1. **State the problem:** Simplify the expression $$\frac{7}{5} \times 1 - \frac{2}{3} - \left(-\frac{4}{5}\right) \cdot \frac{4}{3} + \frac{1}{3}$$. 2. **Rewrite the expression:** $$\frac{7}{5} \times 1 - \frac{2}{3} + \frac{4}{5} \cdot \frac{4}{3} + \frac{1}{3}$$ (Note that subtracting a negative is the same as adding a positive.) 3. **Calculate each term:** - $$\frac{7}{5} \times 1 = \frac{7}{5}$$ - $$\frac{4}{5} \cdot \frac{4}{3} = \frac{16}{15}$$ 4. **Substitute back:** $$\frac{7}{5} - \frac{2}{3} + \frac{16}{15} + \frac{1}{3}$$ 5. **Find a common denominator:** The denominators are 5, 3, and 15. The least common denominator is 15. 6. **Convert each fraction:** - $$\frac{7}{5} = \frac{7 \times 3}{5 \times 3} = \frac{21}{15}$$ - $$\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}$$ - $$\frac{16}{15}$$ (already with denominator 15) - $$\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$$ 7. **Rewrite the expression with common denominators:** $$\frac{21}{15} - \frac{10}{15} + \frac{16}{15} + \frac{5}{15}$$ 8. **Combine the numerators:** $$21 - 10 + 16 + 5 = 32$$ 9. **Final fraction:** $$\frac{32}{15}$$ 10. **Simplify if possible:** 32 and 15 have no common factors other than 1, so the fraction is in simplest form. **Final answer:** $$\boxed{\frac{32}{15}}$$