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We are studying the half-life of certain samples of radioactive materials.
Here is a question that I am trying to understand and solve.
When analyzing a sample of Parent material there are 625 atoms, 19,375
atoms of Daughter material.  The half-life of the Parent is 50 million
years.
Question:  How old is the sample?  How is this age calculated?  Is this
enough data to answer these questions?
 channon

Answer:
Initially the sample was all made of Parent atoms; at that time there were
20,000 (i.e., 625 + 19,375) atoms (since each Parent atom produces one
Daughter). An equation we can use is
 
    N = N_0 * (1/2)^(t/H)   [ or equivalently, N = N_0 * 2^(-t/H) ]
 
where "t" is time (in years) elapsed since the sample was all Parent atoms,
"N" is the number of atoms of Parent that still exist at time t, "N_0"
is the number of Parent atoms initially (i.e., at time 0) and "H" is the
half-life (in years) of the Parent. ("^" means "raised to the power of")
What you want is the particular time t such that N=625. The above equation
can be solved for t; you get
 
 t = H * [log(N/N_0)]/log(1/2), or equivalently t = H * [log(N_0/N)]/log(2)
 
You can use base 10 logs, natural logs, whatever, so long as you use the
same base for both of the log calculations. If we use base 10 log and plug
in N=625, N_0=20,000, and H = 50,000,000 into the first form for t, we get
 
 t = 50,000,000 * log(.03125)/log(0.5) = 50,000,000 * (-1.50515)/(-.30103)
   = 250,000,000 years.

This approach works for the general case. As it happens, there's a shortcut
for your particular problem. Note that the number of surviving Parent atoms
is (625/20,000) = 1/32 of the original. Now, 1/32 = (1/2)^5, so we need 5
half-lives of time in order to decay to this state. Thus, we again get
t = 5*50,000,000 = 250,000,000 years.
 rcwinther