  Certain hadronic two-body decays of $B$ mesons are dominated by penguin
diagrams. The ratios of rates for several such decays, including $\Gamma(B^0
\to \overline{K}^{*0} K^0)/\Gamma(B^0 \to \phi K^0)$, $\Gamma(B^0 \to
\overline{K}^{*0} K^{*0})/\Gamma(B^0 \to \phi K^{*0})$, $\Gamma(B^+ \to
\overline{K}^{*0} K^+)$ $/\Gamma(B^+ \to \phi K^+)$, and $\Gamma(B^+ \to
\overline{K}^{*0} K^{*+})/\Gamma(B^+ \to \phi K^{*+})$, can provide information
on the ratio of Cabibbo-Kobayashi-Maskawa (CKM) elements $|V_{td}/V_{ts}|$ in a
manner complementary to other proposed determinations. SU(3) breaking effects
cancel in some ratios. The cases of neutral $B$ decays are free of corrections
from small annihilation terms.
