For some years I have shared my Excel spreadsheet that calculates the density of alloys. I am pleased that I still get requests for it. Such requests and some questions about the spreadsheet's formula have encouraged me to discuss the derivation. (Thanks to Indium's Bob Jarrett for this simple approach).
Let's assume you have 90 grams of tin (density = 7.29 g/cc) and 10 grams of lead (density = 11.34 g/cc). The volume of the two alloys mixed together is:
90g/7.29g/cc + 10g/11.34g/cc = 12.34 cc + 0.882 cc = 13.23 cc
Hence we have 100g that has a volume of 13.23 cc or a density of 100g/13.23cc = 7.56 g/cc.
A little thought about what we have done above will show that the general formula for calculating densities is:
1/Df = x/Da + (1-x)/Db
Where Df = the final alloy density, x = the weight fraction of metal a, Da the density of metal a, and Db the density of metal b. The weight fraction of metal b has to be 1-x. This formula can be extended for multiple metals.
Note also that this equation only works for alloys in which the atom of one substitutes in the lattice for the other metal. A perfect example being tin substituting for copper in bronze. The formula does not work for interstitial alloys such as carbon in steel.