Add chlorine to a water carrying ammonia and the residual does something no other chemistry does: it rises, then falls, then rises again. The dip is the breakpoint, and until you are past it there is no free chlorine in the water at all.

The first chlorine in reacts with ammonia to make chloramines — combined chlorine. They read on a total-chlorine kit, which is why an operator can watch the residual climb and believe the plant is winning. Keep feeding and the chloramines are destroyed by the chlorine that follows, and the measured residual drops away to almost nothing. That trough is breakpoint. Only past it does every further milligram survive as free chlorine.

The arithmetic is a weight ratio: $R = \dfrac{\mathrm{Cl_2}}{\mathrm{NH_3\text{-}N}}$ — read aloud R equals C-l-two over N-H-three-N. $R$ is the chlorine-to-ammonia weight ratio, a bare number with no unit. $\mathrm{Cl_2}$ is the chlorine dose in mg/L as Cl₂, and $\mathrm{NH_3\text{-}N}$ is the ammonia nitrogen in mg/L as N — which matters, because the same water quoted as NH₃ is a different number and would need a different ratio. Order is built into the name: chlorine on top.

The theoretical breakpoint is 7.6 to 1 by weight, from the stoichiometry of chlorine oxidising ammonia to nitrogen gas. Real waters need more — 8 to 1, sometimes 10 to 1 — because organic nitrogen and other reduced species take their share too. Read the size of that number and the operating lesson falls out: 1 mg/L of ammonia costs you nearly 8 mg/L of chlorine, so a water with 2 mg/L of ammonia needs a 15 mg/L dose before it holds a single tenth of free residual.

Which is why the choice is a real one. Some plants go past breakpoint and hold free chlorine. Others stop deliberately short and run chloramination on purpose — a weaker disinfectant that lasts far longer in the distribution system and makes far fewer trihalomethanes. The worst place to be is neither: stuck on the hump, feeding hard, measuring a total residual, and disinfecting almost nothing.