In this explainer, we will learn how to describe direct variation between two variables and use this to solve word problems.
Letβs begin by defining what is meant by direct variation.
Definition: Direct Variation or Direct Proportion
Two variables are said to be in direct variation, or direct proportion, if their ratio is constant.
This type of relationship is often written as , which is read as is directly proportional to . Since their ratio is constant, we must have that for some constant , provided , where is called the constant of variation or constant of proportionality.
Multiplying both sides of the previous equation through by , we see that
This then allows , since this gives .
There are many examples of real-world phenomena that follow a directly proportional relationship. For example, if a body is moving at a constant velocity of 5 m/s, then its distance traveled after seconds is given by
Hence, the distance traveled by a body (moving at a constant velocity) is directly proportional to the amount of time traveled. We can substitute in values of or and then solve the equation for the missing unknown. For example, when , we see that
In our first example, we will determine the constant of proportionality from the values of two variables that directly vary.
Example 1: Finding the Constant of Proportionality
If and when , determine the constant of proportionality.
Answer
We recall that saying that means that the ratio between corresponding values of and remains constant, so for some constant , where is called the constant of proportionality. We can substitute and into this equation to get
Hence, the constant of proportionality is .
Letβs see an example of using this definition to determine the value of an unknown in a directly proportional relationship.
Example 2: Solving Proportion Equations Involving Direct Variation
If and when , find the value of when .
Answer
We recall that saying that means that the ratio between corresponding values of and remains constant, so for some constant , provided . In particular, we can rearrange this equation to get , which then allows .
We can substitute and into the first equation to get
Substituting this value of into the linear equation gives us
We can then substitute into this equation to find the corresponding value of :
Therefore, the value of when is 10.
In the previous example, we saw that the corresponding pairs of values of and were in proportion, since . This is true for any directly proportional variables, and we can show this from the definition.
If and the variable takes the values and and the corresponding -values are and , then and if we divide the two equations, we get
So, we have that , , , and are in proportion.
We can also recall that a linear function is one in the form , which when graphed has a slope of and a -intercept of . Hence, if , then is a linear function in and its graph is a straight line passing through the origin. It is also worth noting that the same result is true in reverse: if , then . This gives us the following result.
Definition: Direct Variation as a Linear Function
If , then is a linear function in and its graph is a straight line that passes through the origin.
Letβs see an example of how to use this property to correctly determine the graph of a directly proportional relationship.
Example 3: Identifying the Graph of Direct Variation
Which of the given graphs represents the direct variation between and ?
Answer
We recall that saying that means that the ratio between corresponding values of and remains constant, so for some constant , provided . In particular, we can rearrange this equation to get , which then allows .
We then recall that the equation is a straight line with slope and -intercept . So, the graph of any directly proportional relationship is a straight line that has a -intercept of 0, meaning it passes through the origin.
Option b is the only straight line passing through the origin, so it is the only graph of a directly proportional relationship.
Since the graph of a straight line passing through the origin represents two variables in direct variation, with the slope as the coefficient of proportionality, we can determine information about the relationship from this graph. Letβs see an example of this.
By considering the graph below, find the coefficient of proportionality between and , and determine the value of when .
First, we know that the coefficient of proportionality is the slope of the line, so we can find the slope of the line by using two points on the line and , where
We know that the line passes through the origin and we can see in the diagram that the line passes through . Substituting these points into the formula gives
Hence,
We can then find the value of when in two different ways.
- We can substitute into this equation to get .
- We can find the -coordinate of the point on the graph with -coordinate 4. We see that this is 2.
It is also worth recalling that graphs and equations are not the only way of representing linear relations. For example, we can represent these as ordered pairs or as entries in a table. Letβs see an example of how to identify a directly proportional relationship from a table.
Example 4: Recognizing Direct Variation from a Table
Which table does not show varying directly with ?
1 2 3 12 24 36 10 20 30 2 4 6 2 0 8 0 5 3 1 6 10 30 2 4 6 1.5 3 4.5
Answer
We recall that saying that varies directly with means that the ratio between corresponding values of and remains constant, so for some constant provided . We can allow by rewriting the equation as , which gives and .
Hence, we need to determine which table has ratios of corresponding - and -values that do not remain constant. We can calculate the ratios for each table separately. We will do this by adding an extra row to each table calculating the ratio of the terms in the same column.
In table A, we see that , , and , giving us the following.
1 | 2 | 3 | |
12 | 24 | 36 | |
12 | 12 | 12 |
Since the ratio of the corresponding - and -values stays constant, this table represents direct variation with constant of proportionality 12.
In table B, we see that , , and , giving us the following.
