# A rock sample contains 75 atoms of a parent isotope and 25 atoms of a daughter isotope. The half-life of the parent isotope is 100 years. How old is this rock?

##### 1 Answer
Oct 1, 2016

The rock is 41.5 years old.

#### Explanation:

For each half-life, you divide the total amount of the isotope by 2, so

$\frac{\text{Amount remaining" = "original amount}}{2} ^ n$, where $n = \text{the number of half-lives}$.

$A = {A}_{0} / {2}^{n}$

You can rearrange this to

${A}_{0} / A = {2}^{n}$

If there were originally 100 atoms, and 75 of the atoms remain, we have

$\frac{100}{75} = {2}^{n}$

$\frac{4}{3} = {2}^{n}$

$\log \left(\frac{4}{3}\right) = n \log 2$

$n = \log \frac{\frac{4}{3}}{\log} 2 = \frac{0.1249}{0.3010} = 0.415$

$t = 0.415 {t}_{\frac{1}{2}} = \text{0.415 × 100 yr" = "41.5 yr}$

The rock is 41.5 years old.