To determine which table shows a proportional relationship, we need to check if the ratio between "Hours" and "Earnings" is constant for each table. A proportional relationship exists when the ratio between the two variables is the same for all data points.

Let's check each option:

1. Table A:

   • Hours: 4, 6, 8
   • Earnings: 36, 54, 72
   • Ratios: $\frac{36}{4} = 9, \quad \frac{54}{6} = 9, \quad \frac{72}{8} = 9$ The ratio is constant (9), so Table A is proportional.

2. Table B:

   • Hours: 4, 6, 8
   • Earnings: 36, 45, 54
   • Ratios: $\frac{36}{4} = 9, \quad \frac{45}{6} = 7.5, \quad \frac{54}{8} = 6.75$ The ratios are not constant, so Table B is not proportional.

3. Table C:

   • Hours: 2, 3, 4
   • Earnings: 9, 18, 27
   • Ratios: $\frac{9}{2} = 4.5, \quad \frac{18}{3} = 6, \quad \frac{27}{4} = 6.75$ The ratios are not constant, so Table C is not proportional.

4. Table D:

   • Hours: 2, 3, 4
   • Earnings: 18, 27, 54
   • Ratios: $\frac{18}{2} = 9, \quad \frac{27}{3} = 9, \quad \frac{54}{4} = 13.5$ The ratios are not constant, so Table D is not proportional.

Conclusion:

Table A shows a proportional relationship because it has a constant ratio of 9 between hours and earnings.

Would you like further details or have any questions?

Follow-up questions:

1. What is a proportional relationship in mathematics?
2. How can you identify a proportional relationship from a graph?
3. Can a proportional relationship have a constant other than 1?
4. How do unit rates help in identifying proportional relationships?
5. Are there cases where the ratio can slightly change but still be considered proportional?

Tip: When checking for proportional relationships, always verify that the ratio between the two quantities remains constant across all pairs of values.