. load/unload) = 20 miles/hour
One way timing = distance / speed = 10/20 = 0,5 hour
2-way timing = 0,5 *2 = 1 hour
Relocation speed = (Mountain volume / Truck volume) * 2-way timing = 1 191 985 437 300 /40 = 29 799 635 932,50 hours = 3 401 784,92 years
Well, basically we can use 2-3 consepts to solve this problem:
1) The volume of a perfect cone (to simplify calculations)
2) The consept of density (a measure of weight per unit of volume)
3) And a concept of arithmetic progression (optional) (we assume that every subsequent trip of a truck is faster due to the less of a mountain remaining. Hense the lower the mountatain the faster the truck is filled up)
Firstly we need to identify the weight of the mountatiain. To simplify lets say that mountain is a perfect cone with 1 km of height (blind guess) and 6 sq km base. So the volume of the montain would be:
V = 1/3 * base * height = 1/3 * 6 * 1 = 2 km^3
1 km^3 = 1 blm m^3
Let’s assume that the density of rock the mountatain consists of is 200 kg per m^3.
Mass = volume * density = 200* 10^9 = 2 * 10^2 * 10^9 = 2 * 10^11 kg
Now we know the mass of the mountain. We need to assume the mass that the truck can transport. Let’s say that an average truck can transport about 20 tons of rock.
2*10^11 kg = 2*10^8 tons
Number of trips = 2*10^8 tons / 20 tons = 10^7 trips (10 mln trips)
Assuming that the truck travels 25 km (20 miles) per trip (there and back) with average speed of 50 km per hour it would take 0,5 hours per trip.
Now we can estimate the time as
Time = trips*average time = 0,5 h * 10^7 = 5 mln hours
Assuming the truck works ten ours a day we can say:
5 mln hours / 10 hours a day = 500 thousand days
Whih roughly 1370 years.
Mind that I took a bigger mountain and a smaller truck 🙂
Moreover you can modify this logic by intriodusing progression on time as mountain becomes smaller and smaller assume that there are two drivers that work shifts on a single truck. Mind that we assume that truck does not depriciate.
Truck capacity estimated at 50 sq. ft. (4x6x2=48, assuming some mounding of load for approx. 50 sq. ft. per load.)
Time to load, drive, and unload truck is 35 min. (10 min. each to load and unload, 15 min to drive.)
Mountain is 5000 feet tall with a radius of 2500 feet. I calculated a volume of 32,708,333,333 sq. ft.
654,166,667 total truck loads needed to move the mountain. Estimated 27 loads per day allowing for 8 hours sleep.
It will take approximately 66,250 years to move the entire mountain.
Three Tasks:
1. Picking up mountain
2. Transportation
3. Dropping off mountain
Questions?
1. What is size of an average mountain?
2. Assuming unlimited labor capacity
3. Are there any road blocks?
4. Is the mountain being broken down into soil and rocks?
Calculations
1. Picking up: Volume of a cone with radius 100m, height 100m= 1050000 m^3. Assuming labor capacity of 1000 men, who can move 1 m^3 every half hour(30 min), this process would take 1050*0.5 hours=525 hours. Similarly for unloading, so total time is 525*2=1050 Hours
2. Transportation, we can use time=distance/speed.
distance= 10 miles
speed of an average truck=50miles/hour but since heavy weight is involved we can assume the speed reduces by 1/5th for simplicity. Speed=10 miles/hour
Hence time= 1 hour
Total time=1051 hours.
Mountain:
Conical
Height= 2000 metre
Area at the base= 600 sqm
Volume=1/3 x 600 x 2000 = 400000 cum
Truck:
Size = 4.0 x 3.0 x 2.0 = 24.00 cum
No. of trips = 400000/24 = 16000 (approx)
Assuming all other machinery available
Avg. loading time on truck = 0.5 hours
Avg. unloading time on truck = 0.25 hours
Time to travel 10miles (loaded) = 10/25 = 0.4 hours
time to travel 10miles (uloaded) = 10/40 = 0.25 hours
Total time for 1 trip = 0.5 .25 .4 .25 = 1.4 hours
No. of trips in a day = 6 and Total no. of hours required = 16000 x 1.4 = 22500 hours (approx)
Total working hours in a day = 6 x 1.4 = 8.4 hours
No. of days required = 22500/8.4 = 2500 days approx. (approx 7 years)
Assuming the truck requires service after every 1000 miles,
total distance travelled in 16000 trips = 160000 miles
No. of services = 160
therefore the days where no work is done = 160 days
Total days = 2500 160 = 2660 days ( approx 7.25 years)
Assuming other break down days and other contingencies (weather, absence of driver etc.) additional days = 10%
Final ans = 1.10×2660 = 2900 days or 8 years (approx.)
