Answer :
First, we start to separate the whole part from the decimal part
[tex]1+0.888888888\ldots[/tex]then equal the decimal part to x
[tex]x=0.8[/tex]the number of digits that repeat itself is 1, so we multiply the equation by 10
[tex]10x=8.8[/tex]subtract both equations
[tex]\begin{gathered} 10x-x=8.8-0.8 \\ 9x=8 \\ \\ x=\frac{8}{9_{}} \end{gathered}[/tex]then we can write the decimal part as a fraction
[tex]\bar{0.88}=\frac{8}{9}[/tex]then 1.888 can be writen as a mixed number like
[tex]1.8888\ldots=1\frac{8}{9}[/tex]