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To solve this problem, we need to express the given expression \(\frac{y}{2} + \frac{(3 - y)^2}{4y}\) as a single fraction in its simplest form.

### Step-by-Step Solution:

1. **Rewrite the expression:**
   \[
   \frac{y}{2} + \frac{(3 - y)^2}{4y}
   \]

2. **Find a common denominator:**
   The denominators are \(2\) and \(4y\). To combine these terms, the common denominator will be \(4y\).

3. **Rewrite each term with the common denominator:**
   - For \(\frac{y}{2}\): multiply the numerator and denominator by \(2y\) to make the denominator \(4y\).
     \[
     \frac{y}{2} = \frac{2y^2}{4y}
     \]
   - The second term \(\frac{(3 - y)^2}{4y}\) is already over \(4y\), so it remains as it is.

   Now we can rewrite the expression as:
   \[
   \frac{2y^2}{4y} + \frac{(3 - y)^2}{4y}
   \]

4. **Combine the fractions:**
   \[
   \frac{2y^2 + (3 - y)^2}{4y}
   \]

5. **Expand \((3 - y)^2\):**
   \[
   (3 - y)^2 = 9 - 6y + y^2
   \]

   So the expression becomes:
   \[
   \frac{2y^2 + 9 - 6y + y^2}{4y}
   \]

6. **Combine like terms in the numerator:**
   \[
   2y^2 + y^2 - 6y + 9 = 3y^2 - 6y + 9
   \]

   Now we have:
   \[
   \frac{3y^2 - 6y + 9}{4y}
   \]

7. **Simplify the fraction if possible:**
   This is already in its simplest form. Therefore, the answer is:
   \[
   \frac{3y^2 - 6y + 9}{4y}
   \]

### Final Answer:
The correct answer is **D) \(\frac{3y^2 - 6y + 9}{4y}\)**.

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Do you need further details on any part of the solution?

### Related Questions:
1. How do you determine the common denominator when adding fractions?
2. Why is expanding \((3 - y)^2\) necessary for simplification?
3. What are other methods to simplify complex fractions?
4. How can we verify the correctness of our final expression?
5. How can similar expressions be simplified for different denominators?

### Tip:
When combining fractions, always ensure the denominators are the same before adding or subtracting the numerators.

Express \( \frac{y}{2} + \frac{(3 - y)^2}{4y} \) as a single fraction in its simplest form.

Learn how to simplify the algebraic expression \( \frac{y}{2} + \frac{(3 - y)^2}{4y} \) into a single fraction in its simplest form. This problem explores key algebraic concepts such as fraction addition, common denominators, and expanding square terms. Suitable for students in Grade 10.

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