Solving symbolic equation for fraction of variables

You can combine Solve with Eliminate:

Solve[
    Eliminate[{ui-ua r1/(r1+r2)==0,v==ua/ui}, ua],
    v
]

{{v -> (r1 + r2)/r1}}

Although not currently documented, you can include in Solve a list of variables to be eliminated.

Solve[{ui - ua r1/(r1 + r2) == 0, v == ua/ui}, v, {ua}][[1]] // Expand

(* {v -> 1 + r2/r1} *)

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