1. **State the problem:** Multiply the mixed number $2 \frac{2}{3}$ by the fraction $\frac{4}{7}$. 2. **Convert the mixed number to an improper fraction:** $$2 \frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3}$$ 3. **Write the multiplication of fractions:** $$\frac{8}{3} \times \frac{4}{7}$$ 4. **Multiply the numerators and denominators:** $$\frac{8 \times 4}{3 \times 7} = \frac{32}{21}$$ 5. **Simplify the fraction if possible:** $32$ and $21$ have no common factors other than $1$, so the fraction is already in simplest form. 6. **Convert the improper fraction to a mixed number:** Divide $32$ by $21$: $$32 \div 21 = 1 \text{ remainder } 11$$ So, $$\frac{32}{21} = 1 \frac{11}{21}$$ **Final answer:** $$2 \frac{2}{3} \times \frac{4}{7} = 1 \frac{11}{21}$$