A doctor sees each of her patients for 15 minutes. How many patients can she see in 4-1/2 hours?

We know that 1 hour is equal to 60 minutes. So, 4-1/2 hours is equal to 4.5 x 60 = <<4.5*60=270>>270 minutes.

The number of patients a doctor can see in 4-1/2 hours is 270/15 = <<270/15=18>>18 patients. Answer: \boxed{18}.