1. The problem involves understanding the relationship between two variables: $t$ (the number of tickets Patrick purchases) and $r$ (the number of rides Patrick can go on). 2. We are given two statements: - $t$ is the independent variable and $r$ is the dependent variable. - $r$ is the independent variable and $t$ is the dependent variable. 3. In a functional relationship, the independent variable is the input, and the dependent variable is the output that depends on the input. 4. Typically, the number of tickets purchased ($t$) determines how many rides ($r$) Patrick can go on, so $t$ is independent and $r$ is dependent. 5. Conversely, if we consider how many rides Patrick wants to go on ($r$), then the number of tickets needed ($t$) depends on $r$, making $r$ independent and $t$ dependent. 6. Therefore, the roles of independent and dependent variables depend on the context and which variable is considered the input. 7. To summarize: - If $t$ is input, then $r$ depends on $t$. - If $r$ is input, then $t$ depends on $r$. This explains the interchangeable roles of $t$ and $r$ as independent and dependent variables depending on the scenario.