Estimating to an Order of Magnitude

Before doing any arithmetic, look at how far apart the options are. When each is ten times the one before, they are decades apart, and that is a deliberate signal: the question wants the right size, not the right number. You could be forty per cent out and still land comfortably on the correct option. So round hard and round early, because precision here is wasted effort that costs you time you need elsewhere. Checking the spacing of the options first tells you how carefully to work.

Round every quantity to something you can multiply in your head, then commit to it. A step is about three quarters of a metre. A year is about three hundred and fifty or four hundred days, whichever is easier to multiply by. An adult life is about fifty years. There is a small refinement worth knowing: if you round one number up, try to round another down, so the errors work against each other instead of piling up in the same direction. And never carry decimals through an estimate — they add nothing and cost time.

Now the whole calculation, with every assumption written down. A step is about three quarters of a metre, so ten thousand steps is seven and a half thousand metres, which is seven and a half kilometres a day. Multiply by about three hundred and fifty days and you get roughly two thousand six hundred kilometres a year. Over about fifty-five adult years that is around a hundred and forty-three thousand kilometres. The option is a hundred and fifty thousand, and the nearest others are fifteen thousand and one and a half million — nothing else is remotely close, so the estimate is safe.

Two sanity checks are worth running. First, compare with something you know: right round the Earth is about forty thousand kilometres, so a hundred and fifty thousand is roughly four times round in a lifetime, which sounds plausible. Second, test the extreme options backwards. One and a half million kilometres would be about seventy-five kilometres every single day for fifty-five years, which is absurd. Fifteen thousand would be under a kilometre a day, which is far less than ten thousand steps. Working backwards to a daily figure kills two options in seconds, without any real calculation.

The reverse situation matters just as much. If the options are only a few per cent apart, estimating will not separate them — rounding to one digit would leave you sitting between two of them, having settled nothing. Closely spaced options are telling you the question wants real arithmetic. So the spacing is not incidental, it is instruction: decades apart means estimate, a few per cent apart means calculate. Reading the options before choosing your method is the habit that makes both kinds fast. Quiz and exercises now.