let's firstly move the decimal point to the right, thus leaving the recurring part to the right, and then assign the recurring part at right to some variable, let's proceed.
[tex]1.4\overline{55555}\implies \cfrac{14.\overline{55555}}{10}\qquad \qquad \stackrel{\textit{now let's make}}{x = 0.\overline{55555}} \\\\\\ \stackrel{\textit{so then we can say}}{\cfrac{14.\overline{55555}}{10}\implies \cfrac{14+0.\overline{55555}}{10}}\implies \cfrac{14+x}{10} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{so we can say that}~\hfill }{10x~~ =~~5.\overline{55555}}\implies 10x = 5+ 0.\overline{55555}\implies 10x = 5+x \\\\\\ 9x=5\implies x = \cfrac{5}{9}[/tex]
well, now let's plug that into our fraction
[tex]\cfrac{14+x}{10}\implies \cfrac{14+\frac{5}{9}}{10}\implies \cfrac{~~\frac{131}{9}~~}{10}\implies \blacktriangleright\cfrac{131}{90}\implies 1\frac{41}{90} \blacktriangleleft[/tex]
