It sounds like you're saying 0.555... = 0.9. This is not true.
[tex]x=0.555\ldots=0.\overline5\implies 10x=5.555\ldots=5.\overline5[/tex]
[tex]10x-x=9x=5.\overline5-0.\overline5=5\implies 0.\overline5=\dfrac59[/tex]
[tex]\dfrac59=\dfrac{50}{90}[/tex]. Meanwhile, [tex]0.9=\dfrac9{10}=\dfrac{81}{90}[/tex], so 0.555... < 0.9.
I guess you're supposed to be converting 0.0555... into a fraction using the fact that 0.555... = 5/9 (because the above has nothing to do with 0.05). Well,
[tex]0.0\overline5=\dfrac{0.\overline5}{10}=\dfrac{\frac59}{10}=\dfrac5{90}[/tex]
Then for part (b), I think you're supposed to rationalize 0.2555..., and not just 0.25. We have
[tex]0.2\overline5=0.2+0.0\overline5=\dfrac15+\dfrac5{90}=\dfrac{18}{90}+\dfrac5{90}=\dfrac{23}{90}[/tex]
