Decide Whether Each Table Could Represent A Proportional Relationship. If The Relationship Could Be Proportional, What Would Be The Constant Of Proportionality?a. Annie's Attic Is Giving Away \$5 Off
**Deciding on Proportional Relationships: A Guide to Identifying Constants of Proportionality**
Understanding Proportional Relationships
A proportional relationship is a relationship between two variables where one variable is a constant multiple of the other. In other words, if we have two variables, x and y, a proportional relationship exists if y = kx, where k is a constant. This constant, k, is known as the constant of proportionality.
Identifying Proportional Relationships
To determine if a table represents a proportional relationship, we need to examine the relationship between the variables. Here are some steps to follow:
- Examine the table: Look at the table and identify the variables. Are they related in a way that suggests a proportional relationship?
- Check for a constant ratio: If the table represents a proportional relationship, the ratio of the variables should be constant. In other words, if we divide one variable by the other, the result should be the same for all pairs of values.
- Calculate the constant of proportionality: If the table represents a proportional relationship, we can calculate the constant of proportionality by dividing one variable by the other.
Example 1: Annie's Attic
Let's consider the table from Annie's Attic, which gives away $5 off for every 10 items purchased.
| Number of Items | Discount ($5) |
|---|---|
| 10 | 5 |
| 20 | 10 |
| 30 | 15 |
| 40 | 20 |
Is this a proportional relationship?
To determine if this is a proportional relationship, we need to examine the ratio of the number of items to the discount.
| Number of Items | Discount ($5) | Ratio |
|---|---|---|
| 10 | 5 | 0.5 |
| 20 | 10 | 0.5 |
| 30 | 15 | 0.5 |
| 40 | 20 | 0.5 |
As we can see, the ratio of the number of items to the discount is constant, which suggests that this is a proportional relationship.
What is the constant of proportionality?
To calculate the constant of proportionality, we can divide the number of items by the discount.
k = Number of Items / Discount = 10 / 5 = 2
Therefore, the constant of proportionality is 2.
Q&A
Q: What is a proportional relationship?
A: A proportional relationship is a relationship between two variables where one variable is a constant multiple of the other.
Q: How do I identify a proportional relationship?
A: To identify a proportional relationship, examine the table and check for a constant ratio between the variables.
Q: What is the constant of proportionality?
A: The constant of proportionality is a constant that represents the ratio of the variables in a proportional relationship.
Q: How do I calculate the constant of proportionality?
A: To calculate the constant of proportionality, divide one variable by the other.
Q: What if the table does not represent a proportional relationship?
A: If the table does not represent a proportional relationship, it may represent a different type of relationship, such as a linear or quadratic relationship.
Q: Can a proportional relationship have a negative constant of proportionality?
A: Yes, a proportional relationship can have a negative constant of proportionality.
Q: Can a proportional relationship have a zero constant of proportionality?
A: No, a proportional relationship cannot have a zero constant of proportionality.
Conclusion
In conclusion, a proportional relationship is a relationship between two variables where one variable is a constant multiple of the other. To identify a proportional relationship, examine the table and check for a constant ratio between the variables. If the table represents a proportional relationship, we can calculate the constant of proportionality by dividing one variable by the other.