To find the total time of $2$ hours and $30$ minutes plus $1$ hour and $45$ minutes, we can calculate the sum step by step, resulting in a total of $4$ hours and $15$ minutes.

First, we will add the hours together. We have:

Next, we will add the minutes:

Since $75$ minutes exceeds an hour, we need to convert this into hours and minutes. We know that there are $60$ minutes in an hour, so we can perform the conversion:

Now, we combine this extra hour with the previous total of $3$ hours:

Finally, we add the remaining $15$ minutes:

Therefore, the total time amounts to:

This systematic approach—adding hours first, then minutes, and converting when necessary—ensures accuracy and clarity in the calculation process.