Why are reaction rates proportional to the the product of the reactants not their sum?

Where x, y, and a are the number of the reactants (X, Y, and A respectively). The rate is always proportional to the product of the number of reactants, but why not their sum?concentration (or even better chemical acti

'number of the reactants' is an ambiguous term: concentration (or even better chemical activity) is more accurate.

If we take a simple reaction:

$$text{A}+text{B}to text{C}$$

Then in accordance with kinetic theory the rate of reaction is proportional to:

$$frac{text{d}[C]}{text{d}t}propto [A]$$ and: $$frac{text{d}[C]}{text{d}t}propto [B]$$

It follows that: $$frac{text{d}[C]}{text{d}t}propto [A]times [B]$$ And: $$frac{text{d}[C]}{text{d}t}=k [A][B]$$

Where the bracketed quantities are concentrations.