A water analysis lands on the desk: calcium 80, magnesium 25, bicarbonate 180, sulphate 60, all in milligrams per litre. Add them and you have a number that means nothing. Chemistry does not react milligram for milligram; it reacts equivalent for equivalent, one unit of charge against one unit of charge. So the trade converts every result into the same yardstick — milligrams per litre as CaCO₃ — and only then adds, subtracts and compares.

Start with the yardstick itself. $\mathrm{EW} = \dfrac{M}{z}$, read aloud E-W equals M over z. $\mathrm{EW}$ is the equivalent weight in grams per equivalent — the mass that carries exactly one unit of charge. $M$ is the molar mass in grams per mole, and $z$ is the valence: the number of charges, or replaceable hydrogens, one mole carries. It is a bare count with no units. You are solving for $\mathrm{EW}$. Calcium carbonate has $M = 100.09$ and $z = 2$, so its equivalent weight is 50.04 g/eq — and that number sits on top of every conversion in this chapter.

Now the conversion. $C_{\mathrm{CaCO_3}} = C_{\mathrm{ion}} \times \dfrac{50.04}{\mathrm{EW}}$ — C-as-CaCO₃ equals C-ion, times fifty point oh four over E-W. $C_{\mathrm{ion}}$ is what the laboratory reported, in mg/L of that ion itself; $C_{\mathrm{CaCO_3}}$ is the identical amount of substance restated in the common currency; $\mathrm{EW}$ is that ion's own equivalent weight. Calcium's is 20.04, so its factor is $50.04/20.04 = 2.497$. Magnesium's is 12.16, so its factor is $50.04/12.16 = 4.118$.

Those two numbers, pre-computed, are the whole of the hardness relation: $\mathrm{TH} = 2.497\,\mathrm{Ca} + 4.118\,\mathrm{Mg}$ — T-H equals two point four nine seven Ca, plus four point one one eight Mg. $\mathrm{TH}$ is the total hardness in mg/L as CaCO₃; $\mathrm{Ca}$ and $\mathrm{Mg}$ are the calcium and magnesium results as the ions, in mg/L. Solve it forward for the hardness, or backwards for one ion when the total and the other ion are known.

The nugget worth carrying: magnesium counts about 1.65 times harder per milligram than calcium, purely because the magnesium atom is lighter and more of them fit into each milligram. A water with modest calcium and generous magnesium is harder than its calcium result suggests, and a softener sized on calcium alone arrives undersized.

One caution, because it trips people who know some chemistry. Saying a water has 300 mg/L of hardness as CaCO₃ is not a claim that it contains calcium carbonate. It is a statement of equivalents — that the hardness present would take the same amount of anything to react with as 300 mg/L of CaCO₃ would. The compound is a ruler, not an ingredient.