Set of four algebraic expressions demonstrating multiplication and simplification of fractions.

The task involves solving and simplifying the given algebraic expressions. Let's go through each part step by step.

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Part (a):

$$
\frac{a}{2b} \times \frac{a}{c}
$$

#### Step 1: Multiply the fractions.
When multiplying fractions, multiply the numerators together and the denominators together:
$$
\frac{a}{2b} \times \frac{a}{c} = \frac{a \cdot a}{2b \cdot c} = \frac{a^2}{2bc}
$$

#### Final Answer for (a):
$$
\boxed{\frac{a^2}{2bc}}
$$

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Part (b):

$$
\frac{3x}{8y^2} \times \frac{y}{12}
$$

#### Step 1: Multiply the fractions.
Multiply the numerators and the denominators:
$$
\frac{3x}{8y^2} \times \frac{y}{12} = \frac{3x \cdot y}{8y^2 \cdot 12}
$$

#### Step 2: Simplify the expression.
- The numerator is $3xy$.
- The denominator is $8y^2 \cdot 12 = 96y^2$.
So, we have:
$$
\frac{3xy}{96y^2}
$$

#### Step 3: Cancel common factors.
- Both the numerator and the denominator have a factor of $y$. Cancel one $y$ from the numerator and the denominator:
$$
\frac{3x \cancel{y}}{96 \cancel{y} y} = \frac{3x}{96y}
$$
- Simplify the fraction $\frac{3x}{96y}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
$$
\frac{3x}{96y} = \frac{x}{32y}
$$

#### Final Answer for (b):
$$
\boxed{\frac{x}{32y}}
$$

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Part (c):

$$
\frac{4}{3b-1} \times \frac{b}{b-1}
$$

#### Step 1: Multiply the fractions.
Multiply the numerators and the denominators:
$$
\frac{4}{3b-1} \times \frac{b}{b-1} = \frac{4 \cdot b}{(3b-1) \cdot (b-1)}
$$

#### Step 2: Simplify the expression.
The numerator is $4b$, and the denominator is $(3b-1)(b-1)$. There are no common factors to cancel between the numerator and the denominator, so the expression is already in its simplest form:
$$
\frac{4b}{(3b-1)(b-1)}
$$

#### Final Answer for (c):
$$
\boxed{\frac{4b}{(3b-1)(b-1)}}
$$

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Part (d):

$$
\frac{p-q}{p+q} \times \frac{p+q}{2b}
$$

#### Step 1: Multiply the fractions.
Multiply the numerators and the denominators:
$$
\frac{p-q}{p+q} \times \frac{p+q}{2b} = \frac{(p-q) \cdot (p+q)}{(p+q) \cdot 2b}
$$

#### Step 2: Simplify the expression.
- The numerator is $(p-q)(p+q)$.
- The denominator is $(p+q) \cdot 2b$.
- Notice that $(p+q)$ appears in both the numerator and the denominator. Cancel $(p+q)$:
$$
\frac{(p-q) \cancel{(p+q)}}{\cancel{(p+q)} \cdot 2b} = \frac{p-q}{2b}
$$

#### Final Answer for (d):
$$
\boxed{\frac{p-q}{2b}}
$$

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Final Answers:

$$
\boxed{\frac{a^2}{2bc}, \frac{x}{32y}, \frac{4b}{(3b-1)(b-1)}, \frac{p-q}{2b}}
$$