This post provides the answer on how to answer the estimation question example I posted previously.

You'll notice in the comment section of that previous post there were a wide range of answers -- most of which were incorrect.

The first thing to realize in an estimation question is that an acceptable answer MUST mention a specific number.

So the question was how much time does it take to move an average mountain 1 mile (or something along those lines).

If the answer did not include a specific unit of time like X hours, Y days, Z years, then the answer was 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.

You'll notice that amongst the answers given that I labeled "acceptable" the answer given ranged from few hundred hours to a few hundred years.

That's a pretty wide range!

The next thing you'll notice about acceptable answers is they made EXPLICIT ASSUMPTIONS.

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.

So 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. Thats 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 estimate 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.

So 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.

So with some slight rounding, that gets us 60,000,000,000.

So 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**

Then I would move on to estimate the volume of a truck.

The carrying capacity of a cargo truck is the **width x length x height** of 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. So lets 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. So 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.

So 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 cubit 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**

So 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. Lets figure out the travel time first. Lets 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. So the drive time is 10 minutes to the new location, and 10 minutes returning to the old mountain for a total of 20 minutes. Lets 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.

So lets go back to our formula again and update it.

**60,000,000,000 cubic ft mountain / 1,250 cubit 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).

So 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

So 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.

So 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 fairly 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.

**Why Other Answers are Not Acceptable
**

Again, any answer that does not provide an actual number is automatically unacceptable.

An answer like this:

"It is not specified whether or not the mountain is actually on the ground or not. Assuming it is floating in mid air (as this is an assumption based question), then it could take any amount of time varying from one minute or one hundred thousand days. The idea of a mountain of 'average' mass. There is no such thing as an average mountain as we cannot count the mass and size of every mountain in the world. We have no definition of a mountain, so a mountain could be as small as a grain of sand. The mountains we are aware of could be outliers in terms of what we know. An average size truck, by the same token, might not mean what we think it means. There is no time frame for such a question. This question could be as old as trucks are, or it could be a question meant for people in the future. Trucks come in a variety of sizes, and although there is a 'common' truck, there are trucks that could be quite small or quite large in comparison. This is all surrounded by the idea of an '

average'. An average is all things considered and divided by the amount of all things. We cannot know ALL things, and therefore, we cannot judge. There is no wrong answer if the answer is in a time, so the time it would take is five minutes, simply because, until the answer is given, there is no way of getting it wrong because the answer is UNKNOWN knowledge."

... would result in the candidate getting rejected. The candidate's point of view is valid. It isn't possible to know the answer with 100% accuracy. This is true. But the reason it gets the candidate rejected is because clients ask us to answer these kinds of questions at least once a week on the job.

When a candidate or a first year resists answering this kind of question, the phrased insiders use to describe this is:

"Uncomfortable with ambiguity".

Keep in mind the question was to ESTIMATE... and NOT to provide an accurate calculation.

So the first ambiguity is how big is an average sized mountain? There is no hard data on this. There is no explicitly given assumption as you would find on a math test or a problem set in school. So in the real world, we have to make ASSUMPTIONS.

The problem is some people HATE making assumptions. There is nothing wrong with that, but people who hate making assumptions make lousy consultants and rejected during the interview process.

Next if you provided the "answer" but did not disclose your assumptions and "show your work" (like my 8th grade math teacher insisted I do), then that too is an unacceptable answer. Here's why. When you don't show your work, your computations lack transparency. When your computations lack transparency, clients tend not to believe them. This is a problem. Therefore, when you provide an estimate that can not not be justified and COMMUNICATE with EXPLICITLY DEFINED ASSUMPTIONS, you will get rejected.

I hope this explanation gives you a feel for what interviewers are looking for and why they are looking for it.

Hi, I’m a bit confused by how you got the volume of a cone here to be 125,000,000,000 cubic ft:

“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.

So 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.

So then underneath my original formula, I would write the following:

125,000,000,000 cubic ft / volume of a truck * time per truck trip = total time to move mountain”

You say that the volume of a cone would be a little more than *half* the volume of a cube, but isn’t the 125,000,000,000 cubic feet you end up using for the equation the volume of a cube? Not a cone? Wouldn’t you have to half it and end up with 62,500,000,000 cubic feet?

In other words – 5,000 x 5,000 x 5,000 would be the full volume of a cube. But you’d want only about half of that to get the volume of a cone with the same base and height, according to your original estimated formula.

Please explain to me if I’m missing some fundamental simple step here. Thanks!

Erica – Your assessment is correct. The error was on my end and has since been revised in the post.

