In this explainer, we will learn how to create formulas linking two quantities that vary directly and indirectly.
Before we discuss inverse variation, let’s recap what is meant by direct variation and some of the properties of variables that are directly proportional to one another.
Key Term: 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 . Since their ratio is constant, we must have for and some constant , where is called the constant of variation or constant of proportionality.
Multiplying both sides of the previous equation through by , we see that
If , then is a linear function in and its graph is a straight line that passes through the origin.
This is not the only type of proportional relationship. For example, we can recall the relationship between the velocity of a car and the time taken to reach a destination. This is given by the formula .
In this example, the distance the car needs to travel is a constant, so we could say that , with constant of proportionality . This is an example of inverse variation. We say that varies inversely with if varies directly with . We can define this formally as follows.
Definition: Inverse Variation or Inverse Proportion
Two variables and are said to be in inverse variation, or inverse proportion, if is directly proportional to the reciprocal of . In other words, .
This is equivalent to saying that for and some constant ; we call the constant of proportionality.
We can rewrite this equation as . Hence, the product of variables that are inversely proportional to one another remains constant.
We can use this definition to determine unknown values in an inversely proportional relationship given the constant of proportionality and a known value. For example, sharing a fixed amount of money amongst a varying number of people is an inversely proportional relationship. Imagine we need to share $800 among people, then the amount of money, in dollars, that each person gets is given by where
If we are told that after sharing the money equally, each person gets $50, we can determine the corresponding value of by substituting into the equation to get
Then, we multiply the equation through by and divide the equation through by 50 to get
Let’s see an example of how to determine the constant of proportionality in an inversely proportional relationship given two values of the corresponding variables.
Example 1: Finding the Constant of Inverse Proportionality
varies inversely with . Given that when , what is the constant of proportionality?
We recall that two variables and are said to vary inversely if is directly proportional to the reciprocal of . In other words, . This means that there is some constant such that . is called the constant of proportionality.
We can substitute and into this equation to get
Multiplying the equation through by 7 gives
It is worth noting that we could have found directly by noting that can be rearranged to give . In other words, the product of the variables is constant and equal to . Hence, we can always find the constant of proportionality by multiplying the corresponding variables:
In the above example, we used the property that the product of the corresponding variables in an inversely proportional relationship remains constant. A similar statement is true for direct variation; the ratio of the corresponding variables remains constant. These give us useful tests to determine whether a relationship is directly proportional or inversely proportional.
Let’s see an example of how to use these properties to determine the type of relationship given in a table and then solve for an unknown variable using a given value for a variable.
Example 2: Determining Whether the Variation between Two Proportional Quantities Is Direct or Inverse
Decide if varies directly or inversely with and use this to find the value of when .
We recall that varies directly with if their ratio remains constant; however, varies indirectly with if their product remains constant. Therefore, we can determine whether and follow either of these relations by calculating the quotient and product of each pair of the - and -values and checking if these remain constant. We can add these values to the table.
We see that the ratio between the corresponding - and -values varies; however, their product remains constant at 140. Hence, varies inversely with , and .
We can use this equation to determine the value of when by substituting into the equation. This gives
Dividing the equation though by 3 gives
Finally, we can write this as a mixed fraction:
Hence, varies inversely with , and when , .
In the previous example, we saw an example of the product of corresponding variables remaining constant in an indirectly proportional relationship. In general, this means that if and and and and are corresponding values in the relationship, we must have
We can rearrange this equation to get
In other words, , , , and are in proportion, and we can use this to find an unknown in an inversely proportional relationship from three known values of the variables without finding the constant of proportionality.
Before we move on to more examples, consider the graph of an inversely proportional relationship. This will be the graph of an equation of the form ; this is called the reciprocal graph and it has the following shape.
We can see as the value of increases, the value of decreases; similarly, as the value of decreases, the value of increases.
Let’s use this to determine which of several different graphs represents inverse variation.
Example 3: Identifying the Graph of an Inverse Variation
Which of the graphs shown represents inverse variation?
We begin by recalling that, in an inversely proportional relationship, the product of the variables remains constant, so , for some constant . Hence, as the value of increases, the value of must decrease. We can see in the diagram that graphs B, C, and D do not follow this pattern. As the values of increase, we can see that the -values are also increasing, so none of these graphs can represent inverse variation.
In graph A, we can see that as increases, decreases. Similarly, as decreases, increases. This hints to us that graph A represents inverse variation. We can confirm this by noting that the shape of this graph is that of a reciprocal function , which we can rearrange to get .
Hence, only graph A represents inverse variation.
Let’s now see an example of how to use a description of an inversely proportional relationship to find an equation linking the variables.
Example 4: Writing an Equation Describing Inverse Variation
A group of scouts receives a donation of $1 000 to fund places on an international jamboree. The amount each scout receives for their trip varies inversely with the number of scouts from the group going to the jamboree.
- Write an equation for , the amount each scout receives, in terms of , the number of scouts from the group who are going to the jamboree.
- If 25 scouts from the group are going to the jamboree, how much will each scout receive from the donation?
We recall that two variables and are said to vary inversely if is directly proportional to the reciprocal of . In other words, . This means that there is some constant such that , where . is called the constant of proportionality. Therefore, to find an equation for in terms of , we need to determine the value of .
To do this, let’s determine a pair of values for and . We can do this by noting that if there was only one scout, they would receive all of the money since there is no one else to share the money with. Hence, when , we have . Substituting these values into the proportionality equation gives
So, , and we can substitute this into the proportionality equation to get
If 25 scouts from the group are going to the jamboree, then we have , and the amount each scout receives is the corresponding value of . Since we have an equation for in terms of , we can substitute into the equation to find the corresponding value of to get
Therefore, each scout receives $40.
In our next example, we will use an inversely proportional relationship to determine the value of an unknown by using three known values.
Example 5: Using Inverse Variation to Find an Unknown
For a rectangle of fixed area, the length varies inversely with its width . Given that when , determine the value of when
There are two ways we can answer this question. First, we recall that two variables and are said to vary inversely if is directly proportional to the reciprocal of . In other words, . This means that there is some constant such that . We can find the value of by substituting and into the equation to get
Multiplying the equation through by 16 gives us
This is the area of the rectangle. We can substitute this value into the proportionality equation to get
We now substitute into this equation to get .
An easier method would be to use the fact that if two variables vary inversely with each other, then their product remains constant. Therefore, if we call the length we want to find , we have
Dividing the equation through by 44 we get
Hence, the length of the rectangle is 8 cm.
In our final example, we will apply the definitions and properties of inverse variation to determine the time taken for a number of workers to carry out a task given an inversely proportional relationship between the number of hours taken and the number of workers.
Example 6: Solving Word Problems Involving Inverse Variation
The number of hours needed for carrying out a certain task varies inversely with the number of workers who carry out the task. If the task is carried out by 23 workers in 35 hours, what is the time needed for 115 workers to carry out the task?
We first recall that if two variables vary inversely with each other, then their product remains constant. Therefore, if we call the time we want to find , we must have
For some constant , equating the left-hand side of each equation gives us
Dividing the equation through by 115 gives
Hence, it would take 115 workers 7 hours to carry out the task.
Let’s finish by recapping some of the important points from this explainer.
- Two variables and are said to be in inverse variation, or inverse proportion, if . This means that their product remains constant.
- Saying and are inversely proportional is equivalent to saying that for some constant ; we call the constant of proportionality.
- The graph of variables in inverse variation is a reciprocal graph.