A radioactive nucleus emits \(4 \alpha\) particles and \(7 \beta\) particles in succession. The ratio of number of neutrons to that of protons is \([\mathrm{A}=\) mass number, \(\mathrm{Z}=\) atomic number \(]\)

Let us assume, a particle \(X\) having atomic number \(Z\) and mass number \(A\).
When an a-particle is emitted by a nucleus, then its atomic number decreases by 2 and mass number decreases by 4 . So, for given case, \(\mathrm{Z} \mathrm{X}^{\mathrm{A}} \stackrel{4 \text { a-particle }}{\longrightarrow} \mathrm{Z}_{-\mathrm{a}} \mathrm{Y}^{\mathrm{A}-16}\)
When a \(\beta\)-particle is emitted by a nucleus its atomic number increases by one and mass number remains unchanged. So, for given case,
\(\mathrm{Z}-8 \mathrm{Y}^{\mathrm{A}-16} \stackrel{7 \beta-p a r t i c l e}{\longrightarrow} \mathrm{Z}-1 \mathrm{Z}^{\mathrm{A}-16}\)