Solution to #17

Let the two metals be designated "a" and "b" and their densities
be designated Da and Db, repectively. Suppose a composite coin of 
mass M is constructed of layers of metals a and b weighing Ma and 
Mb, repectively.  Thus, M = Ma + Mb.  Let's call X the mass 
fraction of metal a: X = Ma/M. The fraction of metal b 
is therefore 1-X, since there are only 2 metals.

Now, the overall density of the composite coin is just its total
mass divided by its total volume V.  The total volume V is just
the sum of the volumes of the two layers Va + Vb.  

       M          M
D = ------- = ---------
       V       Va + Vb
	   
But from the definition of density Va = Ma/Da and Vb = Mb/Db.  Thus

         M          
D = -----------
     Ma     Mb
    ---- + ----
     Da     Db
	 
But Ma = X*M and Mb = (1-X)*M.  Thus

           M          
D = ---------------
    X*M    (1-X)*M
    ---- + -------
     Da      Db
	 
factoring and cancelling M:

          1          
D = --------------
     X      1-X 
    ---- + ------
     Da      Db
	  
And there you go - given the densities of the two metals and the fraction
of one of the metals, you can calculate the density of the composite coin. 

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