Which table of ordered pairs represents a proportional relationship?

Question

A 2-column table with 3 rows. Column 1 is labeled x with entries 0, 5, 10. Column 2 is labeled y with entries 10, 20, 30.
A 2-column table with 3 rows. Column 1 is labeled x with entries 2, 4, 6. Column 2 is labeled y with entries 10, 20, 30.
A 2-column table with 3 rows. Column 1 is labeled x with entries 1, 2, 3. Column 2 is labeled y with entries 2, 3, 4.
A 2-column table with 3 rows. Column 1 is labeled x with entries 1, 3, 4. Column 2 is labeled y with entries 4, 10, 13.

Answer 1

To determine which table of ordered pairs represents a proportional relationship, we need to check if the ratio of \( y \) to \( x \) is constant for each pair of ordered pairs. A proportional relationship has the form \( y = kx \), where \( k \) is a constant.

Let's evaluate each table:

1. Table 1:

   • \( (0, 10) \) => \( \frac{y}{x} = \frac{10}{0} \) (undefined for \( x = 0\))
   • \( (5, 20) \) => \( \frac{20}{5} = 4 \)
   • \( (10, 30) \) => \( \frac{30}{10} = 3 \)
   • Not proportional (due to undefined ratio).

2. Table 2:

   • \( (2, 10) \) => \( \frac{10}{2} = 5 \)
   • \( (4, 20) \) => \( \frac{20}{4} = 5 \)
   • \( (6, 30) \) => \( \frac{30}{6} = 5 \)
   • Proportional (constant ratio of 5).

3. Table 3:

   • \( (1, 2) \) => \( \frac{2}{1} = 2 \)
   • \( (2, 3) \) => \( \frac{3}{2} = 1.5 \)
   • \( (3, 4) \) => \( \frac{4}{3} \approx 1.33 \)
   • Not proportional (different ratios).

4. Table 4:

   • \( (1, 4) \) => \( \frac{4}{1} = 4 \)
   • \( (3, 10) \) => \( \frac{10}{3} \approx 3.33 \)
   • \( (4, 13) \) => \( \frac{13}{4} = 3.25 \)
   • Not proportional (different ratios).

Based on the evaluations, Table 2 is the only one that represents a proportional relationship.