To model the situation, we start by defining \( t \) as the number of tickets Tim purchased. The total cost for the tickets can be calculated as follows:
- The cost per ticket is $45.
- The convenience charge per ticket is $3.50.
- There is an additional processing fee of $5.50 for the order.
The total cost for \( t \) tickets can be expressed with the following equation:
\[ \text{Total Cost} = (\text{Cost per ticket} + \text{Convenience charge per ticket}) \times t + \text{Processing fee} \]
Substituting the values into the equation gives us:
\[ 199.50 = (45 + 3.50)t + 5.50 \]
Simplifying the equation:
\[ 199.50 = 48.50t + 5.50 \]
Now, we can isolate \( t \) by subtracting the processing fee from both sides:
\[ 199.50 - 5.50 = 48.50t \]
\[ 194.00 = 48.50t \]
Next, we divide both sides by 48.50 to solve for \( t \):
\[ t = \frac{194.00}{48.50} \approx 4 \]
So, Tim purchased 4 tickets.