Convert decimal numerator/denominator to integers

Question

Given any fraction that is defined by a decimal numerator or denominator or both. Which operations can always lead to the correct integer values? Eg: $\frac{0.5}{1}=\frac{1}{2}$

In this multiplying both numbers by 2 worked out, however what is the general procedure?

Answer 1

If the decimal expansions are both terminating, just multiply numerator and denominator with $10$ until you get two integers (You only need to shift the comma to the right).

Then, find out the $\gcd$ and divide numerator and denominator by it to get a fraction with coprime integers.

Answer 2

Convert the decimals into fractions (if possible)

So if $n = \frac ab$ and $m = \frac cd$ then $\frac nm = \frac {\frac ab}{\frac cd} = \frac {ad}{bc}$ and reduce.

If the decimals terminate, then the easiest way to do that with minimum thought. is simply multiply top and bottom by powers of $10$ until they are gone and reduce.

Example: With thought: $\frac {.65}{12.5} = \frac {\frac {13}{20}}{\frac {25}2} = \frac {13*2}{20*25}= \frac {13}{250}$

Without thought: $\frac{.65}{12.5} = \frac {6.5}{125} =\frac {65}{1250} = \frac {13}{250}$.

Must use thought: $\frac {.555555.....}{.125} = \frac {\frac 59}{\frac 18} = \frac {5*8}{9*1} = \frac {40}{9}$.

If the decimal(s) can't be converted to fractions (i.e. aren't rational) this may not be possible unless then are both rational multiples of the same irrational. There's no way to determine that except by recognizing the irrational number (no rule on how to do that) or by doing long division directly (and you can always do long division directly:

Ex; $\frac{3.1415926535897932384626433832795...}{2.3561944901923449288469825374596...}= 3.1415926535897932384626433832795...\div 2.3561944901923449288469825374596... = 1.33333333..... = \frac 43$.

(That was a considence)

$\frac {3.1415926535897932384626433832795...}{2.7182818284590452353602874713527...} = 3.1415926535897932384626433832795...\div 2.7182818284590452353602874713527... = 1.1557273497909217179100931833127....$.

Did long division for umpteen places. No end was in sight. .... So we gave up. .... It happens.