What is the constant of proportionality? Fill in the table below.

To find the constant of proportionality for the relationship between cups of tofu and cups of Sichuan chilis, we need to determine the values that fill in the table based on the relationships provided.

Let’s fill in the table step-by-step:

1. Cups of Tofu = 3

   To find how much it corresponds to Sichuan chilis: \[ 3 × ? = 4 ; 1/3 \]

   To solve for "?", we can express \( 4 ; 1/3 \) as an improper fraction: \[ 4 ; 1/3 = \frac{13}{3} \] So: \[ 3 × ? = \frac{13}{3} \] Now, we divide both sides by 3: \[ ? = \frac{13}{3} ÷ 3 = \frac{13}{3} × \frac{1}{3} = \frac{13}{9} \]

2. Next, fill in: Cups of Tofu = 6

   We now apply the constant of proportionality: \[ 6 × ? = 4 ; 1/3 \] Again, expressing \( 4 ; 1/3 \) as \( \frac{13}{3} \): \[ 6 × ? = \frac{13}{3} \] So: \[ ? = \frac{13}{3} ÷ 6 = \frac{13}{3} × \frac{1}{6} = \frac{13}{18} \]

3. Cups of Tofu = 4 ; 1/3 ÷ 3 = ?

   Here, we know already that \( 4 ; 1/3 = \frac{13}{3} \): \[ ? = \frac{13}{3} ÷ 3 = \frac{13}{3} × \frac{1}{3} = \frac{13}{9} \]

4. Cups of Tofu = 4 ; 1/3 ÷ \frac{1}{3} = ?

   This calculates how many times we can take \( \frac{1}{3} \) from \( \frac{13}{3} \): \[ ? = \frac{13}{3} ÷ \frac{1}{3} = \frac{13}{3} × 3 = 13 \]

5. Cups of Tofu = 4 ; 1/3 · \frac{3}{1} = ?

   Here, we multiply \( \frac{13}{3} \) by 3: \[ ? = \frac{13}{3} × 3 = 13 \]

Now, here is what the full table looks like after calculating everything:

| Cups of Tofu | Cups of Sichuan Chilis | |---------------|-------------------------| | 3 | \( \frac{13}{9} \) | | 6 | \( \frac{13}{18} \) | | \( \frac{4}{3} \) | \( \frac{13}{9} \) | | \( \frac{4}{3} \) | 13 | | \( \frac{4}{3} \) | 13 |

As a final note, the constant of proportionality with 4 ; 1/3 as one value could be identified based on the ratios between cups of tofu and cups of Sichuan chilis. Always verify calculations for correctness.