1. **State the problem:** Solve for the unknown in the equation $$2 \frac{2}{5} \times [\ ] = \frac{2}{5}$$. 2. **Convert the mixed number to an improper fraction:** $$2 \frac{2}{5} = \frac{2 \times 5 + 2}{5} = \frac{10 + 2}{5} = \frac{12}{5}$$. 3. **Rewrite the equation with improper fractions:** $$\frac{12}{5} \times x = \frac{2}{5}$$ where $x$ is the unknown. 4. **Isolate $x$ by dividing both sides by $\frac{12}{5}$:** $$x = \frac{\frac{2}{5}}{\frac{12}{5}}$$ 5. **Divide fractions by multiplying by the reciprocal:** $$x = \frac{2}{5} \times \frac{5}{12}$$ 6. **Simplify the multiplication:** $$x = \frac{2 \times 5}{5 \times 12} = \frac{10}{60}$$ 7. **Reduce the fraction to simplest form:** $$\frac{10}{60} = \frac{1}{6}$$ **Final answer:** $$x = \frac{1}{6}$$