Which half of a gold melt-value quote is arithmetic

Exact conversions derived from the grain, the two karat conventions that disagree, and a consistency check on a published per-karat table. A permanent copy.

A melt-value calculation has exactly two kinds of input. One kind is definitional: the mass units and the fineness conventions are fixed by agreement and have not changed in decades. The other kind is a price, and it changed while you were reading this sentence. Almost every argument about what a piece of scrap is worth is really an argument about the second kind being presented as if it were the first. This note separates them, derives the definitional half from first principles, and then uses a published five-karat table as a worked consistency check.

Stated up front, because it matters: the dollar figures used below come from an undated snapshot and are not a current price. They are used here only to demonstrate internal consistency and the ratio structure between karats, which is the part that stays true when the price moves. Do not quote them.

The mass units are exact, and you can derive them

The base unit is the grain, fixed by the 1959 international yard and pound agreement at exactly 64.79891 milligrams. Everything else in precious-metal weighing is an integer count of grains:

1 grain            = 0.06479891 g                (exact, by definition)
1 pennyweight      = 24 grains = 24 * 0.06479891
                   = 1.55517384 g       (exact)
1 troy ounce       = 480 grains = 480 * 0.06479891
                   = 31.1034768 g      (exact)
1 avoirdupois ounce= 437.5 grains = 28.349523125 g   (exact)
1 troy ounce       = 480/24 = 20 pennyweight   (exact)

Two consequences worth having in mind. First, a troy ounce is 480/437.5 = 1.097143 times the ordinary ounce — about 9.7% heavier. Weighing gold on a kitchen scale set to ounces and multiplying by a spot price quoted per troy ounce understates the value by that margin, and it is a large enough error to notice and a subtle enough one to miss. Second, none of these numbers is a measurement, so none of them carries uncertainty. They can be written into a calculator as exact rationals and never revisited.

Karat fineness has two conventions and they disagree

Karat is a statement about parts in twenty-four, so the naive purity of k karat gold is k/24. Hallmarking practice does not use that. It uses millesimal fineness standards, and for two of the common karats the standard is deliberately not the bare fraction:

The two that matter are the extremes. 14K is carried at 0.585 — the 585 hallmark standard used across much of Europe — rather than 0.583333, a difference of +0.286%. And 24K is carried at 0.999 rather than 1.000, because refined bullion is three-nines fine and not literally pure, a difference of -0.100%. Neither gap is large. Both are systematic, which means they do not average out across a parcel, and both are the sort of thing two calculators can silently disagree about while each looks internally consistent.

A consistency check on a published table

Given the exact constants, a per-karat melt table has almost no freedom left in it. Every row should imply the same underlying spot price, because that is the only free variable:

implied_spot = per_gram / purity * 31.1034768

The five rows imply spot prices spanning 0.24 USD. That looks like an inconsistency and is not one. The per-gram column is published to the cent, so each row carries an unavoidable rounding envelope of 0.005 / purity × 31.1034768 — and because the divisor is the purity, the envelope is worst at the lowest karat, ±0.373 USD at 10K against ±0.156 at 24K. Two rows rounding in opposite directions can differ by 0.529 USD with no error at all. The observed 0.24 sits inside that, so the table is consistent with a single spot price to the limit of its own precision. The derived columns agree too: recomputing per-pennyweight and per-ounce from per-gram and the exact constants reproduces the published values to within 0.008 and 0.094 USD respectively.

That check is worth running on any melt table you did not build. A table where the implied spot wanders by more than its rounding envelope was assembled from figures captured at different times, and the rows are not comparable to each other.

The part that survives the price moving

Spot cancels out of any ratio between karats, so the relative values are stable facts even though the absolute ones are stale the moment they are written:

The two columns agree to 0.00005, which is again rounding. This is the useful form of the table: a gram of 14K is worth 0.5856 of a gram of 24K, today and next year, whatever the price does. Note that it is not 14/24 = 0.5833, for the hallmark reason above.

What actually determines the number on the cheque

Having established that the arithmetic is exact to five decimal places, it is worth being blunt about how little that is worth. Two other terms dominate it.

The first is weighing. A jeweller’s scale reading to 0.1 g has a half-resolution uncertainty of 0.05 g regardless of what is on it, so its relative contribution is entirely a function of how small the item is:

On a 3.5 g chain the scale alone is worth ±1.43%, which is 5 times the 0.286% karat-convention gap that the previous section spent so long on. Any calculator that reports a melt value to the cent on a three-gram item is reporting digits the weighing cannot support.

The second term is the one nobody can compute for you: the fraction of melt value a buyer actually pays. Melt value is a ceiling, not an offer. It is the number the metal is worth if it were already refined and in a vault, and refining, assay, handling and margin all come out of it before a price is quoted. Whatever that fraction is, it multiplies everything above, and it is very much larger than the rounding, the scale and the hallmark convention combined. The reason to get the arithmetic exactly right is not that it is where the money is — it is that getting it right is free, and it is the only way to know which part of a low offer is the metal and which part is the spread.