I’ll start this by saying that this is the very first time I have ever practiced a case question, so please don’t judge too hard!
Let’s assume that the average mountain is 5000 feet tall, is conical, and has a base of diameter 1 mile (5,280 feet). The volume of a cone = (1/3)(pi)(r^2)(h). So (2600^2)(5000) = 33,800,000,000. For simplicity, since the number is so large, let’s just have the 1/3 and pi cancel out. So the average mountain has 33,800,000,000 cubic feet. Let us now assume that in a cubic foot of mountain, there is probably 50 lbs. of material. Let’s say the average truck bed can handle 1500 lbs. That means the average truck can handle carrying 30 cubic feet of mountain in a single trip. So, dividing 33,800,000,000 by 30, we now know that we would have to take 1,126,666,666 trips to move the mountain.
Now let us discuss trip time. Loading the material = 30 min.
Drive time with 1500 lbs. in bed = 17 min.
Unload material = 30 min.
Returning with no material = 13 min.
Total trip time = 1.5 hours
So, multiplying our number of trips by 1.5 hours, we know that the total time for the truck to relocate the mountain (assuming the truck is constantly in use) is 1,689,999,999 hours, or 70,416,666 days and 16 hours. But, we know that this is impossible, because trucks need gas. Let us assume we have to stop and refuel every 200 miles, or every 10 trips. The stop should take an average of 5 minutes (we can also have drivers switch shifts during this break as well). So, 112,666,666.6 times 5 = 563,333,333 minutes (I got this by dividing number of trips by ten, then multiplying that number by 5, to get the total amount of time the truck would spend at the gas station throughout the process). This number represents 391,203 days and 17 hours. So, adding that to our “truck-in-constant-use” total, we get 70,807,870 days and 9 hours. Or, approximately 193,944 years, 60 days, 9 hours.
Would love any feedback, thanks!
7,845,661,800 años
Volume of Mountain = 1/3 * height * Area of base =
= 1/3 * 30* 10000 = 100,000 cubic m
Volume of Tanker = l *b *h = 5 * 5 * 4 = 100 cubic m
Time = Loading Journey Unloading Journey
Journey = 30 minutes (assume speed 20 miles /hr)
Loading = 30 minutes (assuming using good cranes and safety)
Unloading = 30 minutes
Journey 1 round = 30 30 30 30 = 2 Hours
No of rounds required = volume of mountain / volume of tanker capacity = 100,000 /100 = 1000 round
average time per round = 2 hour
no of hours = 1000 * 2 = 2000 hours = 100 days (assume 20 hours per day operations)
Well, firstly, I think the problem should be deconstructed into the following sub-problems:
1. what is the volume of an average size mountain?
2. what is the capacity of an average size truck?
3. what is the speed of an average size truck?
4.what is the time of loading or unloading for one time?
We must find the data of the question above and use this data to caculate the time.
We use m to represent the volume of an average size mountain, n to represent the capacity of an average size truck, v to represent the speed of the truck, t to represent the time of loading or unloading for one time.
we can conclude, T=(20/v 2t)*(m/n).
There are two approaches to do this.
1. Bottom – up: Start at the bottom of the mountain and make your way up.
2. Top – down: Construct a road to top of the mountain. Then start digging at top and make your way down.
With approach 1, as you start digging at the bottom, there is a risk that the top of the mountain may start to collapse. For that reason, I will do my estimation with approach 2.
I am assuming the average size of the mountain is 1 M lb. It will take ~ a week to construct a basic road to access the top of the mountain. Average material we can load into a truck is ~ 10,000 lb. So assuming our crew takes 8 hr to dig 10k lb mountain and 30 min to drive to the location where it being relocated. Say 8 hr to unload the truck. So one trip will take ~ 16.5 hr. It will take 100 trips to relocate the mountain with assumptions we made, which is 1650 hrs i.e. ~206 days (8hr a day) and another week (5 days) to construct the road. So in total, 211 days.