To determine which table represents a direct proportional relationship, let's analyze each table to see if there is a consistent ratio between $x$ and $y$ values. In a direct proportional relationship, $y$ should be equal to $k \times x$ for a constant $k$.

Table Analysis

1. Table A:

   • $k = \frac{y}{x}$
   • Ratios:
     • $\frac{5}{1} = 5$
     • $\frac{10}{2} = 5$
     • $\frac{15}{3} = 5$
     • $\frac{20}{4} = 5$
   • Since $k$ is constant (5), Table A represents a direct proportional relationship.

2. Table B:

   • Ratios:
     • $\frac{3}{1} = 3$
     • $\frac{6}{2} = 3$
     • $\frac{9}{3} = 3$
     • $\frac{12}{4} = 3$
   • $k$ is constant (3), so Table B also represents a direct proportional relationship.

3. Table C:

   • Ratios:
     • $\frac{2}{1} = 2$
     • $\frac{6}{2} = 3$
     • $\frac{8}{3} \approx 2.67$
     • $\frac{10}{4} = 2.5$
   • The ratio changes, so Table C does not represent a direct proportional relationship.

4. Table D:

   • Ratios:
     • $\frac{5}{1} = 5$
     • $\frac{9}{3} = 3$
     • $\frac{13}{5} \approx 2.6$
     • $\frac{17}{7} \approx 2.43$
   • The ratio is inconsistent, so Table D does not represent a direct proportional relationship.

Conclusion

Tables A and B both represent direct proportional relationships. If only one answer is allowed, select A or B based on the instructions.

Would you like more details on how to verify direct proportionality?

Further Questions

1. What defines a direct proportional relationship in mathematical terms?
2. How can we determine if a set of values is inversely proportional?
3. What are other methods to identify proportional relationships besides checking ratios?
4. How would this analysis change if there was an added constant (affine relationship)?
5. What practical examples use direct proportionality?

Tip

When verifying proportionality, always check if the ratio $\frac{y}{x}$ remains constant across all pairs.