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The problem states that the amount of water is directly proportional to the number of cups of rice, and the point \( \left(\frac{1}{2}, 1\right) \) is given as a point on the graph of this proportional relationship. 

Since the relationship is proportional, the form of the equation is:

\[
y = kx
\]

where \( k \) is the constant of proportionality, \( x \) represents the number of cups of rice, and \( y \) represents the amount of water.

To find the constant \( k \), we can substitute the given point \( \left(\frac{1}{2}, 1\right) \) into the equation:

\[
1 = k \times \frac{1}{2}
\]

Solving for \( k \), we get:

\[
k = 2
\]

Now the equation of the line is:

\[
y = 2x
\]

Let's check the other points provided in the multiple choice options to see which ones satisfy this equation:

1. \( \left(\frac{2}{3}, 4\right) \): Substituting \( x = \frac{2}{3} \) into the equation:
   \[
   y = 2 \times \frac{2}{3} = \frac{4}{3} \neq 4
   \]
   This point is **not** on the graph.

2. \( \left(\frac{3}{2}, 3\right) \): Substituting \( x = \frac{3}{2} \) into the equation:
   \[
   y = 2 \times \frac{3}{2} = 3
   \]
   This point **is** on the graph.

3. \( (2, 6) \): Substituting \( x = 2 \) into the equation:
   \[
   y = 2 \times 2 = 4 \neq 6
   \]
   This point is **not** on the graph.

4. \( \left(\frac{1}{4}, 2\right) \): Substituting \( x = \frac{1}{4} \) into the equation:
   \[
   y = 2 \times \frac{1}{4} = \frac{1}{2} \neq 2
   \]
   This point is **not** on the graph.

Therefore, the correct point that lies on the graph is **\( \left(\frac{3}{2}, 3\right) \)**.

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

Here are some related questions:
1. How would you find the constant of proportionality with different points?
2. How can you graph a proportional relationship like this one?
3. What does the constant of proportionality tell you about the relationship?
4. How would the graph change if the proportional constant was different?
5. How do you interpret the slope in a direct proportionality?

**Tip:** In direct proportionality, as one quantity increases, the other increases at a constant rate, represented by the constant of proportionality.

The amount of water is directly proportional to the number of cups of rice. The coordinate (1/2, 1) is on the graph of this proportional relationship. Identify another point on the graph.

Learn how to find a point on the graph of a directly proportional relationship between water and cups of rice. This problem involves using the proportionality constant and linear equations. Suitable for students in Grades 6-8.

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