Given a two-way contingency table of non-negative reals in which the internal entries do not sum to the corresponding marginals, there is often the need to adjust internal entries to achieve additivity. In general, the objective is to have the revised table, in some sense, close to the original table and to have zero entries remain zero and positive entries remain positive. Not all two-way contingency tables can be adjusted to achieve additivity subject to the constraints above and in this paper we present a procedure that will determine whether a given table can be so adjusted.