  We report on a method for measuring the branching ratios of dipole
transitions of trapped atomic ions by performing nested sequences of population
inversions. This scheme is broadly applicable and does not use ultrafast pulsed
or narrow linewidth lasers. It is simple to perform and insensitive to
experimental variables such as laser and magnetic field noise as well as ion
heating. To demonstrate its effectiveness, we make the most accurate
measurements thus far of the branching ratios of both 5P1/2 and 5P3/2 states in
88Sr+ with sub-1% uncertainties. We measure 17.175(27) for the branching ratio
of 5P1/2-5S1/2, 15.845(71) for 5P3/2-5S1/2, and 0.05609(21) for 5P3/2-4D5/2,
ten- fold and thirty-fold improvements in precision for 5P1/2 and 5P3/2
branching ratios respectively over the best previous experimental values.
