${}_{92}^{238}\text{U}$ emits $8$ $\alpha$ -particles and $6$ $\beta$ -particles. The neutron/proton ratio in the product nucleus is:

1. Start: ${}_{92}^{238}\text{U}$ ($P=92,N=238-92=146$).2. Emit $8$ $\alpha$: Each $\alpha$ decreases mass by 4 and atomic number by 2.Decrease in $A=8\times 4=32$. Final $A=238-32=206$.Decrease in $Z=8\times 2=16$. Intermediate $Z=92-16=76$.3. Emit $6$ $\beta$: Each $\beta$ increases atomic number by 1.Increase in $Z=6\times 1=6$. Final $Z=76+6=82$.4. Product nucleus: ${}_{82}^{206}\text{X}$ (which is Lead, $\text{Pb}$).Final Proton ($P$) = $82$.Final Neutron ($N$) = $206-82=124$.$\text{Ratio }N/P=124/82=62/41$.