-Victor

1280 cubic feet is around 36.2 cubic meter. Consider a density of rocks of 2.5 t/cubic meter, it is actually 90 t!

I do not think it is a normal truck, it is a very big truck…

I second Erica.

That would render the estimation to be 25 mil hours,

which is relatively the same with my estimation (30 mil hours) on the estimation page

assumptions:

a truck can hold 100 kilos of soil and an average mountain has 10,000 kilos of soil

a truck will travel the same rate (time and distance) with or without the load from point A to point B. Let us assume that it takes 10hrs for the truck to travel

we would like to find out how many hrs it would take

Thus our equation should be

10,000 / 100 = number of trips

number of trips x (HRStotravelpoint1-2 (x2))

i multiplied it with x2 because it goes back and forth thus doubling the time

This is my answer.. i wonder if this is okay? i did easy numerical assumptions to prevent myself making a mental math blunder….

I used personal experience to base my initial assumptions. Being from Colorado, our mountains are basically two miles high. We live at 6000 feet and the mountains are upwards of 14,000 feet. So I rounded off to two miles high which really skewed the rest of my results and I ended up with 70,000 years.

Victor, I am a bit confused why the time for loading and unloading cargo was not included in the estimation.

In my assumptions I took 10 mins for loading, and 8 mins for unloading (here I include the fact that as mor cargo will be delivered to destination as higher the new mountain will be and as longer the distance will became)

Also you assume 60m/h speed which is almost 100 km/h. Is this a reasonable assumption for a packed truck? I assumed 30 m\h in my solution

In the end I got 70.000 years…

Could you please clarify?

A basic assumption here is that the majority of the time is spent in moving the mountain. However if its a mountain, it has to be broken down, and then rebuilt. I am assuming that is what is meant by “move” or “relocate” the mountain. By my estimates the time required to break down and reassemble the mountain one truck load at a time, is much larger than the time required to moved the same volume by truck. SO I dont agree with this number.

The solution provided by Victor was my first guess as well, but on further thought that is what I came to the conclusion. But I may be wrong and looking too much into the idea of “moving” the mountain.

How good an idea would it be to include loading, off-loading and refueling time into consideration ,while solving this case?

I was think that the whole time. So obviously you would need to figure out the approximate tank size and then also the rate of pour for that tank and how many trips needed before gas is needed. But if you do that you also have to figure out the distance thst each tire can go in normal condition before wear and tear. If you ads gas then you HAVE to add every intangible in as well.

Hi Victor,

How long should a candidate take to answer an estimation question? I ran through one with a partner this morning and rushed through the analysis because it wasn’t a case. Should I regard estimation questions as similar to a case interview? Is it okay to take 15 minutes for an estimation question or should I aim for 5-10 minutes?

Thanks!

Hoa

Hoa,

It depends on if the estimation question takes place within a case. If so, 5-10 min is ideal. If the estimation question is given instead of a case and it’s an elaborate one, then 15 min might be okay. That said, 15min feels long to me. I would take another look at your answer and see if you could have simplified your approach. 5-10min is much safer. The key isn’t to rush, the key is to use a simpler approach that isn’t as time consuming.

Victor

how much time one shud take so as to answer “etimation question” on an avergae?

Did I miss the time estimation of using the excavator to fill the truck….making the assumption of emptying/dumping negligible?

The formula for the volume of a cone is something that you learn in 10th grade math, if you are being considered for a job that requires quantitative skills it will count seriously against you that your estimate of “half the volume of the cube it’s contained in” was so far off. A cone is 1/3 of the cylinder it is contained in, and the cylinder is pi/4 of the box it is contained in, so you should have pi/12 instead of 1/2 which means you are off by a factor of pi/6 already. You very nearly doubled your final estimate simply because you could neither remember the formula nor visualize a cone clearly enough to realize that “1/2 the cube” was a gross overestimate.

1 truck

Length = 10 yards, Breadth = 10 yards, Height = 5 yards

Volume carried by 1 truck = 500 cubic yards

Shape of mountain = Conical

Height of average mountain = 100 yards

Radius of mountain base = 210 yards

Volume of soil in mountain = 22,000 cubic yards

Total truckloads needed to move mountain = 22000/500 = 44

Moving 1 truckload of mountain 10 miles involves = Loading + Transporting + Unloading + Coming back

Assuming time taken to load and unload are the same = 2 hrs

Transporting full truckload 10 miles = 30 mins

Coming back empty = 20 mins

Total time per trip = 2 + 0.5 + 2 + 0.33 = 4.83 hrs

Total trips = 44 x 4.83 = 215 (approximately)

Total time = 9 days (approximately)