To determine which of the tables represents a proportional relationship, we need to check if the ratio of \( y \) to \( x \) is constant for each pair of values in the table.
Table A:
- \( (2, 3) \): \( \frac{3}{2} = 1.5 \)
- \( (3, 6) \): \( \frac{6}{3} = 2 \)
- \( (4, 9) \): \( \frac{9}{4} = 2.25 \)
- \( (5, 12) \): \( \frac{12}{5} = 2.4 \)
Not proportional (different ratios).
Table B:
- \( (2, 3) \): \( \frac{3}{2} = 1.5 \)
- \( (4, 4) \): \( \frac{4}{4} = 1 \)
- \( (6, 5) \): \( \frac{5}{6} \approx 0.833 \)
- \( (8, 6) \): \( \frac{6}{8} = 0.75 \)
Not proportional (different ratios).
Table C:
- \( (2, 3) \): \( \frac{3}{2} = 1.5 \)
- \( (4, 6) \): \( \frac{6}{4} = 1.5 \)
- \( (6, 9) \): \( \frac{9}{6} = 1.5 \)
- \( (8, 12) \): \( \frac{12}{8} = 1.5 \)
Proportional (constant ratio of 1.5).
Table D:
- \( (1, 0) \): \( \frac{0}{1} = 0 \)
- \( (4, 3) \): \( \frac{3}{4} = 0.75 \)
- \( (7, 6) \): \( \frac{6}{7} \approx 0.857 \)
- \( (10, 9) \): \( \frac{9}{10} = 0.9 \)
Not proportional (different ratios).
Conclusion:
Table C represents a proportional relationship.