1. **State the problem:** We need to multiply the fraction $\frac{3}{5}$ by the mixed number $1 \frac{4}{7}$ and simplify the result. 2. **Convert the mixed number to an improper fraction:** $$1 \frac{4}{7} = \frac{7 \times 1 + 4}{7} = \frac{11}{7}$$ 3. **Multiply the fractions:** $$\frac{3}{5} \times \frac{11}{7} = \frac{3 \times 11}{5 \times 7} = \frac{33}{35}$$ 4. **Simplify the fraction:** The numerator 33 and denominator 35 have no common factors other than 1, so the fraction is already in simplest form. **Final answer:** $$\frac{33}{35}$$