Solve the problem.
A train ticket in a certain city is $\$ 3 \mathbf{0 0}$. People who use the train also have the option of purchasing a frequent rider pass for $\$ 16.50$ each month. With the pass, each ticket costs only $\$ 2 \cdot 25$. Determine the number of times in a month the train must be used so that the total monthly cost without the pass is the same as the total monthly cost with the pass.
21 times
Answer
Final Answer: The train must be used \(\boxed{22}\) times in a month for the total cost with the pass to be the same as the total cost without the pass.
Step 1 ：The problem is asking for the number of times the train must be used in a month such that the total cost with the pass is equal to the total cost without the pass.
Step 2 ：The cost without the pass is simply the number of rides times the cost per ride, which is $3.00.
Step 3 ：The cost with the pass is the cost of the pass plus the number of rides times the reduced cost per ride, which is $2.25.
Step 4 ：We can set up an equation to represent this situation and solve for the number of rides. Let's denote the number of rides as x. The equation will be: \(3.00x = 16.50 + 2.25x\)
Step 5 ：We can solve this equation to find the value of x.
Step 6 ：The solution to the equation is 22, which means that the train must be used 22 times in a month for the total cost with the pass to be the same as the total cost without the pass.
Step 7 ：Final Answer: The train must be used \(\boxed{22}\) times in a month for the total cost with the pass to be the same as the total cost without the pass.
