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## 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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