Word problems take place in a graded ring, from (the recently relocated) Mathematics under the Microscope (Alexandre Borovik), via The Unapologetic Mathematician (John Armstrong).
In short, Borovik claims that elementary school word problems take place over , where the xi represent different things that could be added. In this formalism, it makes sense to add, say, apples and oranges, going against the usual rule that you're only allowed to add quantities with the same "dimension". (Indeed, Borovik illustrates the idea with an example of this nature.)
I'm reminded of "dimensional analysis" as taught in, say, introductory physics classes, where we only allow monomials to have meaning, namely that the monomial x ma kgb secc measures some physical quantity with dimensions LaMbTc where L, M, T stand for length, mass, and time. (For example, in the case of speed, a = 1, b = 0, c = -1.) I can't think of situations in physics where one deals with a quantity of the form, say, a kg + b m. Is this because they don't exist, or because I don't know as much physics as some people?