We propose a solution to the puzzle of dimensional reduction in the random field Ising model, asking the following: To what random problem in D=d+2 dimensions does a pure system in d dimensions correspond? For a continuum binary fluid and an Ising lattice gas, we prove that the mean density and other observables equal those of a similar model in D dimensions, but with infinite range interactions and correlated disorder in the extra two dimensions. There is no conflict with rigorous results that the finite range model orders in D=3. Our arguments avoid the use of replicas and perturbative field theory, being based on convergent cluster expansions, which, for the lattice gas, may be extended to the critical point by the Lee-Yang theorem. Although our results may be derived using supersymmetry, they follow more directly from the matrix-tree theorem.