Let's simplify each expression one by one.

1. Expression: $\frac{3}{8} - \left(-\frac{1}{5}\right) + \left(-\frac{1}{2}\right) \div \frac{1}{6} \times \frac{1}{8}$

1. Simplify $\frac{3}{8} - \left(-\frac{1}{5}\right)$: $\frac{3}{8} + \frac{1}{5} = \frac{3 \times 5 + 1 \times 8}{40} = \frac{15 + 8}{40} = \frac{23}{40}$

2. Simplify $\left(-\frac{1}{2}\right) \div \frac{1}{6}$: $-\frac{1}{2} \times \frac{6}{1} = -\frac{6}{2} = -3$

3. Now multiply $-3$ by $\frac{1}{8}$: $-3 \times \frac{1}{8} = -\frac{3}{8}$

4. Combine $\frac{23}{40} + \left(-\frac{3}{8}\right)$: $\frac{23}{40} - \frac{3}{8} = \frac{23}{40} - \frac{15}{40} = \frac{8}{40} = \frac{1}{5}$

So, the simplified expression is $\frac{1}{5}$.

2. Expression: $\frac{3}{8}a - \frac{a}{6} + \frac{2}{3}a$

1. Find a common denominator for the coefficients:

   • The common denominator for $\frac{3}{8}, \frac{1}{6},$ and $\frac{2}{3}$ is 24.

2. Convert each term: $\frac{3}{8}a = \frac{9}{24}a, \quad -\frac{a}{6} = -\frac{4}{24}a, \quad \frac{2}{3}a = \frac{16}{24}a$

3. Combine the like terms: $\left(\frac{9}{24} - \frac{4}{24} + \frac{16}{24}\right)a = \frac{21}{24}a = \frac{7}{8}a$

So, the simplified expression is $\frac{7}{8}a$.

3. Expression: $\frac{4}{5} - \frac{x}{15} + \frac{x}{3} - \frac{1}{5}$

1. Combine the constants $\frac{4}{5}$ and $-\frac{1}{5}$: $\frac{4}{5} - \frac{1}{5} = \frac{3}{5}$

2. Simplify $-\frac{x}{15} + \frac{x}{3}$:

   • The common denominator for $\frac{1}{15}$ and $\frac{1}{3}$ is 15. $-\frac{x}{15} + \frac{5x}{15} = \frac{4x}{15}$

3. Combine the results: $\frac{3}{5} + \frac{4x}{15}$

   Converting $\frac{3}{5}$ to a denominator of 15: $\frac{9}{15} + \frac{4x}{15} = \frac{9 + 4x}{15}$

So, the simplified expression is $\frac{9 + 4x}{15}$.

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Would you like more details or have any questions?

Here are five related questions you might find interesting:

1. How do you find the least common denominator when adding fractions?
2. How can you simplify expressions with different variables?
3. What are some strategies for simplifying complex fractions?
4. How does multiplying and dividing fractions work with algebraic terms?
5. How do you approach solving equations that involve fractions?

Tip: Always look for common denominators when adding or subtracting fractions. This will make simplifying the expression much easier.