The constant of proportionality $k$ between $y$ and $x$ can be found using the formula:

$k = \frac{y}{x}$

We need to determine which table consistently has $k = 10$ across all pairs of $(x, y)$.

Choice A:

• For $(x = 2, y = 20)$, $k = \frac{20}{2} = 10$
• For $(x = 12, y = 132)$, $k = \frac{132}{12} = 11$
• For $(x = 22, y = 220)$, $k = \frac{220}{22} = 10$

The values of $k$ are not consistent (10, 11, 10), so this table is not proportional.

Choice B:

• For $(x = 5, y = 20)$, $k = \frac{20}{5} = 4$
• For $(x = 7, y = 30)$, $k = \frac{30}{7} \approx 4.29$
• For $(x = 10, y = 40)$, $k = \frac{40}{10} = 4$

The values of $k$ are also not consistent (4, 4.29, 4), so this table is not proportional.

Choice C:

• For $(x = 9, y = 90)$, $k = \frac{90}{9} = 10$
• For $(x = 14, y = 140)$, $k = \frac{140}{14} = 10$
• For $(x = 24, y = 240)$, $k = \frac{240}{24} = 10$

The values of $k$ are all consistent (10, 10, 10), so this table is proportional with a constant of proportionality of 10.

Conclusion:

The correct answer is: Choice C

Would you like a detailed explanation of the process? Or do you have any specific questions?

Here are 5 related questions for further understanding:

1. How do you find the constant of proportionality from a graph?
2. Can a table have more than one constant of proportionality?
3. What would a table with a constant of proportionality of 5 look like?
4. How can you verify proportionality using equations instead of tables?
5. If $y = 10x$, what is the value of $y$ when $x = 15$?

Tip: When checking for proportionality in a table, ensure that the ratio $\frac{y}{x}$ is the same for all pairs.