The author considers the relationship between solutions to a given system of ordinary differential equations, numerical approximations to them, and solutions to associated modified equations. Suppose that a consistent one-step numerical method of order \(r\) is applied to a smooth system of ordinary differential equations. Given any integer \(m \geq 1\), the method may be shown to be of order \(r+m\) as an approximation to a certain modified equation. Here, a technique is introduced for proving that the modified equations inherit qualitative properties from the method and the underlying system. The technique uses a straightforward contradiction argument applicable to arbitrary one-step methods and does not rely on the detailed structure of associated power series expansions. The new approach unifies and extends results of this type that have been derived by other means; results are presented for integral preservation, reversibility, inheritance of fixed points, Hamiltonian problems and volume preservation. The technique also applies when the system has an integral that the method preserves not exactly, but to order greater than \(r\). Finally, a negative result is obtained by considering a gradient system and gradient numerical method possessing a global property that is not shared by the associated modified equations.