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!