1. **Stating the problem:** Calculate the value of $$A = \frac{3 - \frac{4}{3}}{3 + \frac{4}{3}}$$ and express it in simplified form. 2. **Formula and rules:** To simplify a fraction with complex numerator and denominator, first simplify numerator and denominator separately, then divide. 3. **Simplify numerator:** $$3 - \frac{4}{3} = \frac{9}{3} - \frac{4}{3} = \frac{9 - 4}{3} = \frac{5}{3}$$ 4. **Simplify denominator:** $$3 + \frac{4}{3} = \frac{9}{3} + \frac{4}{3} = \frac{9 + 4}{3} = \frac{13}{3}$$ 5. **Rewrite A:** $$A = \frac{\frac{5}{3}}{\frac{13}{3}}$$ 6. **Divide fractions:** $$A = \frac{5}{3} \times \frac{3}{13} = \frac{5 \times 3}{3 \times 13} = \frac{15}{39}$$ 7. **Simplify fraction:** Both numerator and denominator are divisible by 3: $$\frac{15}{39} = \frac{15 \div 3}{39 \div 3} = \frac{5}{13}$$ **Final answer:** $$A = \frac{5}{13}$$