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”
9 years
In hours, the total amount of time it would take is = (transport timex2 loading time) x volume of mountain in cubic feet/carrying capacity of truck in cubic feet
Assume that the average truck that can carry a load has dimensions 10 feet by 5 feet by 10 feet. That means it has a capacity of 500 feet.
I know that mountains range anywhere from a couple hundred metres high to thousands of metres high, so for simplicity I will assume that the average mountain is 1000 metres high or 3000 feet high. The height of the mountain is probably less than its length and width, so let’s say that each is twice the height. That yields cubic volume of 108,000,000,000 cubic feet.
Divided by 500, that is equal to 216,000,000 truck loads.
If the average truck travels at 60miles per hour, then a trip both ways is 20 minutes. If the average loading time is 40 minutes, then that’s a total time of 60mins per truck load or 1 hour per truck load. The total time is then 216,000,000 hours to move the mountain 10 miles.
How big is the average mountain?
Let’s assume that the average mountain is shaped as a cone with height 10,000 feet and radius 20,000. The total volume of this mountain is then 3.14×20,000×20,000×10,000/3 =
= 20,000 x 20,000 x 10,000 = 4 x (10,000)^3
How much can the average truck carry?
If I went wrong, this is likely where for two reasons.
1. I am assuming pick up truck.
2. My only constraint is volume and not load.
Compact trucks are probably 2x4x5 = 40 ft^3 whereas bigger trucks are probably 2x5x6 = 60 ft^3, so let’s go with 50 ft^3.
The amount of trips the truck would have to take if filled to the brim is 4 x (10,000)^3 / 50 = 4x2x1000x(10,000)^2 = 8000 x (10,000)^2.
Now with all this load, we can assume that the truck will go at an easy 20 miles per hour to the relocation, making each of those trips 30 minutes. On the way back, the truck will go double the speed at 40 miles per hour, returning in 15 minutes. Note, this does not account for time in loading and unloading. So with 3/4 hour to move each load of mountain, it will take 8000 x (10,000)^2 x 3/4 hours = 6000 x (10,000)^2 hours.
In days this looks like 6000 x (10,000)^2 / 24 =
= 250 x (10,000)^2 = 250 x 100,000,000 = 25,000,000,000 days
Assumptions:
1 – Full capacity of dump truck is 10 tons of material
2 – Dump truck speed when empty 30 MPH
3 – Dump truck speed when full 10 MPH
4 – From visual memory, 10 loads would probably represent a decent sized mound, so a mountain would be at least 1,000 times that.
5 – So moving the mountain would be 10,000 loads
6 – 30 minutes to load the truck
7 – 10 minutes to unload the truck
Workout:
Total roundtrip for one load would be = loading time transport time unloading time return time
This would be = 30 Min transport time 10 Min return time
1 – 10 miles at 10 MPH = 1 hour trip forward = 60 Min
2 – 10 miles at 30 MPH = 1/3 hour return trip = 20 Min
Total roundtrip = 30 60 10 20 = 120 Min
Total mountain move = 10,000 * Total rountrip
Therefore 10,000 * 120 – 20 = 1,200,000 Min; 20,000 Hours
The volume of the mountain can be thought of in terms of the height and width/diameter of base. Thinking of a very typical mountain, let’s assume that it is canonical or pyramid-like in shame (i.e. it has a circular or square base and the shape tapers as you reach the top).
I climbed a mountain in China that was 5000 ft — I think this was neither remarkably high nor tiny so let’s use this as a proxy for an average mountain. I might assume that the base of a mountain is 1 km in diameter. As I am no geometry expert, I’m going to think of the mountain encased in a cube with the same dimensions. That cube would be 1000m x 1000m x 1500m (1ft being 30cm, which gives 150,000cm, which gives 15,000m). The volume of the cube would therefore be 1,500,000,000 cubic metres. Thinking of how much space a pyramid would take up of that cube, I think it would be around a third. So that gives me a mountain of 1,500,000,000 / 3 = 500,000,000 cubic metres.
The next considerations are the logistics of moving 500k cubic metres 10 miles down the road in an average sized truck and the labour involved.
