1. We are given that $\frac{3}{4}$ of a piece of metal has a mass of 15 kg. 2. Let the total mass of the piece of metal be $m$ kg. 3. We can write the equation: $$\frac{3}{4}m = 15$$ 4. To find $m$, multiply both sides by $\frac{4}{3}$: $$m = 15 \times \frac{4}{3} = 15 \times \frac{4}{3} = 20$$ 5. So, the total mass of the piece of metal is 20 kg. 6. Now, we need to find the mass of $\frac{2}{5}$ of the piece. 7. Multiply the total mass by $\frac{2}{5}$: $$\frac{2}{5} \times 20 = \frac{2}{5} \times 20 = 8$$ 8. Therefore, the mass of $\frac{2}{5}$ of the piece of metal is 8 kg.