First, we need to calculate how much money Joseph has left after buying the hat. The hat costs $12.50, so we subtract that from his total amount of $34.25:
\[ 34.25 - 12.50 = 21.75 \]
Next, we find out how many tickets he can buy with the remaining amount. Each ticket costs $7.25, so we divide the remaining amount by the price of a ticket:
\[ \frac{21.75}{7.25} = 3 \]
Therefore, Joseph can buy 3 tickets.