An average truck (I’m thinking of a lorry that might pass you on the motorway which has a square end and is long) might hold 3 x 3 x 10 = 90 cubic metres.
I’m going to assume that the truck might drive along a motorway, given we’ll be in a rural area. So let’s assume tame speed of 60 miles per hour. This means the truck could do the journey in 10 mins. It would have to do this 2-way, which totals 20 minutes.
Then we need to think about the loading and de-loading of the truck. I’m going to assume that we have a decent supply of workers, say 20. Each might be able to collect, load and unload around 2 cubic metres per hour, so 90 cubic metres in 2.5 hours. Taking into account travel, let’s assume that an entire round trip/process with 90 cubic metres takes 3 hours.
Let’s assume there’s a push to get this project completed and the workers are employed Monday to Friday, 8 to 6 including an hour’s lunch break. So each day 3 trips could be made which is equivalent to 210 cubic metres per day. Let’s round this to 200 cubic metres for ease of calculation. 1000 cubic metres could be moved per week. So moving 500,000,000 cubic metres would take 500,000,000/1000 = 500,000 weeks. Or 10,000 years. Seems like an impossible task….
I have assumed that the average mountain circumference would be 1000 m and its height would be 200 m and from here we can calculate the volume of the mountain i.e. 1/3 * pi* r2*h = ~ 5 million cubic m.
Now let’s calculate the volume carrying capacity of an average size truck ( why I am not bothered about the weight is because the trucks are meant to carry objects such as metals which are more dense than soil and trees):
With a length of 10 m and a width of 2.5 m and a Height of 1 m, the carrier of an average size truck can carry 25 cubic meter of load.
Total No. of Trips required = Volume of the Mountain / Truck carrying capacity, i.e. 5 *1000*1000 / 25 = 200,000 Trips.
Time required for the entire process (T) = Time required for one trip (t) * Total number of trips (N)
t = time required for loading (a) journey period to and fro (b) time required for unloading (c)
a = 15 minutes
b = at a speed of 30 miles/hour (keeping in view all the ups and downs of the road), it would take 40 minutes.
c = 5 minutes
So, t = 15 40 5 = 60 minutes or 1 hour
T = 1 * 200,000 = 200,000 hours or ~8000 days
1. define average size of mountain: 1km x 1 km square base and 100 m high.
so total volume is 100 M cubic m
2. an average truck can take 2m x 5m x 2m = 20 cubic m
3. to relocate entire mountain, 100m / 20 = 5m times
4. each time, loading transportation unloading movement = 1 hours
5. total time needed = 1 * 5m = 5 m hours = ~ 600 years
First of all, how long to move a mountain includes the time to fill the trunk, move to somewhere 10 miles away, and unload it. The first part to be considered is the volume of the average size mountain, how long it takes to fill the trunk. The average size mountain is cone shaped with 100 meters high, and 200 meters radius. Therefore, nearly 4,000,000 m3. The excavator fills 1 m3 every time using 3 mins. It takes about 1,300,000 mins to excavate the mountain. In addition, the average size of trunk is 4m*10m*2m, 80 m3. The trunk will carry 50,000 times in total. To move somewhere 10 miles away, it takes an average size trunk 10 mins on one way, totally 20 mins every time. Also, it takes 2 mins each time to unload the soil each time, 100,000 mins in total. In essence, it will take 1,000,000 mins to move the mountain 1,300,000 mins to fill the trunk, 100,000 mins to unload. The answer is 2,400,000 mins, equal to 40,000 hours, 1500 days, 50 months.
Main proxies to drive us to the results here:
– size of an ‘average’ mountain
– size of an ‘average’ truck
– truck’s trip
Assuming, first of all, that we’re gonna break down the mountain, digging it’s dirt and moving it to the truck, and then
moving it 10 miles away to pile it up again. Mountain first:
We could assume that an average size mountain is 1.000m tall. Most mountains, for example,in sky/summer resorts are around 1.500m tall, and less than 1.000m is a small mountain, or even a hill. Also, we must consider that mount Everest is ~8.500m tall, so 1.000m seems a good guess.
