1. **State the problem:** Multiply the mixed number $2 \frac{2}{3}$ by $\frac{1}{4}$. 2. **Convert the mixed number to an improper fraction:** $$2 \frac{2}{3} = \frac{3 \times 2 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3}$$ 3. **Write the multiplication:** $$\frac{8}{3} \times \frac{1}{4}$$ 4. **Multiply the numerators and denominators:** $$\frac{8 \times 1}{3 \times 4} = \frac{8}{12}$$ 5. **Simplify the fraction by dividing numerator and denominator by their greatest common divisor (4):** $$\frac{\cancel{8}^{2}}{\cancel{12}^{3}} = \frac{2}{3}$$ 6. **Final answer:** $$2 \frac{2}{3} \times \frac{1}{4} = \frac{2}{3}$$