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”
Volume of an average mountain
o Base of mountain 2X as long as height
o Assume an average mountain is 3,000ft tall
o Therefore diameter of base is ~6,000ft, radius is 3,000ft
o Mountain is roughly the shape of a cone
o Cone is 1/3 * pi * r^2 = r^2 = 9,000,000ft^3
Volume of an average truck
– We assume a big-rig, moving truck
– Truck bed is rectangular
– ~30ft long x 5ft wide x 6ft tall = 30*30 = 900ft^3
Average speed of truck, time needed to travel 10miles
– Assume we can travel on a freeway, 60mph
– 10miles = 10min
How many trips do we need to take?
– 9,000,000ft^3 / 900ft^3 = 10,000 trips
Time to take that many trips?
– 10,000*10min = 100,000min
Does that make “sense”?
– 100,000min / (60*24) = 100,000 / 1,500 = 70 days
Are there large rocks, or other solid items that could not be transported on an average truck? No.
Do we also need to move the trees on the mountain? No.
Do we also need to move wildlife? No.
Good question. I’ll use metric system to make it easier for me. To solve this problem the variables I’ll need are:
– Average volume of a mountain – V
– Average velocity of a truck – Vm
– Time to load and unload the truck – Tcd
– Maximal capacity of a average truck – Cmc
Assuming that, the time required can be formulated as:
T = (V/Cmc)*(Tcd 32/Vm)
where 32km is the distance to get in the new place and to return to the original position of the mountain. (1 mile ~ 1.6 km)
I don’t have much knowledge about mountains, but assuming that the Everest is about 9km high, an average size mountain must be 1km high. Let us assume that this mountain can be modeled as a cone, we need to know the basis radius, which is 1km as well. Making π ~ 3, the volume is estimated to be around V = (1/3)*(π*(10^3)^2)*10^3 = 10^9 cubic meters.
My neighbor had a truck when I was a child, I must say that I’m pretty familiar with those dimensions. Let us assume that his truck was an average truck and had the dimensions 4m x 3m x 20m, that leads to the maximal capacity of Cmc = 240 cubic meters.
Assuming the truck is loaded by machines and does not need any help to unload, the time to load and unload are 10min and 5min respectively, Tcd = 15min = 0.25h
For simplicity, the average velocity of the truck is Vm = 32km/h.
Since we have estimated all the information required by our model, the time necessary to shift the mountain is:
T = (10^9/240)*(0.25 32/32) = 5,000,000 h ~ 590 years
First, I suppose that when truck moves the rock, it probably drives at 40Miles/hr speed, so it takes 1/4hours to move 10 miles per drive.
And each truck can load rocks at the size which is 2 meter x 1 meter x 1 meter= 2m^3.
Besides, the hight of normal mountain should be probably 1000m.
I suppose the diameter of the bottom of the mountain should be 5km.
so the area of the bottom of the mountain should be like 2.5*2.5*3.14=19(km^3)
Therefore, according to my assumption, the size of the mountain is:
19(km^3)*1(km)*1/3= 6.33(km^3)=6.33*10^9(m^3)
If we divide 6.33*10^9(m^3) with 2(m^3),
the times of drive that truck need to move the rock should be =3.165*10^9
The needed time = 3.165*10^9*0.25(hr)*2(return)
=1.5825*10^9 hrs.
167 years
Assuming there is only one truck to relocate the mountain. We need to estimate the following parameters.
1. Avg. mountain size
2. Avg. Truck size
3. No. of trips = Avg. mountain size/Avg. Truck size
4. Travel time for 10 miles (to and fro) by truck
5. Time to load a truck
6. Time to unload a truck
7. Time per trip = Time for load Time for unload Travel time
8. Working hours in a day
9. No. of trips per day = Working hours per day/Time per trip
10. No. of days = No. of trips needed/no. of trips per day.
So lets assume the size of mountain (pyramid) 10m*10m*100m. Which gives the volume as (1/3*10*10*100)=10000/3=3333.3m3
Lets say the size of truck as 3m*2m*1m, gives volume as 6m3. Lets say the truck while carrying mountain goes with 20mile/hr and while coming it comes with 40mile/hr and it takes 15 minutes to load and unload (consider there are ample number of workers). If we calculate time to transfer 6m3 it gives 1hr (10/20 10/40 15minutes). So it takes (10000/18) 550 hours approximately
I’m going to start off with making a couple of assumptions up front; I’m assuming the height of an average sized mountain is 3000 feet. And that its base diameter is the same as its height, for stability. Further, the average truck has carrier dimensions of 10x5x5 feet, making its volume equal to 250 cubic feet. The volume of the mountain can be calculated using the formula for cones; 1/3*pi*(r^2)*h. Where r is the base radius and h is the height. On calculation, you get approximately 7250 million cubic feet.
