Answer:
82/99
Explanation:
1. Notice the length of the repeating pattern.
the pattern is 82 repeating, a 2-digit pattern
2. Notice where the repeating pattern starts, relative to the decimal point.
the repeating starts at the decimal point
3. If the repeating starts at the decimal point, use a pattern of repeating 9s of the same length as the denominator of the fraction
2 nines are the denominator: 82/99
4. Reduce the fraction if possible
this fraction cannot be reduced
As a simplified fraction, 0.828282... = 82/99
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Comment on more complicated cases
If the repeat does not start at the decimal point, or if the number contains an integer part, then the numerator of the fraction is found in a slightly different way. Let n be the number of repeating digits of the number x. The fraction is then ...
(10^n·x -x)/(10^n -1)
Example:
1.2345345345... a number with a 3-digit repeat not starting at the decimal point
(1234.5345... -1.2345...)/(1000 -1) = 1233.3/999 = 4111/3330
The multiplication by 10^n aligns the repeating parts so they cancel when you do the subtraction.