S varies directly as p and q. If p=6 and q=2 then S=54. Find the constant of proportionality.

The relationship you described is a direct variation, which can be written as:

\[
S = k \cdot p \cdot q
\]

Where \( k \) is the constant of proportionality, and \( S \), \( p \), and \( q \) are the given variables.

To find \( k \), substitute the values of \( S = 54 \), \( p = 6 \), and \( q = 2 \) into the equation:

\[
54 = k \cdot 6 \cdot 2
\]

Simplifying:

\[
54 = k \cdot 12
\]

Now, solve for \( k \):

\[
k = \frac{54}{12} = 4.5
\]

So, the constant of proportionality is \( k = 4.5 \).

Would you like further details or have any questions?

Here are 5 related questions:
1. If \( S \) varies directly as \( p \) and \( q \), how would \( S \) change if both \( p \) and \( q \) double?
2. Given \( S = k \cdot p \cdot q \), how would you express \( p \) in terms of \( S \), \( k \), and \( q \)?
3. If \( S = 100 \), \( p = 8 \), and \( q = 3 \), find the new value of \( k \).
4. If the constant of proportionality is \( k = 5 \), and \( p = 7 \), \( q = 2 \), find \( S \).
5. Can you explain the difference between direct variation and inverse variation?

**Tip:** Always double-check units when working with proportional relationships to ensure consistency throughout your calculations.

Learn how to find the constant of proportionality when S varies directly as p and q with given values p = 6, q = 2, and S = 54. This problem demonstrates the formula S = k * p * q, and is suitable for students in Grades 7-9.

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