1. **State the problem:** Evaluate the expression $\left(4 \frac{1}{2} + 5 \cdot \frac{1}{4} - 1 \frac{1}{2}\right) : 3$. 2. **Convert mixed numbers to improper fractions:** $4 \frac{1}{2} = \frac{9}{2}$, $1 \frac{1}{2} = \frac{3}{2}$ 3. **Calculate inside the parentheses:** $5 \cdot \frac{1}{4} = \frac{5}{4}$ So, $$\frac{9}{2} + \frac{5}{4} - \frac{3}{2}$$ 4. **Find common denominator (4) and rewrite fractions:** $$\frac{9}{2} = \frac{18}{4}, \quad \frac{3}{2} = \frac{6}{4}$$ 5. **Add and subtract:** $$\frac{18}{4} + \frac{5}{4} - \frac{6}{4} = \frac{18 + 5 - 6}{4} = \frac{17}{4}$$ 6. **Divide by 3:** $$\frac{17}{4} : 3 = \frac{17}{4} \times \frac{1}{3} = \frac{17}{12}$$ 7. **Final answer:** $$\boxed{\frac{17}{12}}$$