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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Subtract the cost of the hat from his total amount: \[ 34.25 - 12.50 = 21.75 \]
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Now, divide the remaining amount by the cost of each ticket to see how many tickets he can buy: \[ 21.75 \div 7.25 \approx 3 \]
Therefore, Joseph can buy 3 tickets.