Let's solve the problem step-by-step.

Problem (a):

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

Step 1: Solve inside the parentheses.

$\frac{1}{2} + \frac{1}{6}$ Find a common denominator (LCM of 2 and 6 is 6): $\frac{1}{2} = \frac{3}{6}, \quad \frac{1}{6} = \frac{1}{6}$ $\frac{1}{2} + \frac{1}{6} = \frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$

Step 2: Multiply $\frac{2}{5}$ by $\frac{2}{3}$.

$\frac{2}{5} \times \frac{2}{3} = \frac{4}{15}$

Step 3: Subtract $\frac{4}{15}$ from $\frac{5}{6}$.

Find a common denominator (LCM of 6 and 15 is 30): $\frac{5}{6} = \frac{25}{30}, \quad \frac{4}{15} = \frac{8}{30}$ $\frac{5}{6} - \frac{4}{15} = \frac{25}{30} - \frac{8}{30} = \frac{17}{30}$

Final Answer for (a):

$\boxed{\frac{17}{30}}$

Problem (b):

Expression: $\frac{5}{6} - \frac{2}{5} \times \frac{1}{2} + \frac{1}{6}$

Step 1: Multiply $\frac{2}{5}$ by $\frac{1}{2}$.

$\frac{2}{5} \times \frac{1}{2} = \frac{2}{10} = \frac{1}{5}$

Step 2: Subtract $\frac{1}{5}$ from $\frac{5}{6}$.

Find a common denominator (LCM of 6 and 5 is 30): $\frac{5}{6} = \frac{25}{30}, \quad \frac{1}{5} = \frac{6}{30}$ $\frac{5}{6} - \frac{1}{5} = \frac{25}{30} - \frac{6}{30} = \frac{19}{30}$

Step 3: Add $\frac{1}{6}$ to $\frac{19}{30}$.

Find a common denominator (LCM of 6 and 30 is 30): $\frac{1}{6} = \frac{5}{30}$ $\frac{19}{30} + \frac{5}{30} = \frac{24}{30} = \frac{4}{5}$

Final Answer for (b):

$\boxed{\frac{4}{5}}$

Let me know if you need detailed clarifications or have more questions!

Here are 5 related questions to practice:

1. How do you find the least common multiple (LCM) of two numbers?
2. What is the process of converting fractions to have a common denominator?
3. How do you multiply two fractions?
4. What is the order of operations for fractions?
5. How can fractions be simplified?

Tip: Always perform operations inside parentheses first and simplify fractions wherever possible to avoid mistakes.