These are a few examples of how "forbidden" procedures can lead us to the correct answer:
$$\displaystyle\frac{1\not4^1}{2\not8_2} = \frac{11}{22}=\frac{1}{2}$$
$$\displaystyle\frac{1\not3^1}{3\not9_3} = \frac{11}{33}=\frac{1}{3}$$
$$\displaystyle\frac{{\not3^1}0}{{\not6_2}0} = \frac{10}{20}=\frac{1}{2}$$
I am interested in the question does there exist a fraction such that it satisfies these $5$ requirements and such that it can be "solved" using this "technique"?
- The numerator and denominator have $n$ digits where $5\leq n\leq9$
- The number in numerator doesn`t end in $0$.
- The number in denominator deosn´t end in $0$.
- All digits in number in numerator are different.
- All digits in number in denominator are different.
EDIT1: The user found that it is possible for $n=5$, now I have a somewhat different version of the problem with modified first requirement:
1*. The numerator and denominator have $n$ digits where $6\leq n\leq9$
2.,3.,4.,5. Same as above.
I believe that now such fraction doesn`t exist and would be very pleased if someone shows that it is so or finds a counterexample in any of these two cases:
Case A) Only one cancellation is allowed.
Case B) More than one cancellation is allowed.
EDIT2: It seems that belief in non-existence of such fraction in the modified problem $6\leq n\leq9$ was a failure, below you can see that user Ivan Loh found the solution for the cases $n=6,7$ with one cancellation and solution for the case $n=8$ with two cancellations which is indeed more than I was asking, and, following his examples, I found the case for $n=8$ with one cancellation, it is $\displaystyle\frac {21945703}{65837109}=\frac {21945701}{65837103}=\frac {1}{3}$.
Best Answer
In case you still want a 5 digit example:
$$\frac{14573}{43719}=\frac{14571}{43713}=\frac{1}{3}$$
A 6 digit example:
$$\frac{194573}{583719}=\frac{194571}{583713}=\frac{1}{3}$$
In fact, let's add a $0$, so
$$\frac{1945703}{5837109}=\frac{1945701}{5837103}=\frac{1}{3}$$
If you want more than 1 cancellation, use
$$\frac{19457023}{58371069}=\frac{19457011}{58371033}=\frac{1}{3}$$