$ \begingroup $ The difficulty with this doubt — and it is one that has come up in the comments to several answers, but has not so far been explicitly addressed by any of the answers — is that it appears that the count you are actually concern in is not the number you think you are concern in .
here is a ( wholly hypothetical, made-up ) scenario that I suspect is not besides far off the commemorate from what is going on here. You are trying to solve some kind of problem ; maybe it ‘s “ find the zeroes of such-and-such polynomial ”. Using a graph calculator or some other form of computer algebra system, you graph the polynomial and then use the technology ‘s built-in numeric tools to find the location of the zero. The calculator tells you the answer is $ 0.3760683761 $. But the discipline solution, according to the textbook, is $ 572/1521 $. You confirm, by entering this into your calculator, that the results are adequate, and want to know how you could have found the fraction yourself .
But hera ‘s the thing : those results are not equal. $ 572/1521 $ is not $ 0.3760683761 $. It is entirely approximately that. As Eric pointed out in his answer, $ 572/1521 $ is actually peer to $ 0.376068~376068~376068 \dots $, with a repeating engine block of six digits. If you round this to fit on a calculator display, the leave will look like $ 0.3760683761 $. But that is misleading you : That truncation can not be converted into the imprint $ 572/1521 $, because it is n’t equal to $ 572/1521 $.

so the real number doubt ought to have three parts to it :

• If I see a decimal output that appears to terminate, how can I know if it is really just a truncated form of a repeating decimal with a long block of repeating digits?
• If the “true” decimal value is actually repeating, how do I convert it to a fraction?
• If the “true” decimal value is actually terminating, how do I convert it to a fraction?

For the second depart of the question, see Eric ‘s suffice on how to express a repeating decimal as a fraction.

For the third gear contribution of the motion, see Elliot G ‘s suffice on how to express a displace decimal as a divide.

The substantial trouble is the first gear separate. How do you know if the decimal you see is actually the decimal you want ? The answer, unfortunately, is that if you are relying on some shape of technology to produce your solution, there is no way to know. Calculators are basically finitary devices that work with numeric approximations. A calculator ca n’t understand the actual value of $ 1/3 $, it can merely understand $ 0.333333333 $ to a finite number of places. And if you see $ 0.333333333 $ on a calculator screen, you ca n’t in truth know if it is supposed to be $ 1/3 $, or $ 333333333/1000000000 $, or if possibly there are some other wholly different digits hiding off the blind, buried bass in the decimal fraction expansion .
More precisely, there is no way to tell from a finite string of digits whether you are looking at a separate of a displace decimal, a part of a repeating decimal, or a share of an irrational total. There is fair no manner. This leads to some ( quite absurd ) misconceptions, as for example in this page from a book for kids which happily asserts that $ 8/23 $ is an irrational number because its decimal expansion just goes “ on and on ” with no apparent pattern .
The moral of the floor is, if you are expected ( in a certain classroom context, which I assume is the lawsuit here ) to solve a certain problem and get an answer like $ 572/1521 $, the odds are good that you are supposed to solve it using some method acting that leads directly to that solution, quite than obtain an approximate numeral value using some engineering and then try to reverse-engineer from it the correct measure .

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