In positive beta decay, a proton is transformed into a neutron and a positron is emitted.

${\mathrm{p}}^{+}\to {\mathrm{n}}^0+{\mathrm{e}}^{+}$

Number of neutrons initially was $A-Z.$

Number of neutrons after decay $(A-Z)-3\times 2$ (due to alpha particles)

$+2\times 1$(due to positive beta decay)

The number of protons will reduce by 8.

[as3 × 2(due to alpha particles) + 2 (due to positive beta decay)]

Hence, atomic number reduces by 8.

So, the ratio of number of neutrons to that of protons  $=\frac{A-Z-4}{Z-8}$