Question

Proportional relationships in tables

Answer 1

Proportional relationships can be represented in tables, graphs, equations, or verbal descriptions. In a proportional relationship, two quantities maintain a constant ratio. This means that as one quantity increases or decreases, the other does so in a consistent manner, and they can be expressed in the form:

\[ \frac{y}{x} = k \]

where \( k \) is a constant (the constant of proportionality).

Here’s how you can identify and create a table that showcases proportional relationships:

### Key Features of Proportional Relationships in Tables:

1. **Constant Ratio**: The ratio of the two quantities (often \( y/x \) or the other way around) is constant throughout the table.

2. **Linear Relationship**: When graphed, proportional relationships result in a straight line that passes through the origin (0,0).

### Example of a Proportional Relationship Table:

Let’s take the relationship between the number of hours worked and the amount earned, where a person earns $15 per hour.

| Hours Worked (x) | Earnings (y) |
|-------------------|--------------|
| 0 | 0 |
| 1 | 15 |
| 2 | 30 |
| 3 | 45 |
| 4 | 60 |

### Check for Proportionality:

1. **Calculate Ratios**:
- For each entry, calculate the ratio \( \frac{y}{x} \):
- \( \frac{0}{0} \) is undefined.
- \( \frac{15}{1} = 15 \)
- \( \frac{30}{2} = 15 \)
- \( \frac{45}{3} = 15 \)
- \( \frac{60}{4} = 15 \)

2. **Constant Value**: Here, the constant ratio \( k \) is 15, confirming a proportional relationship.

### Creating Your Own Table:

To create your own table showcasing a proportional relationship:

1. **Choose a Constant Ratio**: For example, you could choose a ratio of 4:1.
2. **Decide on Values for One Variable**: You could select values for \( x \) (e.g., 1, 2, 3, 4, 5).
3. **Calculate the Other Variable**: Use the constant ratio to find \( y \):
- \( y = k \cdot x \)

### Example Table:

For a ratio of 4:1:

| x (Value) | y (Value) |
|-----------|-----------|
| 1 | 4 |
| 2 | 8 |
| 3 | 12 |
| 4 | 16 |
| 5 | 20 |

### Conclusion:

Proportional relationships in tables are an effective way to visualize and understand the relationship between two variables that maintain a constant ratio. Be sure to check that the ratios are the same across all data points to confirm the proportionality!