To determine how many tickets Joseph can buy, we first need to find out how much money he has left after buying the hat.
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Start with the total amount Joseph has: $34.25.
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Subtract the cost of the hat: $12.50.
\[ 34.25 - 12.50 = 21.75 \]
Now, Joseph has $21.75 left to spend on tickets.
Next, we will divide the remaining amount by the cost of each ticket, which is $7.25:
\[ 21.75 \div 7.25 \approx 3 \]
This shows he can buy 3 tickets. We should check the total cost of buying 3 tickets to confirm:
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Cost of 3 tickets:
\[ 3 \times 7.25 = 21.75 \]
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Total cost including the hat:
\[ 21.75 + 12.50 = 34.25 \]
Since he can buy 3 tickets with the remaining money after purchasing the hat and stay within his budget of $34.25, the answer is:
Joseph can buy 3 tickets.