Every softener calculation in this lesson is one idea in three costumes: the resin holds a fixed number of grains of hardness, the water spends them, and the regeneration puts them back. Size the account, know the spending rate, and the schedule falls out. Authored in grains, gallons and cubic feet — the trade's own units, and the ones the exam asks in.

The load. $m = C\,V$ — m equals C V. $m$ is the mass of hardness as CaCO₃ the bed must hold between regenerations, in grains; $C$ is the raw water hardness in grains per gallon; $V$ is the volume treated over the run, in US gallons. This is the one calculation where the imperial units are the tidy ones: grains per gallon times gallons leaves grains, with nothing to convert.

The bed. $V = \dfrac{m_{\text{cap}}}{q}$ — V equals m-cap over q. Here $V$ is the resin volume in cubic feet, $m_{\text{cap}}$ the capacity required per regeneration in grains, and $q$ the resin's rating in grains of hardness per cubic foot. Watch the letters: $V$ meant gallons of water a paragraph ago and means cubic feet of resin now, which is exactly why a symbol has to be named every time it appears.

The clock. $t = \dfrac{m_{\text{cap}}}{C\,Q}$ — t equals m-cap over C Q. $t$ is the run between regenerations in days, $m_{\text{cap}}$ the bed's rated capacity in grains, $C$ the hardness in gpg and $Q$ the average usage in gallons per day. $C$ and $Q$ multiply first, into grains per day — the daily spend — and the capacity is divided by that.

The nugget that surprises people: a resin's capacity is not a property of the resin, it is a purchase. The same cubic foot delivers about 20,000 grains when regenerated with 6 lb of salt, and about 30,000 grains when given 15 lb. Half again the capacity for two and a half times the salt. Which end of that trade a plant sits at is a running-cost decision, made once at commissioning and paid for every month afterwards — and it is why a rating quoted without its salt dosage tells you nothing.