To determine how many tickets Joseph can buy, we can first calculate how much money he has left after purchasing the hat.
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Calculate the money left after buying the hat: \[ \text{Money left} = \text{Total money} - \text{Cost of hat} \] \[ \text{Money left} = 34.25 - 12.50 = 21.75 \]
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Determine how many tickets he can buy with the remaining money: \[ \text{Number of tickets} = \frac{\text{Money left}}{\text{Cost of each ticket}} = \frac{21.75}{7.25} \] \[ \text{Number of tickets} ≈ 3 \]
Since he cannot buy a fraction of a ticket, we can confirm that he can buy a maximum of 3 tickets.
Thus, the answer is 3 tickets.