Constant of Proportionality
Math Skills: What is the Constant of Proportionality? Constant of Proportionality Definition, How to Find Constant of Proportionality
What is the constant of proportionality? Whenever you are working with math topics including ratios, unit rates, and/or direct variation, you will learn about a constant of proportionality.
In this free guide for students, we will explore the constant of proportionality definition and what it means, along with step-by-step examples where will we use tables, equations, and graphs to understand the concept of a constant of proportionality.
Whether you are a student in need of some help or a classroom teacher planning a lesson on constant of proportionality, this page will surely help you to deeply understand this important math topic.
Table of Contents:
Now, let’s start by exploring the answer to the question: what is a constant of proportionality and what does it actually mean?
Preview: What is the constant of proportionality?
Constant of Proportionality Worksheet Preview
What is the Constant of Proportionality?
Let’s start by asking the question, what is the constant of proportionality?
In math, constant of proportionality refers to a fixed value that can be used to describe the relationship between two proportional quantities.
In other words, if two variables change (increase or decrease) at the same rate, then they are considered to be proportional to each other, and the constant of proportionality is the number that relates them. This means that the ratio or the product between the two variables results in a constant.
The constant of proportionality is a value or a number that depends on the two variables and the type of proportion between them.
If you find this explanation of the constant of proportionality to be confusing, that’s ok. This particular math topic can be difficult to grasp at first, but it will start to make more sense as we further explore it through the lens of familiar math vocabulary as well as by exploring tables, equations, and graphs.
For now, the key takeaway for answering, what is the constant of proportionality?, is that it is the number or value you multiply by to go from one value to another in any proportional relationship between two variables.
Figure 01: Constant of Proportionality: What is Direct Variation?
Constant of Proportionality Definition
Now that you have a general idea of what is the constant of proportionality, let’s explore the constant of proportionality definition (and a few other important vocabulary terms related to this topic).
Direct Variation Definition: In math, direct variation is a proportional relationship between two variables that can be expressed as the equation, y=kx, where y and x are variables and k is a constant.
Inverse Variation Definition: In math, inverse variation is a proportional relationship between two variables that can be expressed as the equation, y=k/x, where y and x are variables and k is a constant.
For both direct variation and inverse variation, k is the constant. This value (k) is the constant of proportionality.
Constant of Proportionality Definition: In math, the constant of proportionality is the value of the variable k in the direct variation equation y = kx and the inverse variation equation y=k/x, where y and x are variables and k is a constant ratio.
The image in Figure 02 below illustrates the relationship between direct variation, inverse variation, and the constant of proportionality. Take a close look at the diagram and make sure that you have a good understanding of the constant of proportionality definition before you move onto the next section.
Figure 02: Constant of Proportionality Definition
Now that you know the formal constant of proportionality definition, let’s add some context by exploring a real-world example.
Consider a car that is moving at a rate of 70 miles per hour. In this case, the relationship between the distance traveled (D) and the time (T) is proportional, and it can be modeled using the following equation:
D = 70 × T
In this example, the constant of proportionality is 70 because 70 is the constant that relates the variables D and T, which share a proportional relationship.
How to Find Constant of Proportionality
Next, let’s learn how to find constant of proportionality using a simple formula.
Formula for Finding Constant of Proportionality (k):
k = y/x
So, you can find k by dividing y by x.
As long as the variables x and y share a proportional relationship, then you can use the formula k = y/x to find constant of proportionality.
Let’s take a look at an example of how to find constant of proportionality by looking at a table.
| x | y | k = y ÷ x |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 6 | 3 |
| 3 | 9 | 3 |
| 4 | 12 | 3 |
Notice that the table gives us four different x and y values (1,3), (2,6), (3,9), and (4,12).
You can find the constant of proportionality by selecting the x and y value from any row and then inputting them into the constant of proportionality formula (divide y by x).
If we select the values in the first row (x=1 and y=3), we have:
k = y/x
k = 3/1
k = 3
So, we can conclude that the constant of proportionality, k, is 3.
