No, because 0.9 repeating is 1, which is an integer, not a "normal" fraction.
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First find the length of the repeating section of numbers. eg. For 0.678678678, 678 is the repeating section, and it's length is three. Divide this repeating section by the number with that many nines - for example in this case 678/999 = the fraction. Another example - 0.554755475547 = 5547/9999
89 is an integer, not a fraction. The repeated decimal equivalents are 89.000....(repeating) or 88.999... (repeating).
When something is repeating, you can find the fraction by taking that repeating number and putting over 9s, one nine for each digit. In this case, the number repeating is 4, so only one nine is needed, and the decimal part comes out to be 4/9. Then the 1 in front needs to be added, so the answer is 14/9 or 13/9.
There can be no such thing as a nearest fraction since, given any fraction, it is always possible to find a fraction that is nearer.
The vulgar fraction equivalent to 0.333 is 333/1000, or 333 thousandths. If the number you are looking to find the fraction of is 0.3 recurring (that is, 0.33333..., repeating endlessly), this is equal to 1/3, or one third.