I'm guessing the repeating part is 89 at the end, so that
[tex]x=0.8967\overline89\implies10^4x=8967.\overline{89}[/tex]
Then
[tex]10^4x=8967+\displaystyle89\sum_{i=1}^\infty\frac1{100^i}[/tex]
[tex]10^4x=8967+89\left(\dfrac1{1-\frac1{100}}-1\right)[/tex]
[tex]10^4x=8967+\dfrac{89}{99}[/tex]
[tex]x=\dfrac{8967}{10^4}+\dfrac{89}{99\cdot10^4}[/tex]
[tex]x=\dfrac{443911}{495000}[/tex]
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An arguably quicker way without using geometric series:
[tex]10^4x=8967.\overline{89}[/tex]
[tex]10^6x=896789.\overline{89}[/tex]
[tex]10^6x-10^4x=887822[/tex]
[tex]x=\dfrac{887822}{10^6-10^4}=\dfrac{443911}{495000}[/tex]
