A radioactive nucleus (initial mass number A and atomic number Z) emits 3 $\alpha -$particles and 2 positions. The ratio of number of neutrons to that of protons in the fined nucleus will be

Initial mass number A  and Initial atomic number Z 

Emission $\to 3\alpha \text{ and }2\text{ positron }$

Now atomic number   $=\mathrm{Z}-3\times 2-2\times 1=\mathrm{Z}-8$ 

$\text{ Mass number }=\mathrm{A}-3\times 4=\mathrm{A}-12$

$\text{ Neutrons }=(A-12)-(Z-8)=A-Z-4$

$\frac{\text{ Neutrons }}{\text{ Protons }}=\frac{A-Z-4}{Z-8}$