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The problem presented asks for the constant of proportionality between the number of tickets purchased (denoted by $ x $) and the total cost in dollars (denoted by $ y $). 

To find the constant of proportionality, we divide the total cost by the number of tickets for each pair of values in the table, and if the relationship is proportional, the result should be the same for all data points.

Let's calculate:

- For $ x = 8 $ tickets and $ y = 6.00 $:  
  $ \frac{y}{x} = \frac{6.00}{8} = 0.75 $

- For $ x = 12 $ tickets and $ y = 9.00 $:  
  $ \frac{y}{x} = \frac{9.00}{12} = 0.75 $

- For $ x = 20 $ tickets and $ y = 15.00 $:  
  $ \frac{y}{x} = \frac{15.00}{20} = 0.75 $

- For $ x = 30 $ tickets and $ y = 22.50 $:  
  $ \frac{y}{x} = \frac{22.50}{30} = 0.75 $

- For $ x = 50 $ tickets and $ y = 37.50 $:  
  $ \frac{y}{x} = \frac{37.50}{50} = 0.75 $

Since the ratio $ \frac{y}{x} $ is constant across all the data points, the constant of proportionality is $ 0.75 $. Therefore, the correct answer is:

**Option b: 0.75**

Would you like more details on this solution or have any further questions?

Here are 5 related questions for practice:
1. What is the formula for calculating the constant of proportionality in any proportional relationship?
2. How would the table change if the constant of proportionality was 1.25 instead?
3. If the cost of 60 tickets is $45, is the relationship still proportional? Why or why not?
4. What does the constant of proportionality represent in real-world terms for this arcade example?
5. How can you identify proportionality in a graph?

**Tip**: When dealing with proportional relationships, the constant of proportionality is the unit rate, often a useful way to interpret ratios in real-world contexts.

What is the constant of proportionality that relates the total cost in dollars to the number of tickets purchased?

Learn how to find the constant of proportionality in ticket pricing for arcade games. This problem explores proportional relationships using data on the number of tickets purchased and their corresponding costs. Suitable for students in Grades 6-8.

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