Feed chlorine to raw water and the water eats first: iron, manganese, organics and ammonia consume their share before any protection is left over. The share the water eats is the demand; what survives is the residual, and the residual is what disinfects all the way down the pipe. The balance is $\text{dose} = \text{demand} + \text{residual}$ — read aloud: dose equals demand plus residual.

A sum — the friendliest structure in the chapter. It rotates by plain subtraction: demand is dose minus residual, residual is dose minus demand, and the sanity rule polices all three — the residual can NEVER exceed the dose, because no water carries more chlorine than it was fed. Any rearrangement that breaks that rule has already confessed.