To find the rate at which Sarah's car uses gas in miles per gallon, we need to divide the total distance traveled by the total amount of gas used.
The total distance traveled is 50 miles, and the total amount of gas used is \(2 \frac{2}{3}\) gallons. First, we need to convert \(2 \frac{2}{3}\) into an improper fraction:
\[ 2 \frac{2}{3} = 2 + \frac{2}{3} = \frac{6}{3} + \frac{2}{3} = \frac{8}{3} \text{ gallons} \]
Now we can calculate the miles per gallon (mpg) by dividing the distance by the gallons used:
\[ \text{mpg} = \frac{\text{distance}}{\text{gallons}} = \frac{50 \text{ miles}}{\frac{8}{3} \text{ gallons}} \]
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \text{mpg} = 50 \times \frac{3}{8} \]
Now, we perform the multiplication:
\[ \text{mpg} = \frac{50 \times 3}{8} = \frac{150}{8} = 18.75 \text{ miles per gallon} \]
Therefore, Sarah's car uses gas at a rate of \( \boxed{18.75} \) miles per gallon.