Why 'sevenths' are easy to remember

I love being able to remember how to write fractions as a decimal. Most people think it's impossible to remember sevenths though - but it's quite easy if you know the pattern. When people say "Oh, about 2-in-7 people know their times tables", I can reply, almost straight away, "Hmmmm, that's around 28.57142857%"…

How does it work?

Well, all the sevenths (1/7, 2/7, 3/7 etc. all the way up to 6/7) are recurring decimals (that means they repeat after a while - in this case, 6 digits). Also, the only 6 digits they use are 1 4 2 8 5 & 7 - and it's always in that order.

If you put those 6 digits into 3 pairs, you get 14 28 57, where each pair is (very nearly) double the previous one (watch out for the '57'), and, to make it even easier, there's the link back to 7 - the first pair, 14, is double 7.

The decimal bit of each seventh just starts at a different place in that series of digits, works its way through…and then loops around again at the end.

So 1/7 starts at the beginning (best place to start), which happens to be the smallest digit ('1') with

    1/7 = 0.142857142857142857142857… (forever and ever)

2/7 starts at the 2nd smallest digit in there - the '2', so you get

    2/7 = 0.285714285714285714285714… (etc.)

3/7 starts at the 3rd smallest digit - the '4', giving

    3/7 = 0.42857142857142857142857… (ad infinitum)

And 4/7? Amaingly, that starts with the 4th smallest digit, which is '5' (you could remember that by thinking that 4/7 is just a little bit more than a half, which starts 0.5…)

    4/7 = 0.571428571428571428571428… (ad nauseam)

I'll leave 5/7 and 6/7 as an exercise for the reader :)