1. **Problem statement:** Simplify the expression $\frac{3}{2} + \frac{5}{2} \div \frac{5}{4}$. 2. **Recall the division of fractions rule:** Dividing by a fraction is the same as multiplying by its reciprocal. 3. **Rewrite the division:** $$\frac{5}{2} \div \frac{5}{4} = \frac{5}{2} \times \frac{4}{5}$$ 4. **Multiply the fractions:** $$\frac{5}{2} \times \frac{4}{5} = \frac{5 \times 4}{2 \times 5} = \frac{20}{10} = 2$$ 5. **Add the result to $\frac{3}{2}$:** $$\frac{3}{2} + 2 = \frac{3}{2} + \frac{4}{2} = \frac{7}{2}$$ 6. **Final answer:** $$\boxed{\frac{7}{2}}$$