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The problem asks to find the possible values of the expression \(\frac{b}{2a}\), given the constraints:

\[
4 < a < 5 \quad \text{and} \quad 2 < b < 4.
\]

### Step-by-Step Solution:

1. **Determine the range of values for \(a\):**
   - Since \(4 < a < 5\), we know \(a\) lies between 4 and 5. Therefore, the minimum value of \(2a\) occurs when \(a\) is smallest (i.e., 4), and the maximum value occurs when \(a\) is largest (i.e., 5):
     \[
     2 \times 4 = 8 \quad \text{and} \quad 2 \times 5 = 10.
     \]
     So, \(8 < 2a < 10\).

2. **Determine the range of values for \(b\):**
   - Given \(2 < b < 4\), we can directly say that \(b\) lies between 2 and 4.

3. **Find the range of \(\frac{b}{2a}\):**
   - The expression we are interested in is \(\frac{b}{2a}\). To find its range, divide the range of \(b\) by the range of \(2a\):
     \[
     \frac{b}{2a} = \frac{b}{2a} \quad \text{where} \quad 2 < b < 4 \quad \text{and} \quad 8 < 2a < 10.
     \]

   - For the minimum value of \(\frac{b}{2a}\), take the smallest value of \(b\) and the largest value of \(2a\):
     \[
     \frac{2}{10} = 0.2.
     \]
   
   - For the maximum value of \(\frac{b}{2a}\), take the largest value of \(b\) and the smallest value of \(2a\):
     \[
     \frac{4}{8} = 0.5.
     \]

4. **Final range of \(\frac{b}{2a}\):**
   - Therefore, the possible values of \(\frac{b}{2a}\) lie between 0.2 and 0.5:
     \[
     0.2 < \frac{b}{2a} < 0.5.
     \]

### Conclusion:
The range of the expression \(\frac{b}{2a}\) is \((0.2, 0.5)\).

Would you like any further clarification or more details on any of the steps?

Here are 5 related questions to think about:
1. What happens if the inequality for \(a\) changes to \(4 \leq a \leq 5\)?
2. How would the solution change if the range for \(b\) was different, e.g., \(1 < b < 3\)?
3. What if the expression was \(\frac{2a}{b}\) instead? What would the range be?
4. How would the problem change if both \(a\) and \(b\) could be negative?
5. Can you generalize the process of finding the range of \(\frac{b}{2a}\) for other intervals?

**Tip:** Always ensure that you carefully check the boundaries of inequalities when dealing with expressions involving division!

If 4 < a < 5 and 2 < b < 4, find all possible values of the expression b / (2a).

Learn how to find the range of the expression b / (2a) given that 4 < a < 5 and 2 < b < 4. This problem involves understanding inequalities, fractions, and basic algebraic concepts. Suitable for students in Grades 8-10.

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