Given, initial amount of X and Y be N1 and N2. Let half-life of X be tx and y be ty. According to question, tx=‌
ty
2
=t ⇒‌‌tx=t and ty=2t After 3 half-lives of Y, 3ty=6t As we know that, N=N0e−λt where, N is the number of nuclei left undecayed. and t1∕2‌‌=‌
0.693
λ
λ‌‌=‌
0.693
t1∕2
λ1‌‌=‌
0.693
t
‌ and ‌λ2=‌
0.693
2t
Since, after 3 half-lives of Y number of nuclei of both become equal. ∴∴‌N1e−λ16t=N2e−λ26t ⇒‌N1∕N2=e6t(−λ2+λ1) ⇒‌N1∕N2=e6t(‌