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How do we measure the purity of gold?

In the United States and Canada, the spelling karat is now solely used for the measure of purity, while carat solely refers to the measure of mass weight. As a measure of purity, one karat is purity by mass:

X=24 Mg/Mm

X is the karat rating of the material, Mg is the mass of pure gold or platinum the material, and Mm is the total mass of the material.

Therefore 24 karat gold is pure (100% Au w/w), 18 karat gold is 75% gold, 12 karat gold is 50% gold, and so forth.