Kenneth,

I'm going to get a little philosophical here. Good math definitions are usually not exclusive - that's why mathematicians consider a square to be a (special type of) rectangle, and a circle to be a (special type of) ellipse. Some school textbook writers (not all, I'm glad to say) do not grasp this and try to make definitions as mutually exclusive as possible. This makes 'match the diagrams to the names' tests easier to set but has few other advantages.

So a rate has units that may be different quantities, but do not have to be. Dollars per dollar is just as valid a "rate" as miles per gallon.

(Some people suggest that "rate" should be used only when it's a case of "something-per-time". This is at least consistent but has no special advantages, and rules out many useful usages.)

Good hunting!
RD