https://file.aiquickdraw.com/mathbot/file/173013227548022qoe2jc.jpg

To determine if the table shows a proportional relationship and to find the constant of proportionality, let’s examine the values in the table.

The table values are as follows:

\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 3 & 6 & 9 & 12 \\
y & 12 & 24 & 45 & 60 \\
\hline
\end{array}
\]

For a proportional relationship, the ratio \(\frac{y}{x}\) should be constant across all pairs.

1. For \(x = 3\) and \(y = 12\): \(\frac{y}{x} = \frac{12}{3} = 4\).
2. For \(x = 6\) and \(y = 24\): \(\frac{y}{x} = \frac{24}{6} = 4\).
3. For \(x = 9\) and \(y = 45\): \(\frac{y}{x} = \frac{45}{9} = 5\).
4. For \(x = 12\) and \(y = 60\): \(\frac{y}{x} = \frac{60}{12} = 5\).

Since the ratio \(\frac{y}{x}\) is not consistent (it changes from 4 to 5), the values do not form a proportional relationship.

**Answer:** The correct choice is:
- **No; the quantities are not proportional.**

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

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Here are 5 related questions to deepen understanding:
1. How can you identify a proportional relationship in a table of values?
2. What are common methods to calculate the constant of proportionality?
3. How do proportional relationships differ from linear relationships?
4. What is the significance of a constant ratio in a proportional relationship?
5. How does one solve for missing values in a proportional relationship?

**Tip:** Always check that the ratio remains the same across all pairs when identifying proportional relationships.

Does this table show a proportional relationship? If so, what is the constant of proportionality?

| x | 3 | 6 | 9 | 12 |
|---|---|---|---|----|
| y | 12 | 24 | 45 | 60 |

Determine if the values in a table show a proportional relationship and find the constant of proportionality. This problem covers ratios, constant of proportionality, and is suitable for Grade 7 students.

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