select the tables that show a proportional relationship between x and y.

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2 min read 31-03-2025

select the tables that show a proportional relationship between x and y.

Understanding proportional relationships is crucial in mathematics. This article will guide you through identifying tables that demonstrate a proportional relationship between variables x and y. We'll define what a proportional relationship is, provide examples, and show you how to spot them in tables of data.

What is a Proportional Relationship?

A proportional relationship exists between two variables, x and y, if their ratio remains constant. This means that as x increases or decreases, y increases or decreases proportionally. We can express this relationship mathematically as:

y = kx

where 'k' is the constant of proportionality. This constant represents the ratio between y and x. It's the number you multiply x by to get y.

Identifying Proportional Relationships in Tables

The key to identifying a proportional relationship in a table is to check if the ratio y/x (or x/y, depending on the context) remains the same for all pairs of x and y values.

Let's look at some examples:

Example 1: Proportional Relationship

x	y	y/x
1	2	2
2	4	2
3	6	2
4	8	2

In this table, the ratio y/x is consistently 2 for all pairs of values. Therefore, this table shows a proportional relationship between x and y. The constant of proportionality (k) is 2.

Example 2: Non-Proportional Relationship

x	y	y/x
1	3	3
2	5	2.5
3	7	2.33
4	9	2.25

Here, the ratio y/x changes for each pair of values. There's no constant of proportionality. Therefore, this table does not show a proportional relationship.

Example 3: Another Proportional Relationship (Different Constant)

x	y	y/x
2	1	0.5
4	2	0.5
6	3	0.5
8	4	0.5

This table shows a proportional relationship. The constant of proportionality is 0.5. Notice that even though the numbers are different from Example 1, the ratio remains constant.

Example 4: Zero Values

It's important to note that if either x or y is 0, you cannot calculate the ratio y/x. A proportional relationship can still exist, but you need to examine the remaining ratios. A table with a row where x=0 and y=0 could represent a proportional relationship (with k=0), however, you would need further data points to confirm this.

How to Determine if a Table Shows a Proportional Relationship

Calculate the ratio y/x (or x/y) for each pair of values in the table.
Check if the ratio is consistent across all pairs. If the ratio is the same for all pairs, then the table shows a proportional relationship.
If the ratio is not consistent, the table does not show a proportional relationship.

Real-World Applications of Proportional Relationships

Proportional relationships are prevalent in many real-world situations. For example:

Cost per item: If apples cost $2 per pound, the total cost is proportionally related to the number of pounds purchased.
Speed and distance: If a car travels at a constant speed, the distance traveled is proportionally related to the time spent traveling.
Ingredient ratios in recipes: Doubling or halving a recipe maintains a proportional relationship between ingredient amounts.

Understanding proportional relationships enables us to solve problems involving scaling, rates, and conversions efficiently.

By following these steps and examples, you can confidently identify tables that represent a proportional relationship between x and y. Remember, the key is the consistent ratio between the two variables.