To identify the unit rate in word problems:

Dan earned [tex]\$200[/tex] in 2 days and [tex]\$300[/tex] in 3 days.

[tex]
\[
\begin{array}{l}
\frac{\$200}{2 \text{ days}} = \frac{\$100}{1 \text{ day}} = \$100 \text{ per day} \\
\frac{\$300}{3 \text{ days}} = \frac{\$100}{1 \text{ day}} = \$100 \text{ per day}
\end{array}
\]
[/tex]

A doctor sees 6 patients in [tex]1^{1/2}[/tex] hours. How many patients does the doctor see per hour?

[tex]$\square$[/tex] patients per hour

To find out how many patients the doctor sees per hour, we need to calculate the unit rate of patients per hour.

Here's how you can solve it step-by-step:

1. Determine the Total Number of Patients:
The doctor sees a total of 6 patients.

2. Determine the Total Time:
The doctor works for [tex]$1 \frac{1}{2}$[/tex] hours. In decimal form, this is 1.5 hours.

3. Calculate the Patients per Hour:
To find out how many patients the doctor sees in one hour, divide the total number of patients by the total time in hours.

[tex]\[
\text{Patients per hour} = \frac{\text{Total patients}}{\text{Total hours}} = \frac{6}{1.5}
\][/tex]

4. Compute the Result:
Performing this division gives you 4. Thus, the doctor sees 4 patients per hour.

Therefore, the doctor sees [tex]$\boxed{4}$[/tex] patients per hour.