Solve Mixed Number Division: 1½ ÷ ⅔ Using Ratio Notation

We are required to divide the mixed number $1\frac{1}{2}$ by the fraction $\frac{2}{3}$.

Step 1: Convert the mixed number $1\frac{1}{2}$ into an improper fraction.
1 whole is equal to $\frac{2}{2}$, so $1\frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2}$.
Therefore, $1\frac{1}{2} = \frac{3}{2}$.

Step 2: Find the reciprocal of $\frac{2}{3}$.
The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.

Step 3: Multiply the improper fraction by the reciprocal.
$\frac{3}{2} \div \frac{2}{3} = \frac{3}{2} \times \frac{3}{2} = \frac{9}{4}$.

Step 4: Convert the improper fraction back to a mixed number.
$\frac{9}{4} = 2 \frac{1}{4}$ because 9 divided by 4 is 2 with a remainder of 1. This gives $2\frac{1}{4}$.

Therefore, the solution to the problem is $\boxed{2\frac{1}{4}}$.