Okay, we're going to have to use distance = rate x time. Let's do the following:
For the first part, we know the total distance is 222. We also know the total time is 180 minutes. We can write the following equations:
Note that the wife has to drive the remaining distance, which can be represented by 222-d.Doing a little substitution (plugging in for d in the second equation and simplifying, we get, eventually:
Now, we can set up similar equations in the second part, except the time has changed to 175 minutes. We know the professor drove one-third of the time, of 58 1/3 minutes, and his wife drove the rest of the time, or 116 2/3 minutes. This gives, eventually, the equation:
(I multiplied everything - the times and distances - by 3, thus getting rid of the fractions (and ending up with the ominous 666 on the right.))
We have two equations, two unknowns. Solve it however you like - substitution, linear combinations, etc. You should be home free!
Remember that since we changed the times to minutes, your answers after solving the system will be in miles / minute. Multiply by 60 to convert to miles / hour (since there are 60 minutes in an hour).