10 | 20 | 30 | |
2 | 4 | 6 | |
0.2 | 0.2 | 0.2 |
Since the ratio of the corresponding - and -values stays constant, this table represents direct variation with constant of proportionality 0.2.
In table C, we see that and ; however, we cannot evaluate since we cannot divide by 0. We recall that direct variation between and can be written as the equation , so when , we have . Hence, this still represents direct variation with constant of proportionality 4.
In table D, we see that , , and .
5 | 3 | 1 | |
6 | 10 | 30 | |
1.2 | 30 |
These ratios differ, so this does not represent direct variation.
For due diligence, we can also check table E. We have that , , and , giving us the following table.
2 | 4 | 6 | |
1.5 | 3 | 4.5 | |
0.75 | 0.75 | 0.75 |
Since the ratio of the corresponding - and -values stays constant, this table represents direct variation with constant of proportionality 0.75.
Hence, the relation in table D does not represent direct variation between and .
In our next example, we will determine which of a list of equations represents two variables in direct variation.
Example 5: Identifying an Equation of Direct Variation
Which of the following relations represents a direct variation between the two variables and ?
Answer
We recall that saying that varies directly with means that the ratio between corresponding values of and remains constant, so for some constant , provided . We can rewrite this equation as to allow , and we note that any variables in direct variation must satisfy an equation of this form. So, we need to determine which of the given equations can be written in this form.
In relation A, we can multiply the equation through by to get
We then divide the equation through by to get which is not in the form for a nonzero constant , so this equation does not represent direct variation.
In relation B, we have a linear function in the form , and we note that linear functions only represent direct variation with . Since , this does not represent direct variation.
In relation C, we can divide the equation through by to get
This is not in the form for a nonzero constant , so this equation does not represent direct variation.
In relation D, we multiply the equation through by 4 to get
This is now written in the form , where the constant of variation is .
Hence, only the relation in option D represents direct variation.
Letβs now see a few real-world examples of direct variation and how we can use this relationship to determine the value of unknowns.
Example 6: Finding the Constant of Variation from an Equation
The amount of meat required to feed a captive lion is given by the equation , where is the weight of the meat in kilograms needed to feed a lion for days. What is the unit rate of this proportional relationship?
Answer
We recall that saying that varies directly with means that the ratio between corresponding values of and remains constant, so for some constant , provided ; this constant is called the constant of proportionality.
In the question we are told that , so we can conclude that the ratio of and stays at a constant value of 9. Hence, they are directly proportional and the constant of proportionality is 9.
The question asks for the unit rate of this relationship, and we recall that this is the ratio of two quantities when the second quantity is 1. So, we can find the unit rate by substituting into the equation , which gives
Since, is measured in kilograms and is measured in days, we see that the unit rate is 9 kg/day.
In the previous example, we showed a useful result: the unit rate of a directly proportional relationship is equal to the constant of proportionality, since this is the ratio of any two terms.
In our next example, we will use direct variation to determine the weight of a given object on the moon.
Example 7: Solving Proportion Equations Involving Direct Variation
An object that weighs 120 N on Earth weighs 20 N on the Moon. Given that the weight of an object on Earth is directly proportional to its weight on the Moon, find the weight of an object on the Moon given that its weight on Earth is 126 N.
Answer
We are told in the question that the weight of a body on Earth is directly proportional to its weight on the Moon. We recall that this means that the ratio of any bodyβs weight on the Moon to its weight on Earth is constant. This can be mathematically written as for some constant , and the weight on Earth is nonzero.
We can find the value of by using the first pair of values given in the question and . Hence, using the equation given above and substituting this pair of values, we find
Therefore, we have which can be also written as
We have found that the weight of the given body on the Moon is one-sixth of its weight on Earth. Hence, the weight on the Moon of a body whose weight on Earth is 126 N is
In our final example, we will find the relationship between two variables and , given that varies directly with a given linear function of .
Example 8: Writing an Equation between Two Proportional Quantities
Given that , where when , find the relation between and .
Answer
We recall that means that there exists some nonzero constant such that
We can determine the value of by substituting and into this equation:
We divide the equation through by 20 to get
Substituting this value back into the linear equation gives
Letβs finish by recapping some of the important points from this explainer.
Key Points
- Two variables are said to be in direct variation, or direct proportion, if their ratio is constant.
- Direct variation between two variables and is written as . It is mathematically described as , where is called the constant of variation or constant of proportionality.
- Since in a directly proportional relationship, is also the unit rate.
- A linear relationship between two variables and is described as ; hence, only those linear relationships where correspond to direct variation.
- The graph of direct variation (of a proportional relationship) is a straight line that passes through the origin.