# What is a Proportional Relationship?

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Team Digital
What is a Proportional Relationship?
CCSS.MATH.CONTENT.7.RP.A.2

## Proportional Relationships – Definition

In everyday life, we often encounter situations where understanding the relationship between different quantities is essential, such as when mixing ingredients for a recipe or determining the speed needed to travel a certain distance in a given time. This is where the concept of a proportional relationship becomes crucial, helping us make sense of these everyday scenarios logically and mathematically.

A proportional relationship is a relationship between two variables where their ratios or rates remain constant. This means that as one quantity increases or decreases, the other does so in a directly proportional manner.

Proportional relationships are all about maintaining a consistent ratio which can be represented in the following forms:

• table
• graph
• equation

## Proportional Relationships in Tables

A proportional relationship requires a constant ratio between the two quantities. In this table, for each additional day, the money saved increases by $\$$5. This consistent increase indicates a fixed ratio. This table shows that for each day, the amount of money saved increases by \$$5, maintaining a consistent ratio of 1 day to$\$$5. ### Understanding Direct Proportionality Direct Proportionality: The table shows that the days and the amount of money saved are directly proportional. As the number of days increases, the total amount of money saved increases in a directly proportional manner. Check how well you already understand direct proportionality with some exercises: x y 1 3 2 6 3 9 Is the relationship shown in the table above proportional, and why? x y 1 4 2 7 3 11 Is the relationship shown in the table above proportional, and why? Identifying proportional relationships in tables helps you better understand and interpret data, a key skill in both math and everyday life. ### Inverse Proportional Relationships Inverse proportional relationships occur in many real-world scenarios, offering a different perspective on how quantities relate to each other compared to direct proportionality. In an inverse proportional relationship, two variables behave in a way that when one increases, the other decreases proportionally, such that their product is always constant. This relationship is represented by the formula y = \frac{k}{x}, where k is a constant. Consider a scenario involving the brightness of a projector screen as you move further away. 1. Closer Distance: When you are 2 meters away from the screen, the brightness is at a certain level. 2. Increasing Distance: As you move further away, say to 4 meters, the brightness decreases. 3. Relationship: If the brightness at 2 meters is B, at 4 meters it would be \frac{B}{2}. In this case, the brightness of the screen is inversely proportional to the distance from it. The further you move away, the less bright the screen appears. The product of the distance and brightness remains constant, illustrating an inverse proportional relationship. ## Proportional Relationships in Graphs Imagine a scenario where you save 5 every day. The graph of your savings over time would show a straight line, where the horizontal axis represents the days and the vertical axis represents the amount of money saved. In a graph depicting a proportional relationship, the key feature is a straight line that intersects the origin. This line is a visual interpretation of the equation y = kx, where k stands for the constant of proportionality. It demonstrates that every value of y is directly determined by multiplying the corresponding x value by the constant factor k, thus showcasing a precise proportional relationship between the two variables. Let’s apply your knowledge of proportional relationships in graphs to practice now. Is this relationship proportional, and how do you know? Is this relationship proportional, and why or why not? Interpreting graphs of proportional relationships will also help you better understand the importance of the slope intercept form. ## Proportional Relationships in Equations An equation represents a proportional relationship if it can be expressed in the form y = kx, where k is a constant. The relationship between the days and the amount of money saved can be described by the equation y = 5x, where y is the total amount of money saved, and x is the number of days. This equation shows that the amount saved is directly proportional to the number of days. Is the relationship described by the equation, y= 5x proportional, and how do you know? Is the relationship described by the equation, y = 2x + 3 proportional, and why or why not? Representing proportional relationships with equations is a method that will directly lead to understanding what the slope of a line is. ## Proportional Relationships – Real-World Problems Let's work through some examples of real-world problems and determine whether there are proportional relationships. Josh gets paid \$$10 per hour for his part-time job. The relationship between his total earnings ($E$) and hours worked ($h$) can be represented by the equation $E = 10h$. Is this relationship proportional?
Hours (h) Distance (km)
1 60
2 120
3 180
Amy is traveling by train, which moves at a constant speed. Is the relationship between hours and distance traveled proportional? Refer to the table above.

Is the relationship between time and pages read proportional? Refer to the graph above.