Denote the mass of the older sample as `m_1` and of the modern as `m_2` . The older sample contains `x_1` grams of Carbon-14 and the modern `x_2` . Also denote the half-life as T and the age of the older sample as E years. And denote the number of counts per second as `n_1` and `n_2.`
After the time T half of Carbon-14 atoms remains, after 2T 1/4 remains and after time t `1/2^(t/T)` remains (this is true for t less than T also).
Therefore E years ago there was `x_1*2^(E/T)` atoms in older sample, and because materials are the same,
Now use the number of counts per second. Denote one second in years as s. Then
Thus `x_1/x_2=n_1/n_2` and from the first equation
`2^(E/T)=m_1/m_2 * n_2/n_1` , or
In numbers it is approximately 12200 years.