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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