constant of proportionalityis a number that relates two variables. The two quantities can be directly or inversely proportional to each other. If the two variables are directly proportional to each other, the other variable also increases.
If the two variables are inversely proportional to each other, when one variable increases, the other decreases. For example, the relationship between two variables, $x$ and $y$, is represented as $y = kx$ when they are directly proportional to each other, and $y =\frac{k }{x}$ when they are inversely proportional . Here"k" is the constant of proportionality.
constant of proportionalityis a constant number denoted "k" that is either equal to the ratio of two quantities if they are directly proportional or the product of two quantities if they are inversely proportional.
You should refresh the following concepts to understand the material discussed on this topic.
- Basic Arithmetic.
- graphics
What is the constant of proportionality
The constant of proportionality is the constant produced when two variables are directly or inversely related. The value of the constant of proportionality depends on the type of relationship. The value of "k" always remains constant, regardless of the nature of the relationship between two variables. The proportionality constant is also known as the proportionality coefficient. We have two types of proportions or variations.
Directly proportional: If you specify two variables "y" and "x", then "y" is directly proportional to "x" if an increase in the value of the variable "x" is a proportional increase in the value of "yes." You can represent the direct relationship between two variables as .
$y \,\, \alpha \,\,x$
$y = kx$
For example, You want to buy 5 chocolates of the same brand, but you have not yet decided which chocolate brand you want to buy. Let's say the brands available in the shop are Mars, Cadbury and Kitkat. The variable "x" is the price of a chocolate, while "k" is the constant of proportionality and will always be equal to 5 since you decided to buy 5 chocolates. In contrast, the variable "y" is the total cost of the 5 chocolates. Let's assume the prices of the chocolates are
$Mars = 8\hspace{1mm}Dollar$
$Cadbury = 2 \hspace{1mm}Dollar$
$Kitkat = 6 \hspace{1mm}Dollar$
As we can see, the variable "x" can be equal to 5, 2 or 6 depending on which brand you want to buy. The value of "y" is directly proportional to the value of "x", when you buy the expensive chocolate, the total cost also increases and is higher than the other two brands. You can calculate the value of "y" using the equation $y = 5x$
X | K | Y |
$8$ | $5$ | $8\times 5 =40$ |
$2$ | $5$ | $2\times 5 =10$ |
$6$ | $5$ | $6\times 5 =30$ |
Invers proportional:The two given variables "y" and "x" are inversely proportional to each other if an increase in the value of variable "x" causes a decrease in the value of "y". You can represent this inverse relationship between two variables as.
$y \,\, \alpha \,\, \dfrac{1}{x}$
$ y = \dfrac{k}{x} $
Let's take the example of Mr. Steve driving his car from destination "A" to destination "B". The total distance between "A" and "B" is 500 km. The maximum speed on the motorway is 120 km/h. In this example, the speed at which the car is moving is variable "x" while "k" is the total distance between target "A" and "B" since it is constant. The variable "y" is the time in "hours" to reach the final goal. Mr. Steve can drive at any speed below 120 km/h. Let's calculate the time to go from destination A to B if the car is moving at a) 100 km/h b) 110 km/h c) 90 km/h.
X | K | Y |
$100$ | $500$ | $\dfrac{500}{100} =5Std$ |
$110$ | $500$ | $\dfrac{500}{110} =4.5 hours$ |
$90$ | $500$ | $\dfrac{500}{100} =5,6 Std.$ |
As we can see in the table above, when the car is moving at higher speed, it takes less time to reach the destination. As the value of variable "x" increases, the value of variable "y" decreases.
How to find the constant of proportionality
We have developed our knowledge of both types of proportions. The constant of proportion is easy to find once you have analyzed the relationship between the two variables.
First, let's take the chocolate examples discussed earlier. In this example, we have set the value of "k" to 5. Let's change the values of the variables and draw a chart. Suppose we have 5 chocolates priced at $2,4,6,8, and $10 respectively. The value of "x" increases in increments of 2 while the value of "k" stays constant at 5, and by multiplying "x" by "k" we get the values of "y". When we draw the graph, we can observe that a straight line is created that describes a direct relationship between the two variables.
The constant of proportionality "k" is the slope of the line drawn using the values of the two variables. In the graph below, the slope is labeled as a constant of proportionality.
The above example illustrated the concept of constant of proportionality using a graph, but the value of "k" was predetermined by us. So let's take an example where we need to find the value of "k".
example 1:The following table contains the values of the two variables "x" and "y". Determine the nature of the relationship between the two variables. Do you also calculate the value of the constant of proportionality?
X | Y |
$1$ | $3$ |
$2$ | $6$ |
$3$ | $9$ |
$4$ | $12$ |
$5$ | $15$ |
Solution:
The first step is to determine the nature of the relationship between the two variables.
