A detailed description of a new analysis, in which the correlation of the electric charge of the b-quark to that of the lepton is utilized to isolate different decay modes of the b-hadrons, is presented. Using the data collected by the DELPHI detector during the 1994 and 1995 data-taking periods, the inclusive b-hadron semileptonic branching fraction is measured to be B R b→ℓ % =10.75+/-0.15stat +/-0.28syst -0.24+0.43model . Along with the semileptonic branching fraction, the branching fractions for the cascade mode, b→c→ℓ , and the upper-vertex mode, b→c&d1;→ℓ , are also measured: BR b→c→ℓ %=7.99+/-0.27 stat+/-0.28 syst-0.21+0.10 model BR b→c→ℓ %=1.34+/-0.30 stat+/-0.29 syst+0.36-0.58 model.