jmc

Let $a=0.\overline{43}$. Then $100a=43.\overline{43}$. Subtracting the left-hand sides $100a$ and $a$, and subtracting the right-hand sides $43.\overline{43}$ and $0.\overline{43}$ yields $$\begin{array}{rl}100a-a& =43.\overline{43}-0.\overline{43}\\ \Rightarrow 99a& =43\\ \Rightarrow a& =\boxed{\frac{43}{99}}.\end{array}$$