Solve the problem.

A train ticket in a certain city is $\mathrm{\$}3\mathbf{0}\mathbf{0}$. People who use the train also have the option of purchasing a frequent rider pass for $\mathrm{\$}16.50$ each month. With the pass, each ticket costs only $\mathrm{\$}2\cdot 25$. 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

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 $\boxed{{\displaystyle 22}}$ times in a month for the total cost with the pass to be the same as the total cost without the pass.