JEE Main · 2010 · 11 Apr 2010 · Offline · Modern Physics

Each alpha particle carries off mass number $4$ and charge $2$, so three of them lower the mass number by $12$ and the atomic number by $6$. Each positron emission turns a proton into a neutron, lowering the atomic number by $1$ while leaving the mass number unchanged, so two positrons lower the charge by a further $2$. The final nucleus has mass number $A-12$ and atomic number $Z-8$, hence a neutron count $N=(A-12)-(Z-8)=A-Z-4$. The ratio of neutrons to protons is $\frac{A-Z-4}{Z-8}$.