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To solve this problem, we need to find the constant of proportionality between the amount of concentrate \( x \) (in quarts) and the amount of water \( y \) (in quarts).

### Step-by-Step Solution:
1. **Understand the Graph**: The graph shows a linear relationship between \( x \) and \( y \).
   - The linear relationship means that \( y = kx \), where \( k \) is the constant of proportionality.
   - To find \( k \), we can use the slope formula from the graph: \( k = \frac{\Delta y}{\Delta x} \).

2. **Select Two Points**: From the graph, let’s take two points to determine the slope:
   - Point 1: \( (0, 0) \)
   - Point 2: \( (1, 0.5) \)

3. **Calculate the Slope (k)**:
   \[
   k = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0.5 - 0}{1 - 0} = 0.5
   \]

So, the constant of proportionality \( k \) is \( 0.5 \).

### Answer:
The correct answer is \( \frac{1}{2} \).

Would you like further details or explanations on this problem? Here are some related questions that can deepen your understanding:

1. What does the constant of proportionality represent in this context?
2. How would the graph look if the constant of proportionality was 1?
3. What would happen to the amount of water if we doubled the amount of concentrate?
4. How can you determine the constant of proportionality from a table of values?
5. What does it mean if a graph has a constant of proportionality greater than 1?

**Tip:** The constant of proportionality represents the rate of change between two variables in a direct variation. In a graph, it’s the slope of the line that passes through the origin.

Gabriel is making lemonade from concentrate. The relationship between the number of quarts of concentrate, x, and the number of quarts of water, y, is represented by the graph below. What is the constant of proportionality as shown in the graph?

Learn how to find the constant of proportionality between two variables using a graph. This problem involves determining the slope between points on a graph representing a linear relationship between concentrate and water. Ideal for students in Grades 6-8.

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