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We investigate in detail the qualitative behaviour of the class of Bianchi type B spatially homogeneous cosmological models in which the matter content is composed of two non-interacting components; the first component is described by a barotropic fluid having a gamma-law equation of state, whilst the second is a non-interacting scalar field with an exponential potential V ()= e k. In particular, we study the asymptotic properties of the models both at early and late times, paying particular attention to whether the models isotropize (and inflate) to the future, and we discuss the genericity of the cosmological scaling solutions.