Printable math worksheet for practicing the constant of proportionality with tables and missing values.

Let's solve each problem step by step. The key concept here is proportionality.

When two variables $ x $ and $ y $ are proportional, their relationship can be written as:

$$
y = kx
$$

where $ k $ is the constant of proportionality, and it can be found using:

$$
k = \frac{y}{x}
$$

We'll use this to find $ k $ from known values and then fill in missing values.

---

1)

Given:
| x | 9 | | 5 | |
|---|---|---|---|---|
| y | 36| 16| | 8 |

- Use first pair: $ x = 9, y = 36 $
$$
k = \frac{36}{9} = 4
$$

Now use $ k = 4 $ to find missing values.

- For $ y = 16 $, find $ x $:
$$
x = \frac{y}{k} = \frac{16}{4} = 4
$$

- For $ x = 5 $, find $ y $:
$$
y = kx = 4 \times 5 = 20
$$

- For $ y = 8 $, find $ x $:
$$
x = \frac{8}{4} = 2
$$

Completed table:
| x | 9 | 4 | 5 | 2 |
|---|---|---|---|---|
| y | 36| 16| 20| 8 |

✔ Constant of proportionality: $ k = 4 $

---

2)

| x | | 6 | 15 | 9 |
|---|---|---|----|---|
| y | 4 | 2 | | |

Use $ x = 6, y = 2 $:
$$
k = \frac{2}{6} = \frac{1}{3}
$$

Now apply $ k = \frac{1}{3} $:

- For $ y = 4 $, find $ x $:
$$
x = \frac{y}{k} = \frac{4}{1/3} = 12
$$

- For $ x = 15 $, find $ y $:
$$
y = kx = \frac{1}{3} \times 15 = 5
$$

- For $ x = 9 $, find $ y $:
$$
y = \frac{1}{3} \times 9 = 3
$$

Completed table:
| x | 12 | 6 | 15 | 9 |
|---|----|---|----|---|
| y | 4 | 2 | 5 | 3 |

✔ $ k = \frac{1}{3} $

---

3)

| x | 12 | | 30 | 48 |
|---|----|---|----|----|
| y | | 3 | | 8 |

Use $ x = 12 $, $ y = ? $ — wait, we don’t know both. But look at second column: $ y = 3 $, but $ x $ is missing.

Wait — let’s check for consistency.

Try using $ x = 30 $, $ y = ? $, but no value. Wait — third column has $ x = 30 $, but $ y $ missing.

But fourth column: $ x = 48, y = 8 $. Let’s use that:

$$
k = \frac{y}{x} = \frac{8}{48} = \frac{1}{6}
$$

Check with $ x = 30 $, $ y = ? $:
$$
y = \frac{1}{6} \times 30 = 5
$$

Now $ x = 12 $, $ y = \frac{1}{6} \times 12 = 2 $

Now $ y = 3 $, so $ x = \frac{3}{1/6} = 18 $

So:

- $ x = 12 \Rightarrow y = 2 $
- $ x = ? \Rightarrow y = 3 \Rightarrow x = 18 $
- $ x = 30 \Rightarrow y = 5 $
- $ x = 48 \Rightarrow y = 8 $

Completed table:
| x | 12 | 18 | 30 | 48 |
|---|----|----|----|----|
| y | 2 | 3 | 5 | 8 |

✔ $ k = \frac{1}{6} $

---

4)

| x | | | 4 | |
|---|---|---|---|---|
| y | 25 | 40 | 20 | 60 |

Use $ x = 4, y = 20 $:
$$
k = \frac{20}{4} = 5
$$

Now use $ k = 5 $:

- $ y = 25 \Rightarrow x = \frac{25}{5} = 5 $
- $ y = 40 \Rightarrow x = \frac{40}{5} = 8 $
- $ y = 60 \Rightarrow x = \frac{60}{5} = 12 $

