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Let's analyze the second problem step-by-step.

### Given Information:

1. \( p \) and \( q \) are numbers such that their ratio is the same as the ratio of \(9\) to \(12\). This means:

\[
\frac{p}{q} = \frac{9}{12}
\]

2. It is given that:

\[
p^2 - q^2 = 28
\]

3. We need to determine the value of \(N\) where:

\[
N = \sqrt{2pq \cdot 3q - p \cdot q}
\]

### Step-by-Step Solution:

1. **Find the relationship between \(p\) and \(q\):**

   The given ratio \(\frac{p}{q} = \frac{9}{12}\) simplifies to:

\[
\frac{p}{q} = \frac{3}{4}
\]

   This means:

\[
p = \frac{3}{4}q
\]

2. **Use the equation \(p^2 - q^2 = 28\):**

   Substitute \(p = \frac{3}{4}q\) into the equation:

\[
\left( \frac{3}{4}q \right)^2 - q^2 = 28
\]

   Simplify the equation:

\[
\frac{9}{16}q^2 - q^2 = 28
\]

   Combine like terms:

\[
\frac{9}{16}q^2 - \frac{16}{16}q^2 = 28
\]

\[
-\frac{7}{16}q^2 = 28
\]

   Multiply both sides by \(-\frac{16}{7}\):

\[
q^2 = 28 \times -\frac{16}{-7} = 64
\]

   Solve for \(q\):

\[
q = \sqrt{64} = 8
\]

3. **Find the value of \(p\):**

   Substitute \(q = 8\) into \(p = \frac{3}{4}q\):

\[
p = \frac{3}{4} \times 8 = 6
\]

   So, \(p = 6\) and \(q = 8\).

4. **Calculate \(N\):**

   Substitute \(p = 6\) and \(q = 8\) into the formula for \(N\):

\[
N = \sqrt{2pq \cdot 3q - p \cdot q}
\]

   First, calculate each term:

\[
2pq = 2 \times 6 \times 8 = 96
\]

\[
3q = 3 \times 8 = 24
\]

\[
2pq \cdot 3q = 96 \times 24 = 2304
\]

\[
p \cdot q = 6 \times 8 = 48
\]

   Substitute these into the expression for \(N\):

\[
N = \sqrt{2304 - 48} = \sqrt{2256}
\]

   Finally, calculate the square root:

\[
N = 48
\]

### Conclusion:

The value of \(N\) is **48**.

Would you like more details on any step, or do you have any questions?

### Related Questions to Expand Your Understanding:

1. How would the problem change if the ratio between \(p\) and \(q\) was different?
2. What is the geometric interpretation of \(p^2 - q^2 = 28\)?
3. Can you solve the expression for \(N\) using different methods?
4. What would happen if the expression for \(N\) had different coefficients?
5. How do we solve similar problems involving different equations for \(p\) and \(q\)?

### Tip:

Always make sure to simplify ratios and equations as much as possible to reduce calculation complexity.

Explore the solution to a ratio and proportion problem involving quadratic equations. Learn how to find values using ratio simplification and quadratic equation solving methods. Suitable for Grades 9-10.

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