First, convert the mixed fractions to improper fractions: $2\frac{2}{5} = \frac{(2 \times 5) + 2}{5} = \frac{12}{5}$ and $2\frac{1}{5} = \frac{(2 \times 5) + 1}{5} = \frac{11}{5}$. Now, perform the division: $\frac{12}{5} \div \frac{11}{5} = \frac{12}{5} \times \frac{5}{11} = \frac{12}{11}$. Since $\frac{12}{11} \neq 2$, the statement is false.