So:

| x | 5 | 8 | 4 | 12 |
|---|---|---|---|----|
| y | 25 | 40 | 20 | 60 |

✔ $ k = 5 $

---

5)

| x | 7 | 11 | 4 | |
|---|---|----|---|---|
| y | | | 32 | 24 |

Use $ x = 4, y = 32 $:
$$
k = \frac{32}{4} = 8
$$

Now:

- $ x = 7 \Rightarrow y = 8 \times 7 = 56 $
- $ x = 11 \Rightarrow y = 8 \times 11 = 88 $
- $ y = 24 \Rightarrow x = \frac{24}{8} = 3 $

Completed table:
| x | 7 | 11 | 4 | 3 |
|---|---|----|---|---|
| y | 56 | 88 | 32 | 24 |

✔ $ k = 8 $

---

6)

| x | | 14 | | |
|---|---|----|---|---|
| y | 21 | 6 | 15 | 3 |

Use $ x = 14, y = 6 $:
$$
k = \frac{6}{14} = \frac{3}{7}
$$

Now:

- $ y = 21 \Rightarrow x = \frac{21}{3/7} = 21 \times \frac{7}{3} = 49 $
- $ y = 15 \Rightarrow x = \frac{15}{3/7} = 15 \times \frac{7}{3} = 35 $
- $ y = 3 \Rightarrow x = \frac{3}{3/7} = 3 \times \frac{7}{3} = 7 $

So:

| x | 49 | 14 | 35 | 7 |
|---|----|----|----|---|
| y | 21 | 6 | 15 | 3 |

✔ $ k = \frac{3}{7} $

---

7)

| x | | 5 | | 30 |
|---|---|---|---|----|
| y | 14 | | 6 | 12 |

Use $ x = 30, y = 12 $:
$$
k = \frac{12}{30} = \frac{2}{5}
$$

Now:

- $ y = 14 \Rightarrow x = \frac{14}{2/5} = 14 \times \frac{5}{2} = 35 $
- $ x = 5 \Rightarrow y = \frac{2}{5} \times 5 = 2 $
- $ y = 6 \Rightarrow x = \frac{6}{2/5} = 6 \times \frac{5}{2} = 15 $

Completed table:
| x | 35 | 5 | 15 | 30 |
|---|----|---|----|----|
| y | 14 | 2 | 6 | 12 |

✔ $ k = \frac{2}{5} $

---

8)

| x | 3 | 4 | | 7 |
|---|---|---|---|---|
| y | 18 | | 54 | |

Use $ x = 3, y = 18 $:
$$
k = \frac{18}{3} = 6
$$

Now:

- $ x = 4 \Rightarrow y = 6 \times 4 = 24 $
- $ y = 54 \Rightarrow x = \frac{54}{6} = 9 $
- $ x = 7 \Rightarrow y = 6 \times 7 = 42 $

Completed table:
| x | 3 | 4 | 9 | 7 |
|---|---|---|---|---|
| y | 18 | 24 | 54 | 42 |

✔ $ k = 6 $

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✔ Final Answers:

| Problem | Constant (k) | Completed Table |
|--------|--------------|------------------|
| 1 | 4 | x: 9, 4, 5, 2; y: 36, 16, 20, 8 |
| 2 | 1/3 | x: 12, 6, 15, 9; y: 4, 2, 5, 3 |
| 3 | 1/6 | x: 12, 18, 30, 48; y: 2, 3, 5, 8 |
| 4 | 5 | x: 5, 8, 4, 12; y: 25, 40, 20, 60 |
| 5 | 8 | x: 7, 11, 4, 3; y: 56, 88, 32, 24 |
| 6 | 3/7 | x: 49, 14, 35, 7; y: 21, 6, 15, 3 |
| 7 | 2/5 | x: 35, 5, 15, 30; y: 14, 2, 6, 12 |
| 8 | 6 | x: 3, 4, 9, 7; y: 18, 24, 54, 42 |

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