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”
Average mountain: 1km*1km*1km = 1km^3 volume
Average truck: 5m*2m*2m = 20 m^3
Return trips: 1km^3/20m^3 = 5*10^7
Truck speed when empty: 100 mph
Trip time when empty: 10 miles/100mph = 6 mins
Truck speed when full: 50mph
Trip time when full: 12 mins
Total trip time: 18 mins
Load truck time: 22 mins
Unload time: 20 mins
Total time for one trip: 1h
Total time: 50 million h ~ 5000 years
To tackle this question we need to look at:
1. volume of an average mountain
2. volume of an average truck
3. speed of that truck
4. time taken to load a truck
We know that Mt. Everest is about 8000m in height. Let us assume, on average, a mountain is 5000m tall. Assuming a pyramidal structure, let’s assume a base diameter of 4000m. This means that the volume of an average mountain is 1/3*pi*(2000)^2*5000 = 20 billion cubic meters (higher end)
Let’s assume that the average dump truck has dimensions of 10m*5m*5m giving us a volume of 250 cubic meteres
An average truck would require to make 20,000,000,000/250 = 80 million trips to the mountain and back.
Total distance travelled= 80,000,000*32,000m (or 4 miles, 2 miles*2)=2560 billion m = 2560 million km
Let’s assume the truck moves at a pace of 80km/hour
Therefore time taken to make all the trips= 32 million hours
Additionally, let’s assume it takes 30 mins to load a truck = 40 million hours spent loading the truck
Total hours= 72 million hours = 3 million days = 833 years (approx)
13100 days
2600years
4,000 years.
My reasoning:
Total time = (Size of Montain / Size Truck) * 10 miles / speed Truck.
I have no idea what could be the average size of a montain. I know Mont Blanc is about 5,000m high. According to that, I assume that the height of an average montain is 1/5 of the height of Mont Blanc, i.e. 1 km. The assumptions being anyway very rough, I assume a cubic shape for the montain => 1 km^3.
For the truck, I assume a size of 2 m x 2 m x 5 m = 20 m^3.
For the average speed of a truck, I assume 60 miles per hour. Since there is also the time of loading and unloading, I divide this number by 3 and I end up with a speed of 20 miles per hour.
Replacing these numbers in the above formula, I get 25 million hours, which gives about 1 million days. Assuming 250 working days per year, my final answer is 4,000 years
Mountains are cone shaped, so the volume of a cone is pi*r^2*h. they are usually wider than they are high. h=300 ft r=500 ft so the volume of the mountain is 3*2500*300. 3.14 round down to three (point finger down). 3*300 is 900, bump it to 1000 for good simplicity (hand flat). 1000*2500= 2,500,000 ft cubed. the size of the mountain.
A truck bed is about 6x4x2 or 48 ft cubed, round up to 50 ft cubed. how many trips would it take to do 5,000,000 ft cubed, and divide that in half. 5 mil divided by 50 is 100k, so it would take 50k trips. how long does each trip take?
Assuming the mountain is a easily shovel-able pile of dirt (no demolition), it would take 10 minutes to completely load/unload the truck, 20 minutes total. driving 60 miles per hour for 20 miles (there and back), it would take 20 minutes. 20 20 = 40.
50,000 trips at 40 minutes each. 2 million minutes. then convert this into days. 60×24 is 240 1200=1440 minutes in a day. round up 1500 minutes in a day (point down bc fewer days included). 2 mil divided by 1500. It goes into 1000 times to get 1.5 million. Now how many times does it go into half a mil? 100×1500 is 150,000, three times (300) gets 450,000 (low again). So it takes 1,300 days. 365 days in a year. 1,300 divided by 365. Round up to 380. 380 380 = 760 380 = 1140 380 = 1520. Above 300 (up again, so neutral).
About 4 years.
divide 4 years by 3 because I messed up in my very first formula, it is one third the side of the cylinder I calculated.. so 1 year and 4 months if it moves it continuously.
Assume that average size of a mountain is 1km*1km*500m/3 = 0.17km3
= 170,000,000m3, and the capacity of a truck is 2m*2m*3m = 12m3,
the average velocity of the truck is 50km/hour, and it takes 6 minutes (0.1 hour) to load or unload stuff. So for a round trip, it takes (16/50＋0.1)*2 = 0.85hours, and there are totally 170,000,000/12 = 14,000,000 rounds, so , 14,000,000* 0.85 hours = 12,000,000 hours, which is 500,000 days, or about 140 years. But the truck does not work without rests. So assume it works 8 hours per day. so we triple the outcome and get the outcome of 420 years.
a truck of height 2ft, length 20ft and breadth 5 feet has an area of 200 cubic feet. Similarly a mountain of height 2000ft and radius of 1000 ft would had an area of 1b cubic feet. Hence it will take almost 5 million trips to move the mountain.
Assuming loading unloading and to and fro travel time to be an hour, it will take 5 million hours to transport.
1. Size of an average mountain. 500 meters X 500 meters X 200 meters = 250,000 X 200 = 50,000,000 m3.
2. How much m3 can a truck carry. Regular cars are four meters long. Truck is 3 regular cars long = 12 meters. Front part is 2 meters, so the back side is 10 meters long, width 3 meters. Height of the back side is 3 meters tall. So the amount of soil it can carry is 10 meters X 3 meters X 3 meters = 60 m3.
3. Depending on the road quality, 10 miles of driving with 50 miles/hour back and forth may take 20 minutes (for the sake of simplicity). Time to put and take out soil may take 10 minutes. In total we will spend 30 minutes per one round.
4. Lets assume that truck works 10 hours a day. So it will carry 10 hours / 0,5 hours X 60 m3 = 1200 m3 per day.
5. It will take 50,000,000 m3 / 1200 m3 = 4,200,000 days. If we work 300 days per year, then it will be 4,200,000 days / 300 days = 15,000 years.
The answer is 15,000 years
I would take a half of the first step, because mountains are usually have a pyramid shape and pyramid equals to half of a cube.
they actually equal one third of a cube..
if have an area where height is 25 meters and length is 18 meters and width 1 meters how many H frames required of 2 X 1 meters
225
2000 years