To find out how many tickets Joseph can buy, we first need to determine how much money he has left after purchasing the hat.
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Subtract the cost of the hat from his total amount:
\[ 34.25 - 12.50 = 21.75 \]
Joseph has $21.75 left after buying the hat.
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Next, we need to find out how many tickets he can buy with the remaining amount. Each ticket costs $7.25. We can find the number of tickets by dividing the remaining amount by the price of each ticket:
\[ \frac{21.75}{7.25} \approx 3 \]
To confirm, we can multiply the number of tickets by the price per ticket:
\[ 3 \times 7.25 = 21.75 \]
This means Joseph can buy exactly 3 tickets.
Thus, Joseph can buy 3 tickets.