$$\alpha\,\mathrm{A}+\beta\,\mathrm{B}+\cdots\rightleftharpoons \rho\,\mathrm{R}+\sigma\,\mathrm{S}+\cdots$$ The equilibrium constant for this reaction can be calculated using the following formula:
$$\mathrm K_c = \frac{{{[\mathrm{R}]}}^\rho {{[\mathrm{S}]}}^\sigma ... } {{{[\mathrm{A}]}}^\alpha {{[\mathrm{B}]}}^\beta ...}$$
In this formula for the equilibrium constant of a reaction, why are the quantities R and S (and likewise A and B) multiplied?
I understand the equilibrium constant to be a ratio of the amounts of the products and the amounts of the reactants. But if this is the case, wouldn't a formula like $\mathrm K_c = \frac{[\mathrm{R}] + [\mathrm{S}] ... }{[\mathrm{A}] + [\mathrm{B}] ...}$ make more sense? I think this question is really a result of not understanding the meaning of the equilibrium constant.