We could have chosen the x and y-values from any other row as well. For example, if we chose the second row where x=2 and y=6:
k = y/x
k = 6/2
k = 3
Since x and y share a proportional relationship, the value of k (the constant of proportionality) will always be 3.
Final Answer: The table represents the equation y=3x where the constant of proportionality is 3.
So, how do you find the constant of proportionality? Simply input x and y into the equation k = y/x and solve for k. Now, let’s gain some more experience by working through a few more examples.
Constant of Proportionality Examples
In the previous section, we learned how to find the constant of proportionality from a table. Now, let’s try finding it from a graph.
Example #1:
Find the constant of proportionality.
In this example, we are given a graph of a linear function.
We know that x an y share a proportional relationship because the line passes through the origin at (0,0), and it is increasing at a constant rate.
And, that constant rate, or constant of proportionality, is what we need to find.
To find the constant of proportionality, we have to take a point on the line with (x,y) coordinates and plug them into the equation:
k = y/x
Example #1: How to find the constant of proportionality from a graph.
Since the line passes through the point (4,2), we can plug x=4 and y=2 into the constant of proportionality equation as follows:
k = y/x
k = 2/4
k = 1/2
In this case, the constant of proportionality is 1/2.
Notice that, for a linear equation, the constant of proportionality is the slope of the line, y = 1/2x.
Final Answer: The constant of proportionality is 1/2.
For our next example, we will find the constant of proportionality from a given equation.
Example #2:
Find the constant of proportionality for y=7x
In this example, we are given an equation of a linear function, and we can find the constant of proportionality in a few ways.
The easiest way to determine the constant of proportionality for this equation is to notice that it is already in y=kx form, where k, the constant of proportionality, is 7.
However, if you would prefer to use the equation k = y/x to find the constant of proportionality, you would just have to select a point that the line that represents the equation passes through (y=7x) and plug it into the formula k = y/x.
| x | y |
|---|---|
| 0 | 0 |
| 1 | 7 |
| 2 | 14 |
| 3 | 21 |
| 4 | 28 |
We can find the value of k by choosing the point (1,7):
k = y/x
k = 7/1
k = 1
In this case, the constant of proportionality is 7.
Again, notice that the constant of proportionality is the slope of the line, y =7x.
Final Answer: The constant of proportionality is 7.
Example #3:
A line passes through the points (0,0) and (3, 9). What is the constant of proportionality?
For this final example, we have to determine the value of the constant of proportionality of a line that passes through two given points.
Since one of the given points is (0,0), we know that x and y share a proportional relationship.
We can again use the formula k = y/x to solve for k. However, we can not use the point (0,0) since we can not divide by zero, so we can use (9,3) instead:
k = y/x
k = 3/9
k = 1/3
In this case, we can simplify k = 3/9 down to k = 1/3.
Again, notice that the constant of proportionality is the slope of the line, y =1/3x.
Final Answer: The constant of proportionality is 1/3.
Example #3: How to find the constant of proportionality using two given points.
Constant of Proportionality Worksheet
Are you looking for some extra practice with how to find the constant of proportionality? Click the text links below to download a free Constant of Proportionality Worksheet (Version A and Version B) PDF file.
Each constant of proportionality worksheet includes an answer key so you can check your work and assess your progress. If you discover that you are still answering questions incorrectly, we recommend going back and working through the practice problems above.
Preview: Constant of Proportionality Worksheet
Conclusion: The Constant of Proportionality
Understanding how to find the constant of proportionality between two variables with a proportional relationship is an important math skill that will help you with solving a variety of problems involving ratios, unit rates, and linear relationships with direct variation.
Here are a few key takeaways to remember:
A constant of proportionality is a number (k) that describes the relationship between two proportional quantities.
Any proportional linear relationship with direction variation will be of the form y=kx, where k is the constant of proportionality (and the slope of the line). All lines of the form y=kx will pass through the origin.
You can find the constant of proportionality (k) between x and y by using the formula: k = y ÷ x
So, if you came to this guide wondering what is constant of proportionality, you now know everything you need to understand the concept, definition, and the formula for finding constant of proportionality.
More Math Resources You Will Love:
