The half-life of a radio-active element is 9 days, what fraction of atoms has decayed in 36 days?

The number of half-lives ($n$) is the total time divided by the half-life.

$$n = \frac{\text{Total time}}{\text{Half-life}}$$ $$n = \frac{36 \text{ days}}{9 \text{ days}}$$ $$n = 4$$

After $n$ half-lives, the fraction of atoms remaining is given by $(\frac{1}{2})^n$.

$$\text{Fraction remaining} = \left(\frac{1}{2}\right)^n$$ $$\text{Fraction remaining} = \left(\frac{1}{2}\right)^4$$ $$\text{Fraction remaining} = \frac{1^4}{2^4}$$ $$\text{Fraction remaining} = \frac{1}{16}$$

The fraction decayed is $1 - \text{fraction remaining}$.

$$\text{Fraction decayed} = 1 - \frac{1}{16}$$ $$\text{Fraction decayed} = \frac{16}{16} - \frac{1}{16}$$ $$\text{Fraction decayed} = \frac{15}{16}$$

The fraction of atoms that has decayed in 36 days is $\boxed{\frac{15}{16}}$.