One equation, everything in parts per thousand
Fineness is the fraction of the total weight that is the precious metal, written per thousand. Karat is the same fraction written out of 24, and percent is the same fraction out of a hundred. Pick one and stay in it.
Every component of a melt brings some precious metal with it, equal to its weight times its fineness. Add those up and they have to equal the batch weight times the target fineness:
(weight1 x fineness1) + (weight2 x fineness2) + ... = total weight x target fineness
With two components and a known total, that is one equation and one unknown, and it rearranges to:
weight of high source = total x (target - low fineness) / (high fineness - low fineness)
At the defaults on this page — 100 g at 585 fineness from fine metal at 999 and a master alloy carrying none — that is 100 x 585 / 999, which is 58.56 g of fine metal and 41.44 g of master alloy. Check it: 58.56 x 0.999 gives 58.50 g of the precious metal in 100 g, which is 585 per thousand.
Karat numbers and the bare fraction
Fourteen twenty-fourths is 583.3 per thousand, and 585 is what a great deal of 14 karat stock is actually made to. Eighteen twenty-fourths is 750 exactly. Nine twenty-fourths is 375. The gap between the arithmetic fraction and the number a trade works to is small but it is not zero, and it exists for reasons that have to do with tolerance and with what the local rules require rather than with arithmetic.
Which number you should be aiming at is not something this page decides. Enter the fineness you are working to and the mass balance follows from it. If you enter a karat number instead, this page divides by 24 and tells you it has done so, because that is the only thing arithmetic can honestly do with a karat figure.
Melting scrap into the batch
Scrap makes the problem more interesting because its weight is fixed — the whole lot is going in — and its fineness is usually a claim rather than a measurement. The equation is the same, with the scrap contribution moved to the other side:
high source = (total x target - scrap weight x scrap fineness - remainder x low fineness) / (high - low)
Take 30 g of 585 scrap into a 100 g batch at 585, with fine metal and a plain master alloy. The scrap brings 17.55 g of precious metal, the remaining 70 g has to bring 41.0 g, and that works out to 41.0 g of fine metal and 29.0 g of master alloy. Sensible, because the scrap is already on target and the other 70 g just has to be too.
Now take 30 g of 750 scrap into the same 100 g batch at 585. The scrap brings 22.5 g, the remaining 70 g needs to bring 36.0 g, so 36.0 g of fine metal and 34.0 g of master alloy. The higher-fineness scrap has bought you a little of the fine metal back.
When the target is out of range
A mass balance can only land between the lowest and highest fineness you put into it, and once a fixed weight of scrap is in the pot the range narrows further. Put 60 g of 750 scrap into a 100 g batch and the lowest fineness reachable is 450 per thousand, no matter how much master alloy you add, because the scrap alone is carrying 45 g of precious metal into a 100 g batch. This page says so rather than returning a negative weight, and where the problem is too much scrap it works out how large the batch would have to be.
Two things the arithmetic cannot know
The first is whether the fineness figures are true. A stamp on a piece of scrap is somebody else assertion about metal that has since been worn, repaired and possibly plated, and a touchstone comparison is a judgement rather than a number. A melt built on either is exactly as good as those inputs, and a large one is worth an assay first.
The second is melting loss. Metal is lost across a melt and it does not come off the components evenly, since they do not all oxidise or volatilise at the same rate. This page will scale the whole charge up by whatever loss figure you enter so the finished weight lands where you want it, and it assumes the loss is proportional, which is convenient rather than correct. Weighing in and weighing out over a run of melts is the only way to know your own number.
What a finished piece may be stamped, described or sold as is not an arithmetic question at all. It belongs to the assay office or authority wherever the work is sold, and to an independent test of the metal that actually came out of the crucible.
Questions people ask
How much fine gold and master alloy makes 100 grams of 14 karat?
Working to 585 fineness from fine metal at 999 and a master alloy carrying none, it is 58.56 g of fine metal and 41.44 g of master alloy. Working to the bare fraction of 583.3 instead it is 58.39 g and 41.61 g. Which target you should use is set by the rules where the work is sold and by your own testing, not by this page.
What is the formula for mixing an alloy to a fineness target?
Weight of the high source equals the total weight times the target fineness minus the low fineness, divided by the difference between the two source finenesses. Everything must be in the same units — parts per thousand throughout is the least error-prone. The rest of the batch is the low-fineness material.
Can I mix scrap of a different karat into a batch?
Yes, and the equation handles it by treating the scrap as a fixed weight carrying a fixed amount of precious metal, with the fine metal and master alloy flexing around it. What it cannot do is verify the fineness you claim for the scrap. A stamp is an assertion made by somebody else about metal that has since been repaired and worn, and a large melt is worth an assay before it goes in the crucible.
Why does the calculator say my target is not reachable?
Because a mass balance can only land between the highest and lowest fineness materials you gave it. Usually the cause is too much high-fineness scrap for the batch weight — 60 g of 750 scrap in a 100 g batch cannot go below 450 fineness however much alloy is added. The page prints the range that is actually available and, where the scrap is the constraint, the batch weight that would work.
How do I account for melting loss when alloying?
Scale every charge weight up by one divided by one minus the loss fraction, so a 2 percent loss means multiplying by 1.0204. That keeps the finished weight on target and assumes the loss falls evenly across the components, which is an approximation because they do not all oxidise at the same rate. Your own figure comes from weighing in and weighing out across a run of melts and keeping the records.