Usually a mountain is flatter near its base and very steep at its summit, so we could assume that its average steepness is 45 degrees. And why do we use it? To calculate the horizontal distance between the base and its peak. Since its a triangle with 45 degrees, we know that in a right triangle with its degrees being 45, its height is equal to its base. So the base would measure 1.000m as well. To calculate the mountain’s volume, we could consider that it has a cone-like shape. The volume of a cone is 1/3*pi*r²*h. Since r = h = 1.000m and considering pi =~3, the volume would be 1.000³ = 10^9 m³.
Now the truck:
It’s loading part has a rectangular shape. It’s height can be considered 1.5m. Its width is practicaly the width of a street lane, say 3m. To estimate its lenght, we could imagine a real situation where we are driving with our car and see right next to us an average size truck. Its lenght would be around 2 times the lenght of a car. Considering that a car has ~4m of lenght, the lenght of the loading part of the truck would be 8m. So its volume would be 1.5 (height) x 3 (width) x 8 (lenght) = 36 m³. Since its height could be a little higher than that, we could round this number to 40 m³.
Cool. So we have 1 billion cubic meters to be moved by a 40 cumbic meters ‘container’. So, the total number of trips that we’d need to make is 1.000.000.000/40 = 25.000.000 trips.
Alrighty. So now let’s see how much time we would need to go there, take 40 m³ from the mountain and move it 10 miles away. Since it’s a mountain, we can consider that the road from and to the mountain is a nice, traffic-free rural road. So while it’s rolling, we can consider that the truck travels at 50 mph. To move 10 miles, it would need 0,2 hours = 12 min. To load the truck, let’s say we could use a bulldozer to do that (without a bulldozer, we’d have to do it by hand. Ugh!). So let’s say it would take somewhere close to 10 min to do it. Let’s say 9 minutes. Moving 10 miles away from the mountain would take the same 12 minutes, and unloading the weight is faster, the truck would just need to lean its loading part. So say 3 minutes to do that.
So the total amount of time to go there, load the truck, move it and unload it would be 12 9 12 3 = 36 minutes.
Since we’d need to make 25.000.000 trips, the total time would be 25.000.000*36, in minutes. Instead of multiplying it already, since we know it’s a huge amount of minutes, we know that it would take months to do it.. maybe years! So let’s divide this number by 60(minutes to hours)*24(hours to days)*30 (days to months). So that would be 25.000.000*36/(60*24*30). 25mi/24 = ~1mi. So 36.000.000/60*30 = 360.000/18 = 20.000 months. So, that’s a lot of years. More precisely, that’s 20.000/12 = 5.000/3 = ~1670 years.
Answer: 1.670 years.
– First of all, I would like estimate the size of an average mountain.
A mountain typically has the shape of a pyramid, with almost square-shape base, and converge to a point on the top. Let’s say the base of a mountain is of a size of 5 km x 10 km, so the base has a size of 50 kmˆ2. About the height, we have an average mountain around 3000-5000 m, with the upper limit about 8000 m. So let’s assume average mountain height is 4500 m, thus the volume will be 1/3 x 50 x 4.5 = 75 kmˆ3 = 75 x 10ˆ9 mˆ3.
– Estimating size of container.
Let’s now assume average truck size has one container, with, length between 5-10 m, so I’ll pick 7.5 m. Then the wide and height, say 2 m each, so the size of a container is 7.5 x 2 x 2 = 30 mˆ3.
– # of delivery needed.
divide the volume of mountain by the container, as we have one average-size truck, and assuming only one container for the truck.
= (75 x 10ˆ9) / 30 = 25 x 10ˆ8 times.
– time needed per delivery.
I’m familiar using km, so let’s transpose 1 mile to 1.6 km so 10 miles is equivalent to 16 km for one delivery. Assuming the truck will travel at 50km/h, then it takes about 20 minutes per delivery, 40 minutes including the return travel. As we have no limitation on number of employees we can hire to move the mountain, I would assume reasonably the employees are able to load the truck in 10 minutes, another 10 to unload. So, 1 hour for 1 delivery.
-total time required.
So total time required is 1 hour x 2,500,000,000 delivery = 2,500,000,000 hours
~obviously much faster if we get more trucks.