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!