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.

This solution simplifies the algebraic expression (b/30 + 3b/30) ÷ b + (5b/2 - 1/2) step by step. Key concepts covered include combining fractions, dividing algebraic expressions, and simplifying fractions. Suitable for students in Grades 9-10.

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