Worksheet for finding the constant of proportionality from tables of x and y values.

Let's solve each problem step by step.

We are asked to determine the constant of proportionality $ k $ for each table and write the proportional relationship between $ x $ and $ y $.

In a proportional relationship, $ y = kx $, where $ k $ is the constant of proportionality. To find $ k $, we divide $ y $ by $ x $ for each pair:
$$
k = \frac{y}{x}
$$
If the ratio $ \frac{y}{x} $ is the same for all pairs in the table, then the relationship is proportional, and that common value is $ k $.

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1)

| x | 2 | 5 | 7 | 9 |
|---|---|---|---|---|
| y | 16 | 40 | 56 | 72 |

Check ratios:
- $ \frac{16}{2} = 8 $
- $ \frac{40}{5} = 8 $
- $ \frac{56}{7} = 8 $
- $ \frac{72}{9} = 8 $

✔ All equal → $ k = 8 $

Relationship: $ y = 8x $

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2)

| x | 5 | 10 | 15 | 20 |
|---|---|----|----|----|
| y | 4 | 8 | 12 | 16 |

Ratios:
- $ \frac{4}{5} = 0.8 $
- $ \frac{8}{10} = 0.8 $
- $ \frac{12}{15} = 0.8 $
- $ \frac{16}{20} = 0.8 $

✔ All equal → $ k = 0.8 $ or $ \frac{4}{5} $

Relationship: $ y = 0.8x $ or $ y = \frac{4}{5}x $

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3)

| x | 5 | 9 | 11 | 13 |
|---|---|---|----|----|
| y | 10 | 18 | 22 | 26 |

Ratios:
- $ \frac{10}{5} = 2 $
- $ \frac{18}{9} = 2 $
- $ \frac{22}{11} = 2 $
- $ \frac{26}{13} = 2 $

✔ All equal → $ k = 2 $

Relationship: $ y = 2x $

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4)

| x | 1 | 3 | 4 | 5 |
|---|---|---|---|---|
| y | 9 | 27 | 36 | 45 |

Ratios:
- $ \frac{9}{1} = 9 $
- $ \frac{27}{3} = 9 $
- $ \frac{36}{4} = 9 $
- $ \frac{45}{5} = 9 $

✔ All equal → $ k = 9 $

Relationship: $ y = 9x $

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5)

| x | 6 | 8 | 10 | 12 |
|---|---|---|----|----|
| y | 18 | 24 | 30 | 36 |

Ratios:
- $ \frac{18}{6} = 3 $
- $ \frac{24}{8} = 3 $
- $ \frac{30}{10} = 3 $
- $ \frac{36}{12} = 3 $

✔ All equal → $ k = 3 $

Relationship: $ y = 3x $

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6)

| x | 2 | 5 | 6 | 8 |
|---|---|---|---|---|
| y | 12 | 30 | 36 | 48 |

Ratios:
- $ \frac{12}{2} = 6 $
- $ \frac{30}{5} = 6 $
- $ \frac{36}{6} = 6 $
- $ \frac{48}{8} = 6 $

✔ All equal → $ k = 6 $

Relationship: $ y = 6x $

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7)

| x | 9 | 18 | 27 | 36 |
|---|---|----|----|----|
| y | 7 | 14 | 21 | 28 |

Ratios:
- $ \frac{7}{9} \approx 0.777... $
- $ \frac{14}{18} = \frac{7}{9} \approx 0.777... $
- $ \frac{21}{27} = \frac{7}{9} $
- $ \frac{28}{36} = \frac{7}{9} $

✔ All equal → $ k = \frac{7}{9} $

Relationship: $ y = \frac{7}{9}x $

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8)

| x | 10 | 11 | 15 | 20 |
|---|----|----|----|----|
| y | 50 | 55 | 75 | 100 |

Ratios:
- $ \frac{50}{10} = 5 $
- $ \frac{55}{11} = 5 $
- $ \frac{75}{15} = 5 $
- $ \frac{100}{20} = 5 $

✔ All equal → $ k = 5 $

Relationship: $ y = 5x $

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✔ Final Answers:

1) $ k = 8 $, $ y = 8x $
2) $ k = 0.8 $, $ y = 0.8x $ or $ y = \frac{4}{5}x $
3) $ k = 2 $, $ y = 2x $
4) $ k = 9 $, $ y = 9x $
5) $ k = 3 $, $ y = 3x $
6) $ k = 6 $, $ y = 6x $
7) $ k = \frac{7}{9} $, $ y = \frac{7}{9}x $
8) $ k = 5 $, $ y = 5x $

All tables represent directly proportional relationships, as the ratio $ \frac{y}{x} $ is constant in each case.