To find Sadie's average speed in miles per hour, we can use the formula for speed, which is:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
Given that Sadie drove a distance of \(13 \frac{1}{2}\) miles, let's first convert this mixed number into an improper fraction:
\[ 13 \frac{1}{2} = \frac{26}{2} + \frac{1}{2} = \frac{27}{2} \text{ miles} \]
Her time of travel is given as \(\frac{2}{3}\) hours. Now we can plug these values into the speed formula:
\[ \text{Speed} = \frac{\frac{27}{2}}{\frac{2}{3}} \]
To divide by a fraction, we multiply by the reciprocal:
\[ \text{Speed} = \frac{27}{2} \times \frac{3}{2} = \frac{27 \times 3}{2 \times 2} = \frac{81}{4} \text{ miles per hour} \]
Next, we can express \(\frac{81}{4}\) as a mixed number:
\[ \frac{81}{4} = 20 \frac{1}{4} \]
Thus, Sadie's average speed is:
\[ \boxed{20 \frac{1}{4}} \text{ miles per hour} \]
We can also leave it in improper fraction form as \(\frac{81}{4}\), which is also an appropriate answer. So, in simplest form, her average speed is:
\[ \boxed{\frac{81}{4}} \text{ miles per hour} \]
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