First, let's try to develop an inverse relationship between these two variables. We know that the inverse relationship is shown as
$ y = \dfrac{k}{x} $
$k = y. x $
X | Y | K |
$1$ | $3$ | $k = 3\times 1 = 3$ |
$2$ | $6$ | $k = 2\times 6 = 12$ |
$3$ | $9$ | $k = 3\times 9 = 27$ |
$4$ | $12$ | $k = 4\times 12 = 48$ |
$5$ | $15$ | $k = 5\times 15 = 75$ |
As we can see, the value of "k" is not constant, so the two variables are not inversely proportional to each other.
Next we will see if they have a direct relationship between them. We know that the formula for the direct relation is given as.
$y = kx$
X | Y | K |
$1$ | $3$ | $k = \dfrac{3}{1} = 3$ |
$2$ | $6$ | $k = \dfrac{6}{2} = 3$ |
$3$ | $9$ | $k = \dfrac{9}{3} = 3$ |
$4$ | $12$ | $k = \dfrac{12}{4} = 3$ |
$5$ | $15$ | $k = \dfrac{15}{5} = 3$ |
We can see that the value of "k" remains constant; therefore both variables are directly proportional to each other. You can plot the slope of the given relationship as .
example 2:The following table contains the values of the two variables "x" and "y". Determine the nature of the relationship between the two variables. Do you also calculate the value of the constant of proportionality?
X | Y |
$10$ | $\dfrac{1}{5}$ |
$8$ | $\dfrac{1}{4}$ |
$6$ | $\dfrac{1}{3}$ |
$4$ | $\dfrac{1}{2}$ |
$2$ | $1$ |
Solution:
Let's determine the nature of the relationship between the two variables.
We know that the inverse relational formula is given as.
$ y = \dfrac{k}{x} $
$k = y. x $
X | Y | K |
$10$ | $\dfrac{1}{5}$ | $k = \dfrac{10}{5} = 2$ |
$8$ | $\dfrac{1}{4}$ | $k = \dfrac{8}{4} = 2$ |
$6$ | $\dfrac{1}{3}$ | $k = \dfrac{6}{3} = 2$ |
$4$ | $\dfrac{1}{2}$ | $k = \dfrac{4}{2} = 2$ |
$2$ | $1$ | $k = \dfrac{2}{1} = 2$ |
From the table it can be seen that the value of "k" remains constant; therefore both variables are inversely proportional. You can plot the slope of the given relationship as .
Two variables can be either directly or inversely proportional to each other. Both relationships cannot exist at the same time. Since they are inversely proportional to each other in this example, they cannot be directly proportional.
Definition of the constant of proportionality:
The constant of proportionality is the ratio between two variables that are directly proportional to each other and is generally represented as
$\mathbf{k=\dfrac{y}{x}}$
Example 3:The following table contains the values of the two variables "x" and "y". Determine if there is a relationship between these two variables. If so, find the nature of the relationship between the two variables. Also calculate the value of the constant of proportionality.
X | Y |
$3$ | $6$ |
$5$ | $10$ |
$7$ | $15$ |
$9$ | $18$ |
$11$ | $33$ |
Solution:
The relationship between the two variables can be either direct or inverse.
First, let's try to establish a direct relationship between given variables. We know that the direct relational formula is given as.
$y = kx$
X | Y | K |
$3$ | $3$ | $k = \dfrac{3}{3} = 1$ |
$5$ | $6$ | $k = \dfrac{6}{5} = 1.2$ |
$7$ | $9$ | $k = \dfrac{9}{7} = 1.28$ |
$9$ | $12$ | $k = \dfrac{12}{9} = 1.33$ |
$11$ | $15$ | $k = \dfrac{15}{11} = 1.36$ |
As we can see, the value of "k" is not constant, so the two variables are not directly proportional to each other.
Next, let's try to develop an inverse relationship between them. We know that the formula for the inverse relationship is given as.
$ y = \frac{k}{x} $
$k = y. x $
X | Y | K |
$3$ | $3$ | $k = 3\times 3 = 9$ |
$5$ | $6$ | $k = 6\times 5 = 30$ |
$7$ | $9$ | $k = 9\times 7 = 63$ |
$9$ | $12$ | $k = 12\times 9 = 108$ |
$11$ | $15$ | $k = 15\times 11 = 165$ |
The variables are therefore not directly or inversely related to one another, since the value of "k" does not remain constant in either case.
example 4:When 3 men do a job in 10 hours. How much time do 6 men need for the same task?
Solution:
As the number of men increases, the time it takes to complete the task decreases. So it is clear that these two variables have an inverse relationship. So let's represent the men by the variable "X" and the working time by the variable "Y".
X1= 3, Y1= 10, X2 = 6 and Y2 =?
We know that the formula for the inverse relationship is given as
$ Y1 = \dfrac{k}{X1} $
$ k = Y1. X1 $
$ k = 10\times 3 = 30 $
$Y2 = \dfrac{k}{X2}$
We know k = 30
$ Y2 = \dfrac{30}{6} $
$Y2 = 5$