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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?

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### 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.

Choose the table that represents a direct proportional relationship among four given tables, each with values for x and y.

Analyze tables of x and y values to determine which table represents a direct proportional relationship. Learn how to verify direct proportionality by examining if the ratio y/x remains constant. Suitable for middle school students (Grades 6-8).

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