After emitting three $\alpha$ - particles, the radioactive nucleus (initial mass number $A$ and atomic number $Z$) will be rendered into a nucleus having mass number $(A-12)$ and atomic number $(Z-6)$. Further, after emitting two positrons, the mass number will remain unchanged i.e., $(A-12)$ but atomic number will become $(Z-6-2)$ i.e., $(Z-8)$.

Therefore, number of protons in the final nucleus $=\left(A-12\right)-\left(Z-8\right)=\left(A-Z-4\right)$

Therefore, the ratio of number of neutrons to that of protons in the final nucleus

$=\frac{A-Z-4}{Z-8}$