Keep an 8-sided roll or switch to a d12

You throw a fair eight-sided die once. After seeing the result you may either stop and be paid that value, or throw a fair twelve-sided die once and be paid whatever it shows instead. Playing optimally, what is your expected payout?

$27/4$. A fresh twelve-sided die is worth $6.5$ on average, so keep the first roll only if it is $7$ or $8$. The expected payout is $\tfrac{7+8}{8}+\tfrac68\cdot\tfrac{13}{2}=\tfrac{27}{4}$.