To determine how many tickets Joseph can buy, we first need to calculate how much money he has left after buying the hat.
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Subtract the cost of the hat from the total amount of money Joseph has: \[ 34.25 - 12.50 = 21.75 \]
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Now, with the remaining $21.75, we can find out how many tickets he can buy. Each ticket costs $7.25, so we divide the remaining amount by the ticket price: \[ 21.75 \div 7.25 = 3 \]
Joseph can buy 3 tickets.