If the half-lives of a radioactive element for α and β decay are 4 years and 12 years respectively, the ratio of its initial activity and that after 12 years will be:

The decay constant for α decay = k₁ = ln2/4
The decay constant for β decay = k₂ = ln2/12
The radioactive element decays both ways simultaneously.
Thus, dN/dt = −(k₁ + k₂)N ⇒ N = N₀e^−(k₁+k₂)t
The ratio of the activities at t = 12 and at t = 0 is
(k₁ + k₂)N / (k₁ + k₂)N₀ = N/N₀ = e^−(k₁+k₂)(12) = 0.0625 = 6.25