1. State the problem: Solve for $x$ in the equation $$\frac{5}{8} - \frac{3}{5} = \frac{x}{10}.$$\n\n2. Find the least common denominator (LCD) of the fractions on the left side. The denominators are $8$ and $5$. The LCD is $$\text{lcm}(8,5) = 40.$$\n\n3. Convert each fraction to an equivalent fraction with denominator 40:\n$$\frac{5}{8} = \frac{5 \times 5}{8 \times 5} = \frac{25}{40},$$\n$$\frac{3}{5} = \frac{3 \times 8}{5 \times 8} = \frac{24}{40}.$$\n\n4. Subtract the two fractions:\n$$\frac{25}{40} - \frac{24}{40} = \frac{25 - 24}{40} = \frac{1}{40}.$$\n\n5. Now the equation is:\n$$\frac{1}{40} = \frac{x}{10}.$$\n\n6. Cross multiply to solve for $x$:\n$$1 \times 10 = 40 \times x,$$\nwhich simplifies to\n$$10 = 40x.$$\n\n7. Divide both sides by 40 to isolate $x$:\n$$x = \frac{10}{40} = \frac{1}{4} = 0.25.$$\n\nFinal answer: $$x = 0.25.$$