1. The problem is to multiply the mixed number $2 \frac{3}{4}$ by the fraction $\frac{4}{5}$. 2. First, convert the mixed number to an improper fraction. $$2 \frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}$$ 3. Now multiply the improper fraction by $\frac{4}{5}$. $$\frac{11}{4} \times \frac{4}{5} = \frac{11 \times 4}{4 \times 5} = \frac{44}{20}$$ 4. Simplify the fraction by canceling the common factor 4 in numerator and denominator. $$\frac{\cancel{44}^{11 \times \cancel{4}}}{\cancel{20}^{5 \times \cancel{4}}} = \frac{11}{5}$$ 5. Convert the improper fraction back to a mixed number. $$\frac{11}{5} = 2 \frac{1}{5}$$ 6. Therefore, the product of $2 \frac{3}{4}$ and $\frac{4}{5}$ is $2 \frac{1}{5}$.