1. **State the problem:** Solve for $x$ in the equation $$\frac{x}{63+x} = 0.15.$$\n\n2. **Formula and rules:** This is a rational equation where $x$ is in the numerator and $63+x$ in the denominator. To solve, we multiply both sides by the denominator to clear the fraction.\n\n3. **Multiply both sides by $63+x$: $$x = 0.15(63+x)$$**\n\n4. **Distribute $0.15$: $$x = 0.15 \times 63 + 0.15x = 9.45 + 0.15x$$**\n\n5. **Isolate $x$ terms on one side: $$x - 0.15x = 9.45$$**\n\n6. **Simplify left side: $$0.85x = 9.45$$**\n\n7. **Divide both sides by $0.85$: $$x = \frac{9.45}{0.85}$$**\n\n8. **Calculate the division: $$x = 11.1176470588 \approx 11.12$$**\n\n**Final answer:** $$x \approx 11.12$$