I’m assuming the truck’s average speed is 20 miles per hour (Trucks travel significantly slower than lighter vehicles). That amounts to a total time taken of 1 hour, for one round trip. Finally, the number of trips it has to make is 2900000 (725000000/250) which means the truck will take about 2900000 hours to relocate the entire mountain (121000 days approx.)
By the way, to pick the height of the mountain, I just sort of very roughly averaged out the height of the tallest and the shortest mountains. I hope that’s alright.
Due to stability reasons I assume that an average mountain’s base is twice as large as its hight. Also I assume the average mountain to bei 2000 m high. Also the average mountain can be interpreted as a cone.
The Volume of an average mountain is therefore:
4000x4000x2000x1/3 m³ = roughly 10 Billion m³
The biggest trucks in Germany can carry 40 t of weight. Assuming an average mountain density of 1 g/cm³, this truck can carry 40 m³ of dirt. The Truck therefore needs 250 Mio round trips to carry the whole mountain.
Assuming optimized loading and unloading and neglecting the additional time the truck would need to drive upwards the new mountain, the time for a round trip will be:
Loading: 10 min
Driving with load: 10 miles / 40 miles / h = 15 min
Unloading: 3 min
Driving without load : 10 miles / 50 miles / h = 12 min
makes a total of 40 minutes.
Therefore the total time will be 250 Mio trips * 40 min / trip = 10 Billion min = 1/6 Billion h = 1/(6*25h) Billion day = 1 / (150 x 350 days/year) Billion years = roughly 19000 years.
Seems an aweful lot of work 🙂
The 10 mile distance information is irrelevant in this example.
Usually 60 story building with a rigorous delivery schedule take around 30 months to completion. Constructing a tower is broken down into the following: transferring the material to the construction site-building in the site . Assuming in a month,it takes one week to transfer the material needed to build for the whole month.Total weeks to transferring rock to site is 30 weeks (30*4*(1/4).Howveer, these buildings usually have at least 3 trucks. So if they had one average sized truck, it will take them 90 weeks.
Assuming an average sized mountain has available rock to build 5 60 story building. Therefore the total needed time is 90 weeks * 5 = 450 weeks. 450 weeks/4,therefore 9.3 years would be needed.
My starting estimates to solve the case were:
1) Assumption that the mountain is conical, since it is of average size, I assumed that it would be roughly 3m high and have a base radius of about 4m. Giving a rough volume of 50 m3.
2)I estimated that we would have roughly 10 people to collect the mountain in the first location and deposit in the next location. Lets also assume that each person would take roughly 2 minutes to use a shovel and put some mud into the truck and vice versa.
3) Given that we have spoken about the shovel, let’s assume it’s a cuboid with a dimensions of .1m length and a height and breadth of .05 meter, giving a total volume of .1 x .05 x .05 m3= 25 x 10-5 m3.
4) I assumed the truck to be another cuboid of roughly 2m length, 2m depth and 2m height giving it’s voulme to be 8m3.
5) Therefore, given that the truck was roughly 8 m3 in volume and the shovel is of 25 x 10-5 m3 volume, we would require the mud to be shoveled (8/(25x 10-5)) = 32000 times. Given that each shoveling act takes roughly 2 minutes per person, we will require a total of 64000 minutes to fill or un-fill the truck once. Given that we have 10 people to fill and un-fill the truck, it will take 64000/10 minutes since all the workers are assumed to be working simultaneously. Therefore time required to fill or unfill the truck will be roughly 6400 minutes.
6) Given the volume of the mountain to be 50 m3 and the volume of the truck to be 8 m3, let us assume it will take (50/8) which is roughly 6, journeys (to and fro) to relocate the mountain. So it will take 6400 minutes to fill the truck once and another 6400 minutes to un-fill the truck at the desired location. So a total of 12800 minutes.
7) Let us assume the truck to have a speed of 50 miles per hour, therefore to cover 10 miles once, it will take (10/50)x60 minutes for a one way trip. Therefore it will take roughly 12 minutes for a one way trip. Therefore, for a 2 way trip it will take 24 minutes. Given that we have to make 6 such trips, we will take roughly 6×25 minutes= 150 minutes. Given there may be some traffic let’s assume 200 minutes.
8) Therefore,a total time would be 200 12800= 13000 minutes= 217 hours.