https://file.aiquickdraw.com/mathbot/file/17278807395624i2eruvl.jpg

To determine which representation shows a proportional relationship between \( x \) and \( y \), we need to analyze each option.

A proportional relationship means that the ratio \( \frac{y}{x} \) is constant for all values of \( x \) and \( y \). Let's evaluate each of the given options:

**A:**  
For the table in A, we calculate the ratios \( \frac{y}{x} \):
- \( \frac{3}{5} = 0.6 \)
- \( \frac{5}{7} \approx 0.714 \)
- \( \frac{19}{21} \approx 0.905 \)
- \( \frac{22}{24} \approx 0.917 \)

The ratios are not constant, so this is **not** a proportional relationship.

**B:**  
For the table in B, we calculate the ratios \( \frac{y}{x} \):
- \( \frac{5}{3} \approx 1.667 \)
- \( \frac{20}{12} \approx 1.667 \)
- \( \frac{25}{15} \approx 1.667 \)
- \( \frac{45}{27} \approx 1.667 \)

The ratios are constant, so this is a **proportional relationship**.

**C:**  
The graph in C shows a parabolic curve, which represents a quadratic function. Quadratic functions are not proportional relationships.

**D:**  
The graph in D is a straight line through the origin. Linear relationships through the origin represent proportional relationships. This could also be a proportional relationship.

### Conclusion:
Both **B** and **D** represent proportional relationships, but **B** fits the typical format of a proportional table, and **D** is a graphical representation of the same concept.

Answer: **B**

Would you like more details or have any other questions?

Here are 5 related questions to explore further:
1. How can you identify proportional relationships in equations?
2. What is the difference between proportional and linear relationships?
3. Why must a proportional relationship pass through the origin in a graph?
4. Can quadratic equations ever represent proportional relationships?
5. How do real-life scenarios apply proportional relationships?

**Tip:** In a graph, a proportional relationship always passes through the origin and forms a straight line.

Which representation shows a proportional relationship between x and y?

Learn how to identify proportional relationships between x and y using tables and graphs. This problem covers key concepts like ratios, linearity, and proportionality. Suitable for students in Grades 6-8.

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