1. **State the problem:** Solve for $x$ in the equation $$\frac{1}{40} + \frac{1}{16} = \frac{1}{60} + \frac{1}{x}.$$\n\n2. **Write the formula and rules:** To solve for $x$, first combine the fractions on the left side by finding a common denominator, then isolate $\frac{1}{x}$ and solve for $x$.\n\n3. **Find common denominator for left side:** The denominators are 40 and 16. The least common multiple (LCM) is 80.\n\n$$\frac{1}{40} = \frac{2}{80}, \quad \frac{1}{16} = \frac{5}{80}.$$\n\n4. **Add the fractions on the left:**\n$$\frac{2}{80} + \frac{5}{80} = \frac{7}{80}.$$\n\n5. **Rewrite the equation:**\n$$\frac{7}{80} = \frac{1}{60} + \frac{1}{x}.$$\n\n6. **Isolate $\frac{1}{x}$:**\n$$\frac{1}{x} = \frac{7}{80} - \frac{1}{60}.$$\n\n7. **Find common denominator for right side:** The denominators are 80 and 60. The LCM is 240.\n\n$$\frac{7}{80} = \frac{21}{240}, \quad \frac{1}{60} = \frac{4}{240}.$$\n\n8. **Subtract the fractions:**\n$$\frac{1}{x} = \frac{21}{240} - \frac{4}{240} = \frac{17}{240}.$$\n\n9. **Solve for $x$ by taking reciprocal:**\n$$x = \frac{240}{17}.$$\n\n10. **Final answer:**\n$$x = \frac{240}{17} \approx 14.12.$$