Question 278777
Let's work on 0.22222.
Then we'll just add the 1 back. 
{{{x=0.22222}}} repeating
{{{10x=2.22222}}} repeating
Now subtract {{{10x-x=2.2222-0.22222=2}}}
{{{9x=2}}}
{{{x=2/9}}}
Now add the 1,
{{{1.2222=2/9+1}}}
{{{1.2222=2/9+9/9}}}
{{{1.2222=11/9}}}