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Can my precious metal be counterfeit despite having the "right" density?

Yes, gold has a density of 19.32 g/cm³, but the popular counterfeit material wolfram has a density of 19.30 g/cm³. For silver, which has a density of 10.49 g/cm³, lead-tin or molybdenum alloys with a similar density are often used. Thus, a counterfeit cannot be ruled out with a density measurement alone.