This problem can be worked with units of feet in the vertical direction and units of inches in the horizontal direction, as long as you maintain that relationship.
I find it convenient to draw a graph of a line that represents the ladder. I choose to have the y-intercept be where the ladder touches Catie's head (5.5 feet above the ground), and the x-intercept to be where the ladder touches the ground, 16 inches away from Catie.
Then the wall is 4.5 feet (54 inches) from the ladder's base, or 38 inches in the other direction from Catie. The graph shows the ladder hits the wall at a height of about 18.6 feet.
The graph shows the calculations for the position of the wall and for the equation of the line. (The line is written in "intercept form.") The graphing calculator shows the intercept points, so no "work" is required except to highlight them.