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 has the height of 1000 m and the base in the shape of circle with radius 50 m. Then, the volume of the mountain is equal to 1/3*3,14*50^2*1000. When we round it, it’s about 2500000m 3. he average truck has dimension: 10x3x2=120m3. Then, we know that we need around 21000 trucks. To load and unload the truck we will need 20 minutes and to go back and forth(so it’s 20 miles) we need 40. So it’s one hour for one truck, hence it’s 21000 hours/24=875 days = 2 years and 4 months (rounded)
730 days
Average size of the mountain = 1000^3 ft
Average size car’s carrying load = 5^3 ft
Distance for one trip = 10 miles * 2 (for return trip) = 20 miles
Estimated speed = 60mph
Estimated time to travel for one trip = 20 minutes
Time to unload = 10 minutes
Total time for one trip = .5 hours (20 10 minutes)
One trip bring 5^3 feet, so taking 1000^3 feet would take (1000/5 )= 200 trips
Each trip takes a half hour, so it would take about 100 hours of consistent work. Factoring in 10 hour work days, it would take 10 days to complete
It’s only 10 miles. So you can move the mountain just by moving part of its bottom. Let’s cut the mountain into horizontal layer, and then shift the most bottom layer 10 miles to any direction. The layer shape would be cylinder. All we have to do is to move half of its side to the opposite side. Let’s guess the volume of such part. Let’s say the mountain have r=500m. Half of the cylinder length is (pi)*r = 3.14*500 = 1,570 m. Say the height of the cylinder is 20m, and the depth (towards inside the cylinder) is around 5 m. Thus, 1,570*20*5 = 157,000 m3 to move.
Now let’s guess the volume of a dump truck, let’s say it’s 5 m * 3 m * 1.5 m = 22 m3.
It depends on how many trucks do we have. Given 10 trucks, it’ll take 714 times to move the soil volume.
If one truck can move back and forth 10 times in one day (include load and unload, and smooth road, and rest time – we have relief crew, it’ll take 71 days to move the mountain 10 miles.
Oops- Sent submit too early for the previous post.
Answer would be: Time for breaking the mountain time for loading the truck time for truck’s transportation and back
I would do the following:
1. Estimate the size of an average mountain and divide it into smaller geometrical shapes- rectangles or smaller triangles
2. Estimate the size of an average truck and estimate the volume it can transport. Volume would be in iterms of rectangles or triangles
Make assumptions:
1. Assume an avg speed of the truck
2. Assume the time of loading of the truck
3. Assume there are no roadblocks in the path of the truck
4. Assume there is enough resource availability
Answer would be: Time for breaking the mountain for the first delivery time for loading
I’ve tried to do this in as few moves as possible. As no follow up questions are allowed, and little detail is provided (and also from reading the above responses) it is clear that endless variations in assumptions are available, and is it normal to to sit in front of an interviewer for 15 minutes in silence while you mull over these in your head? Plus I am not a maths or science person!
Remembering that MC is the best solution available in the time available with the information available, I assumed that anything we haven’t been given details on is outside scope, and all that is at your disposal is one average truck. Therefore all that really matters is the volume of the truck, the volume of the mountain, and the amount of time it takes to move that volume 10 miles.
Average mountain – 3,500 metres.
As its triangular, I guessed its volume from halving that of a cube with sides all equalling 3,500m – i.e. 3,500 cubic metres / 2 = 1,750 cubic metres.
Assumed each cubic metre of dirt/gravel would weigh 1 tonne i.e. 1000 kg.
Your average moving truck has a load of 3 tonnes.
1,750 / 3 = approx 580 truckloads.
10 miles equals approx 20 kilometres.
Say the truck travels at 60km/h, making a one way trip 20 minutes and return 40 minutes.
580 x 40 minutes = 23,200 –> i.e. approx 400 hours.
Taking into account an average 8 hour work day – 400/8 = 50 days.
I.E. 50 working days of one truck in constant movement, not taking into account loading time, quarrying, labour constraints etc etc etc.
Let’s assume the following:
average size truck:
Load 8m3
speed 30 mph
average size mountain:
Piramid shape
height 1000m
basis square 1000m x 1000m
Volume to be transported:
V = 1/3 (B x H) = 1/3 (1.000.000.000) m3 = 333.333.333 m3
Time needed:
Load/Unload = 10 min
Load unload go to go from = 10 10 20 20 = 60 min = 1h
Time needed is one hour per one truck load
Total time needed with one truck = (1h x 333.333.333 / 8) = 41.666.667h
To determine the duration of the move, we need to know the duration of each trip and the number of trips needed:
Total duration = (# trips) * (duration of a trip)
The first element; the # of trips can be calculated by dividing the size of the mountain by the size of the truck:
# trips = (size of mountain) / (size of truck)
The second element; the duration of a trip, is the sum of several components:
– Duration of filling the truck
– Duration of the trip to the new location
– Duration of unloading the truck
– Duration of the trip back
– Overhead (truck breaks down, truck has to get gas, getting the materials ready in the morning, preparing for the night after the work day etc.)
So let’s look at each of the two main components individually to get an estimate.
1. # of trips necessary to move the mountain
a. Size of truck
Let’s say the bed of the truck has a volume of 5m*3m*2m = 30m^3. For efficiency, we will pile up high and get about 50m^3 of earth into the truck.
b. Size of the mountain.
Let’s assume the mountain is a cone shape. The height is about 1400m and the radius is therefore 1000m. The volume of a cone is 1/3*pi*r^2*h, which is 1/3 * 3.14 * 1E6 m^2 * 2E3 m. This is about 2E9 m^2.
===> Total number of trips necessary is 2E9 / 50 = 40 million
2. Duration of a trip
Here we have to make a couple of strong assumptions. First of all, I’m going to assume that the new location of the mountain is a little under an hour’s drive away from the old one. I am going to assume a 8-hour work day. I will assume the following numbers for the different parts of the trip:
– Duration of filling the truck: 0.1 workday
– Duration of the trip to the new location: 0.1 workday
– Duration of unloading the truck: 0.1 workday
– Duration of the trip back: 0.1 workday
– Overhead: 0.1 workday
The total of a trip then comes to 0.5 workday, which means we can make 2 trips per workday.
===> T0tal number of working days necessary to move the mountain (with 1 truck): 20 million
First we need to determine how much material there is in the mountain and how much the truck can hold. Let’s assume that an average mountain is approximately cone-shaped and is 1,000 feet high with a radius of 500. If the cone formula is 1/3 base * height, this gives us an area of approximately 500,000 feet squared (1/3 * 1,000*500). An 18-wheeler truck could hold perhaps 500 square feet of material. [Note: I chose these numbers because they are relatively easy to work with but they could be way off — I would verify that my assumptions are reasonable with the interviewer.] This means that it would take 1,000 trips for the truck to move the mountain.
Each trip can be broken into 4 parts: loading the truck, traveling the 10 miles, unloading the truck, and traveling back. We don’t know how many people we have doing the loading — let’s assume that we have a big crew and they can load up the truck in 1 hour. Since the truck is driving to a mountain, it’s probably not on a highway, so I’ll assume it’s going 40 miles an hour and can make the 10-mile trip in 15 minutes. Unloading the truck would also take 1 hour, and traveling back would be another 15 minutes. This gives us a total of 2.5 hours per trip. Multiplied by 1,000 trips = 2500 hours, or approximately 104 days. (2400 hours = 100 days, 100 leftover hours = ~4 days.)