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We report and demonstrate 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 to species with metastable lambda systems and can be generalized to find the branching of any state to lowest states. It does not use ultrafast pulsed or narrow linewidth lasers and is 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 5P[subscript 1/2] and 5P[subscript 3/2] states in [superscript 88]Sr[superscript +] with sub-1% uncertainties. We measure 17.175(27) for the 5P[subscript 1/2]–5S[subscript 1/2] branching ratio, 15.845(71) for 5P[subscript 3/2]–5S[subscript 1/2], and 0.056 09(21) for 5P[subscript 3/2]–4D[subscript 5/2]. These values represent the first precision measurement for 5P[subscript 3/2]–4D[subscript 5/2], as well as ten- and thirty-fold improvements in precision respectively for 5P[subscript 1/2]–5S[subscript 1/2] and 5P[subscript 3/2]–5S[subscript 1/2] over the best previous experimental values., National Science Foundation (U.S.). Center for Ultracold Atoms, United States. Intelligence Advanced Research Projects Activity. Multi-Qubit Coherent Operations