1. **State the problem:** Simplify the expression $$\frac{\frac{3}{2a}}{\frac{4}{3a} - \frac{a}{4}}$$. 2. **Recall the formula:** To divide by a fraction, multiply by its reciprocal. Also, to subtract fractions, find a common denominator. 3. **Simplify the denominator:** $$\frac{4}{3a} - \frac{a}{4} = \frac{4 \cdot 4}{3a \cdot 4} - \frac{a \cdot 3a}{4 \cdot 3a} = \frac{16}{12a} - \frac{3a^2}{12a} = \frac{16 - 3a^2}{12a}$$ 4. **Rewrite the original expression:** $$\frac{\frac{3}{2a}}{\frac{16 - 3a^2}{12a}} = \frac{3}{2a} \times \frac{12a}{16 - 3a^2}$$ 5. **Cancel common factors:** $$= \frac{3}{\cancel{2} \cancel{a}} \times \frac{12 \cancel{a}}{16 - 3a^2} = \frac{3}{2} \times \frac{12}{16 - 3a^2}$$ 6. **Multiply numerators and denominators:** $$= \frac{3 \times 12}{2 \times (16 - 3a^2)} = \frac{36}{2(16 - 3a^2)}$$ 7. **Simplify the fraction:** $$= \frac{\cancel{36}{}^{18}}{\cancel{2}{}^{1}(16 - 3a^2)} = \frac{18}{16 - 3a^2}$$ **Final answer:** $$\boxed{\frac{18}{16 - 3a^2}}$$