1. **State the problem:** Simplify the expression $\left(\frac{3}{5} - \frac{2}{5} + \frac{1}{4}\right) + \frac{1}{6}$.\n\n2. **Combine the fractions inside the parentheses:** Since $\frac{3}{5}$ and $\frac{2}{5}$ have the same denominator, subtract them first:\n$$\frac{3}{5} - \frac{2}{5} = \frac{3-2}{5} = \frac{1}{5}.$$\n\n3. **Add $\frac{1}{5}$ and $\frac{1}{4}$:** Find a common denominator, which is 20:\n$$\frac{1}{5} = \frac{4}{20}, \quad \frac{1}{4} = \frac{5}{20}.$$\nSo,\n$$\frac{4}{20} + \frac{5}{20} = \frac{4+5}{20} = \frac{9}{20}.$$\n\n4. **Add $\frac{9}{20}$ and $\frac{1}{6}$:** Find a common denominator, which is 60:\n$$\frac{9}{20} = \frac{27}{60}, \quad \frac{1}{6} = \frac{10}{60}.$$\nSo,\n$$\frac{27}{60} + \frac{10}{60} = \frac{27+10}{60} = \frac{37}{60}.$$\n\n5. **Final answer:**\n$$\boxed{\frac{37}{60}}.$$