  Given a closed ideal $I$ in a C*-algebra $A$, we develop techniques to bound
the real rank of $A$ in terms of the real ranks of $I$ and $A/I$. Building on
work of Brown, Lin and Zhang, we obtain complete solutions if $I$ belongs to
any of the following classes:
  (1) C*-algebras with real rank zero, stable rank one and vanishing
$K_1$-group; (2) simple, purely infinite C*-algebras; (3) simple,
$\mathcal{Z}$-stable C*-algebras with real rank zero; (4) separable, stable
C*-algebras with an approximate unit of projections and the Corona
Factorization Property.
