Answering "could this possibly fit?" by estimating in stages

0:00Here is a question that sounds like it cannot be answered. Could an entire city fit inside a lakh of buses? A lakh is one hundred thousand, so picture a hundred thousand buses in a line. And picture a city of more than one crore twenty four lakh people. Your instinct is probably that this is not a real question. It is. It becomes a real question the moment you turn it into two numbers and compare them.

0:31But notice what is missing. Nobody has told us how many people fit in one bus. That number is not given, so we have to supply it ourselves. A value you supply is called an assumption. Hold on to that word, because it is going to matter more than any of the arithmetic. So let us choose one, and say it out loud: fifty people to a bus. Now build the first number in easy stages.

1:01One bus takes fifty. Ten buses take five hundred. A hundred buses take five thousand. A thousand buses take fifty thousand. And a lakh of buses take fifty lakh people. Fifty lakh. That is the whole capacity of every bus in the line. The second number is the population itself, just over one crore twenty four lakh. The impossible-sounding question has become an ordinary one. Is fifty lakh as big as one crore twenty four lakh?

1:37It is not, and it is not close. The buses fall short by about seventy four lakh people. Put the two side by side and the buses hold about two fifths of the city. So the answer is no. But look at what we just did there. We did not only say no. We said how far off. That second part is the margin, and a verdict without one cannot really be checked.

2:05Missing by two fifths is a different kind of no from missing by a hair. One of them survives being slightly wrong about the numbers. The other does not. So the honest answer is: no, at fifty to a bus. That last phrase is carrying the whole thing. Watch what happens when we change nothing except the number we chose ourselves. Suppose a bus takes a hundred and twenty five people instead.

2:34A lakh of buses now hold one crore twenty five lakh. That is more than the population. The same question now answers yes. Same city, same lakh of buses, opposite verdict. So the answer was never a fact about the city. It was a fact about the city and our assumption together. You can find the exact turning point by dividing: one hundred and twenty four point four people per bus.

3:04Which is why a hundred and twenty five works. It is not a rounding, it is the next whole seat. Now the same method with something bigger. A large passenger ship might carry two and a half thousand people. Could that same city fit into five thousand such ships? Five thousand ships, two and a half thousand each, gives one crore twenty five lakh. That is bigger than the population, so yes, it fits.

3:35But the margin this time is fifty seven thousand, six hundred and twenty seven people. Against a city of over a crore, that is under half of one per cent. Compare the two results. The buses failed by seventy four lakh. The ships succeed by fifty seven thousand. The first verdict is safe. The second is only true if every number we used is very nearly exact. Here is a different kind of fitting question.

4:09If you could travel a hundred kilometres every single day for ten years, would you reach the Moon? The Moon is three lakh eighty four thousand, four hundred kilometres away. Do it in stages. First, one year. A hundred kilometres times three hundred and sixty five days is thirty six thousand five hundred kilometres. Now ten of those years: three lakh sixty five thousand kilometres. So you fall short by nineteen thousand four hundred kilometres. About five per cent.

4:44That is far closer than most people expect, and they guess wrong in both directions. Which is exactly why you compute. And before you ask about leap days: they add three hundred kilometres against a shortfall of nineteen thousand. You would arrive after about ten and a half years. Notice how that was done. At no point was there one enormous multiplication. There were three small steps, and each one meant something.

5:17Kilometres in a day. Kilometres in a year. Kilometres in ten years. Do it as one long calculation and you get a single number with nothing to compare it against. Do it in stages and every halfway number is a fact you can check on its own. Thirty six thousand five hundred kilometres in a year is worth knowing even if you never finish the question. It is also where a mistake shows up. If a stage looks wrong, you have narrowed the error to one step.

5:51The buses worked the same way: five hundred, five thousand, fifty thousand, fifty lakh. Staging is not the slow way of doing this. It is the checkable way. Same method, much wilder answer. The distance to the Sun is about twenty one hundred times seventy thousand kilometres. Which comes to fourteen crore seventy lakh kilometres. Now travel ten times faster than before: a thousand kilometres every day. A year of that is three lakh sixty five thousand kilometres.

6:30Divide one by the other and the journey takes about four hundred and three years. Not a lifetime. About five of them. Nothing about the method changed between the Moon and the Sun. Only the numbers did. That is what makes it worth learning. It does not care how big the question is. Three fast ones, to see how quick this gets. Could you carry a lakh of sheets of paper, if each sheet weighs five grams?

7:01Five lakh grams, which is five hundred kilograms. No, you could not. Could a million babies be born in a single day, at two hundred and fifty births a minute? Two hundred and fifty, times sixty, times twenty four, is three lakh sixty thousand. So no. But look carefully at that one, because it is sitting in an interesting place. Three lakh sixty thousand is under a million, but it is comfortably over a lakh. Change the word in the question and the answer reverses.

7:38And the last: counting coins at one a second, all day, gets you eighty six thousand four hundred. Under a lakh. Three questions, three noes, and three completely different margins. Which brings us back to the word from the beginning. This method is only ever as good as the numbers you feed it. So an assumption is not just any value that makes the sum work out. It is one you could say out loud and have somebody else accept.

8:10Some are not really assumptions at all. A day has eighty six thousand four hundred seconds. That is a fact. A bus seats fifty is a choice, but a defensible one, and you can name why. A useful way to hold it: the quantity, the value, the reason, and what would change your mind. And how careful you need to be depends entirely on the margin. When you miss by two fifths, a rough assumption is fine. When you win by half a per cent, it is not.

8:44So here is the method turned loose on a few more. Coins one millimetre thick, stacked to the top of a statue a hundred and eighty metres tall. A hundred and eighty metres is a hundred and eighty thousand millimetres, so a hundred and eighty thousand coins. A bird crossing twelve thousand kilometres, covering nine hundred to a thousand a day, is flying for twelve days to a little over thirteen.

9:13Another flies thirteen thousand five hundred and sixty kilometres in about eleven days: over twelve hundred kilometres a day, about fifty one an hour. Eagles have been seen at four and a half to six thousand metres. Against a forty metre building, that is a hundred and thirteen to a hundred and fifty of them. Everest is about two hundred and twenty one buildings. A plane cruises at two hundred and fifty to three hundred and twenty, so it really does fly over the mountain.

9:49None of those needed anything you do not already have. So here is one to try. Pick any question of your own, say the assumption out loud first, build both numbers in stages, and give your answer with its margin.