Repeating decimal
A shortcut

The procedure below can be applied in particular if the repetend has n digits, all of which are 0 except the final one which is 1. For instance for n = 7:

So this particular repeating decimal corresponds to the fraction 1 / 10 − 1, where the denominator is the number written as n 9s. Knowing just that, a general repeating decimal can be expressed as a fraction without having to solve an equation. For example, one could reason:

or

It is possible to get a general formula expressing a repeating decimal with an n -digit period (repetend length), beginning right after the decimal point, as a fraction:

More explicitly, one gets the following cases:

If the repeating decimal is between 0 and 1, and the repeating block is n digits long, first occurring right after the decimal point, then the fraction (not necessarily reduced) will be the integer number represented by the n -digit block divided by the one represented by n 9s. For example,

If the repeating decimal is as above, except that there are k (extra) digits 0 between the decimal point and the repeating n -digit block, then one can simply add k digits 0 after the n digits 9 of the denominator (and, as before, the fraction may subsequently be simplified). For example,

Any repeating decimal not of the form described above can be written as a sum of a terminating decimal and a repeating decimal of one of the two above types (actually the first type suffices, but that could require the terminating decimal to be negative). For example,

An even faster method is to ignore the decimal point completely and go like this

It follows that any repeating decimal with period n, and k digits after the decimal point that do not belong to the repeating part, can be written as a (not necessarily reduced) fraction whose denominator is (10 − 1)10.

Conversely the period of the repeating decimal of a fraction c / d will be (at most) the smallest number n such that 10 − 1 is divisible by d.

For example, the fraction 2 / 7 has d = 7, and the smallest k that makes 10 − 1 divisible by 7 is k = 6, because 999999 = 7 × 142857. The period of the fraction 2 / 7 is therefore 6.
combination:two_responses