Let [tex]x=0.23737\ldots=0.2\overline{37}[/tex]. Then
[tex]10x=2.\overline{37}[/tex]
[tex]100x=237.\overline{37}[/tex]
[tex]\implies100x-10x=237.\overline{37}-2.\overline{37}[/tex]
[tex]\implies90x=235[/tex]
[tex]\implies x=\dfrac{235}{90}=\dfrac{47}{18}[/tex]
and the sum of the numerator and denominator is 65.
