For a binary mixture you can start with the fact that volumes are additive and the volume of any component is equal to the density divided by its mass.

For a mixture of A and B, the total volume

[tex]V_t = V_A + V_B[/tex]

Therefore

[tex]V_t = \frac{m_A}{\rho_A} + \frac{m_B}{\rho_B}[/tex]

Where m_{A} and m_{B} are the masses of components A and B, respectively, and ρ values are the respective densities. But the masses are also additive, so the mass of B is just difference between the total mass (m_{t}) and the mass of A.

[tex]V_t = \frac{m_A}{\rho_A} + \frac{m_t-m_A}{\rho_B}[/tex]

You are looking to find the mass of A in the mixture. Therefore the mass of A is equal to:

[tex]m_A=\frac{V_t \rho_A \rho_B}{\rho_B - \rho_A} - \frac{\rho_A m_t}{\rho_B - \rho_A}[/tex]

The densities are known. You can measure the total mass with a scale and the total volume using submersion in a liquid.

Let's assume your solid measured 10 mL in volume and weighed 100 g. Using densities for gold of 19.3 g/mL and for the other component (quartz) of 2.5 g/mL*, I find that the mass of the gold component is 86.16 g. This seems about right because 100 grams of pure gold would be ~5 mL and of pure quartz would be 40 mL. The true volume is closer to that of pure gold, so we should have a majority of gold.

Note: this only works for binary mixture. If you have more than two components you would need more information, but you can approximate this as a binary mixture because quarts and calcite have similar densities and they are very different from gold. There are a few other areas where this would break down (if the two phases are really well mixed, such that the bulk densities aren't appropriate), so as I said: rough estimate. Also, be careful of your units.

(Someone check all that, I did it rather quickly.)

(I took a value around the densities of quartz and calcite.)