Worksheet on finding the constant of proportionality in proportional relationships through graphs, tables, equations, and word problems.

Problem Overview:

The task involves finding the constant of proportionality for each given scenario and then using it to complete a table or equation. The constant of proportionality is the ratio between two directly proportional quantities, often represented as \( k \) in the equation \( y = kx \).

Let's solve each part step by step.

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Part 1:

#### Given:
- Table:
| \( x \) | 1 | 2 | 3 | 4 |
|---------|---|---|---|---|
| \( y \) | 2 | 4 | 6 | 8 |

#### Solution:
1. Identify the relationship: The values of \( y \) are increasing linearly with \( x \). This suggests a direct proportionality.
2. Find the constant of proportionality (\( k \)):
- From the table, when \( x = 1 \), \( y = 2 \).
- The constant of proportionality is \( k = \frac{y}{x} = \frac{2}{1} = 2 \).
3. Verify with other points:
- When \( x = 2 \), \( y = 4 \): \( k = \frac{4}{2} = 2 \).
- When \( x = 3 \), \( y = 6 \): \( k = \frac{6}{3} = 2 \).
- When \( x = 4 \), \( y = 8 \): \( k = \frac{8}{4} = 2 \).
- The constant \( k \) is consistent across all points.
4. Equation:
- The equation is \( y = 2x \).

#### Final Answer for Part 1:
\[ \boxed{2} \]

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Part 2:

#### Given:
- Table:
| \( x \) | 1 | 2 | 3 | 4 |
|---------|---|---|---|---|
| \( y \) | 1 | 2 | 3 | 4 |

#### Solution:
1. Identify the relationship: The values of \( y \) are increasing linearly with \( x \). This suggests a direct proportionality.
2. Find the constant of proportionality (\( k \)):
- From the table, when \( x = 1 \), \( y = 1 \).
- The constant of proportionality is \( k = \frac{y}{x} = \frac{1}{1} = 1 \).
3. Verify with other points:
- When \( x = 2 \), \( y = 2 \): \( k = \frac{2}{2} = 1 \).
- When \( x = 3 \), \( y = 3 \): \( k = \frac{3}{3} = 1 \).
- When \( x = 4 \), \( y = 4 \): \( k = \frac{4}{4} = 1 \).
- The constant \( k \) is consistent across all points.
4. Equation:
- The equation is \( y = x \).

#### Final Answer for Part 2:
\[ \boxed{1} \]

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Part 3:

#### Given:
- Table:
| \( x \) | 1 | 2 | 3 | 4 |
|---------|---|---|---|---|
| \( y \) | 3 | 6 | 9 | 12 |

#### Solution:
1. Identify the relationship: The values of \( y \) are increasing linearly with \( x \). This suggests a direct proportionality.
2. Find the constant of proportionality (\( k \)):
- From the table, when \( x = 1 \), \( y = 3 \).
- The constant of proportionality is \( k = \frac{y}{x} = \frac{3}{1} = 3 \).
3. Verify with other points:
- When \( x = 2 \), \( y = 6 \): \( k = \frac{6}{2} = 3 \).
- When \( x = 3 \), \( y = 9 \): \( k = \frac{9}{3} = 3 \).
- When \( x = 4 \), \( y = 12 \): \( k = \frac{12}{4} = 3 \).
- The constant \( k \) is consistent across all points.
4. Equation:
- The equation is \( y = 3x \).

#### Final Answer for Part 3:
\[ \boxed{3} \]

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Part 4:

#### Given:
- Table:
| \( x \) | 5 | 10 | 15 | 20 |
|---------|---|----|----|----|
| \( y \) | 2 | 4 | 6 | 8 |

#### Solution:
1. Identify the relationship: The values of \( y \) are increasing linearly with \( x \). This suggests a direct proportionality.
2. Find the constant of proportionality (\( k \)):
- From the table, when \( x = 5 \), \( y = 2 \).
- The constant of proportionality is \( k = \frac{y}{x} = \frac{2}{5} \).
3. Verify with other points:
- When \( x = 10 \), \( y = 4 \): \( k = \frac{4}{10} = \frac{2}{5} \).
- When \( x = 15 \), \( y = 6 \): \( k = \frac{6}{15} = \frac{2}{5} \).
- When \( x = 20 \), \( y = 8 \): \( k = \frac{8}{20} = \frac{2}{5} \).
- The constant \( k \) is consistent across all points.
4. Equation:
- The equation is \( y = \frac{2}{5}x \).

#### Final Answer for Part 4:
\[ \boxed{\frac{2}{5}} \]

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Part 5:

#### Given:
- Table:
| \( x \) | 1 | 2 | 3 | 4 |
|---------|---|---|---|---|
| \( y \) | 1.25 | 2.5 | 3.75 | 5 |

#### Solution:
1. Identify the relationship: The values of \( y \) are increasing linearly with \( x \). This suggests a direct proportionality.
2. Find the constant of proportionality (\( k \)):
- From the table, when \( x = 1 \), \( y = 1.25 \).
- The constant of proportionality is \( k = \frac{y}{x} = \frac{1.25}{1} = 1.25 \).
3. Verify with other points:
- When \( x = 2 \), \( y = 2.5 \): \( k = \frac{2.5}{2} = 1.25 \).
- When \( x = 3 \), \( y = 3.75 \): \( k = \frac{3.75}{3} = 1.25 \).
- When \( x = 4 \), \( y = 5 \): \( k = \frac{5}{4} = 1.25 \).
- The constant \( k \) is consistent across all points.
4. Equation:
- The equation is \( y = 1.25x \).

#### Final Answer for Part 5:
\[ \boxed{1.25} \]

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Final Answers:

1. \( \boxed{2} \)
2. \( \boxed{1} \)
3. \( \boxed{3} \)
4. \( \boxed{\frac{2}{5}} \)
5. \( \boxed{1.25} \)