Why am I not seeing the pattern here???

Question

State a general rule describing which fractions have decimal forms that terminate and which forms have ones that repeat?

I did all the fractions into decimals from 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, and 1/12. I am not finding the answer to this question. Some repeat some terminate but I don't see why they do or don't???

Answer 1

If one or more factors of the denominator of the fraction is not a factor of 10, then the corresponding decimal repeats.

For example: 1/3, 5/6, 11/13

If all the factors of the denominator of the fraction are factors of 10, then the corresponding decimal terminates.

For example 1/4, 1/5, 3/20

Why is this important? Well, first, it's probably your first brush with number theory. But it also explains why 10 is such a rotten number base and 100 is such a rotten number of small coins to make a big unit (e.g. 100 cents to the dollar, 100 kopeks to the rouble).

There's an important corollary of this. The old pounds, shillings and pence currency of Britain and the British Empire was better than the rotten decimal currency that we're stuck with today. Since one pound was divisible by 240 (960 if you go back to the 1950s when a penny was four farthings) and one guinea was divisible by 252, you could divide a sum of money by 2, 3, 4, 5, 6, 7 (convert it to guineas first, then one seventh of a guinea is 3/-), 8, 10 and 12. More or less every schoolchild could perform these calculations instantly, accurately, and in their heads. I can still do it.

Which proves that people who don't understand maths shouldn't be left in charge of anything more complicated than a tea trolley with stabilisers.

Answer 2

I don't know if it helps, but numbers in the 3x table repeat, up to 6x3 1/3 1/6 1/9 1/12 1/15 1?18.