To determine how many tickets the student bought, we start by analyzing the cost function given:
\[ c = 24.50 + 12.50t \]
where:
- \( c \) is the total cost,
- \( t \) is the number of tickets.
The student spent a total of $110.50 on tickets, so we can set up the equation:
\[ 24.50 + 12.50t = 110.50 \]
Now, we can solve for \( t \).
- Subtract $24.50 from both sides:
\[ 12.50t = 110.50 - 24.50 \]
\[ 12.50t = 86.00 \]
- Now, divide both sides by \( 12.50 \):
\[ t = \frac{86.00}{12.50} \]
\[ t = 6.88 \]
Since the number of tickets must be a whole number, and you can't buy a fraction of a ticket, let's round \( t \) down to 6 (if we assume they have to be whole tickets).
So, since we can’t have partial tickets, the student likely bought 6 tickets. To confirm:
Calculating the total cost for 6 tickets:
\[ c = 24.50 + 12.50(6) = 24.50 + 75.00 = 99.50 \]
This is below $110.50.
So let's try 7 tickets:
Calculating the total cost for 7 tickets:
\[ c = 24.50 + 12.50(7) = 24.50 + 87.50 = 112.00 \]
This exceeds $110.50.
Thus, the closest valid solution is purchasing 6 tickets for $99.50, as purchasing 7 tickets goes over budget. Hence, the actual answer to the number of tickets should account for the maximum within the budget, leading to the conclusion that 6 tickets were assuredly purchased.