Let's simplify the expression step by step.

Given expression:

$\left(\frac{b}{30} + \frac{3b}{30}\right) \div b + \left(\frac{5b}{2} - \frac{1}{2}\right)$

Step 1: Simplify inside the parentheses

Combine like terms inside the first set of parentheses: $\frac{b}{30} + \frac{3b}{30} = \frac{b + 3b}{30} = \frac{4b}{30} = \frac{2b}{15}$

Now the expression becomes: $\left(\frac{2b}{15}\right) \div b + \left(\frac{5b}{2} - \frac{1}{2}\right)$

Step 2: Simplify the division by $b$

$\frac{2b}{15} \div b = \frac{2b}{15} \times \frac{1}{b} = \frac{2}{15}$

Now the expression becomes: $\frac{2}{15} + \left(\frac{5b}{2} - \frac{1}{2}\right)$

Step 3: Simplify the second term

Combine the two fractions: $\frac{5b}{2} - \frac{1}{2} = \frac{5b - 1}{2}$

Step 4: Final expression

Now the expression is: $\frac{2}{15} + \frac{5b - 1}{2}$

This is the simplified expression.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you combine fractions with different denominators?
2. What is the process for dividing algebraic expressions?
3. Can you simplify further if we know the value of $b$?
4. What happens if $b = 0$ in this expression?
5. How do we multiply fractions in algebra?

Tip: Always factor out common terms in fractions to simplify them quickly.