1. **State the problem:** We are given that the sum of two numbers is 84, and one number is $\frac{2}{5}$ of the other. We need to find the smallest number. 2. **Set variables:** Let the larger number be $x$. Then the smaller number is $\frac{2}{5}x$. 3. **Write the equation for the sum:** $$x + \frac{2}{5}x = 84$$ 4. **Combine like terms:** $$\frac{5}{5}x + \frac{2}{5}x = \frac{7}{5}x = 84$$ 5. **Solve for $x$:** $$x = 84 \times \frac{5}{7} = 12 \times 5 = 60$$ 6. **Find the smaller number:** $$\frac{2}{5} \times 60 = 24$$ 7. **Answer:** The smallest number is $24$.