To find out how many hours the plumber worked, you can use the formula:
\[ \text{Hours worked} = \frac{\text{Total Charge}}{\text{Hourly Rate}} \]
In this case, the plumber charged Luke $180 and her hourly rate is $45.
\[ \text{Hours worked} = \frac{180}{45} = 4 \]
So, the plumber worked 4 hours.