RETURN to "A Penny for your Thoughts"

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

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

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

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. 

RETURN to "A Penny for your Thoughts"