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To determine if \( x \) and \( y \) have a proportional relationship, we need to check whether the ratio \( \frac{y}{x} \) is constant across all values in the table.

Let's calculate the ratio for each pair:

1. For \( x = 36 \) and \( y = 26 \):
   \[
   \frac{y}{x} = \frac{26}{36} \approx 0.722
   \]
2. For \( x = 60 \) and \( y = 32 \):
   \[
   \frac{y}{x} = \frac{32}{60} \approx 0.533
   \]
3. For \( x = 64 \) and \( y = 33 \):
   \[
   \frac{y}{x} = \frac{33}{64} \approx 0.516
   \]
4. For \( x = 100 \) and \( y = 42 \):
   \[
   \frac{y}{x} = \frac{42}{100} = 0.42
   \]

Since the ratios are not the same for each pair of values, the relationship between \( x \) and \( y \) is **not proportional**.

Would you like further explanation, or do you have any questions?

Here are five follow-up questions:
1. What does it mean for a relationship to be proportional?
2. Can you find a linear equation that describes this situation?
3. How would the graph of this relationship look?
4. What role does the flat fee play in affecting proportionality?
5. How could you check for proportionality using only the graph?

**Tip:** In a proportional relationship, all ratios between corresponding values must be the same, which translates to a graph that passes through the origin (0, 0).

According to the values in the table, do x and y have a proportional relationship?

Analyze whether the relationship between distance traveled and cost in a train ride is proportional. This example explores ratios, linear relationships, and the formula for proportionality with a step-by-step breakdown. Suitable for students in Grades 6-8.

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