In this explainer, we will learn how to create formulas linking two quantities that vary directly and indirectly. Before we discuss inverse variation, letβs recap what is meant by direct variation and some of the properties of variables that are directly proportional to one another. 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 π¦βπ₯. Since their ratio is constant, we must have
π¦π₯=π
for π₯β 0 and some constant π
β 0, where π is called the constant of variation or constant of proportionality. Multiplying both sides of the previous equation through by
π₯, we see that π¦=π
π₯. If π¦βπ₯, then
π¦ is a linear function in π₯ 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 π‘=ππ£.. In this example, the distance the car needs to travel is a constant, so we could say that
π‘β1π£,
with constant of proportionality π. This is an example of inverse variation. We say that π‘ varies inversely with π£ if π‘ varies directly with
1π£. We can define this formally as follows. Two variables π¦ and π₯ are said to be in
inverse variation, or inverse proportion, if π¦ is directly proportional to the reciprocal of π₯
. In other words, π¦β1π₯. This is equivalent to saying that π¦=ππ₯ for π₯β 0 and some constant
πβ 0; we call π
the constant of proportionality. We can rewrite this equation as π₯π¦=π. 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 π people, then the amount of money, in dollars, that each person gets is given by π¦ where π¦=800π. If we are told that after sharing the money
equally, each person gets $50, we can determine the corresponding value of π by substituting π¦
=50 into the equation to get 5
0=800π. Then, we multiply the equation through by π and divide the equation through by 50 to get π=80050=16. Letβs 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 Proportionalityπ¦ varies inversely with π₯. Given that π¦=8 when π₯=7, what is the constant of proportionality? AnswerWe recall that two variables π¦ and π₯ are said to vary inversely if π¦ is directly proportional to the reciprocal of π₯. In other words, π¦β1π₯. This means that there is some constant πβ 0 such that π¦=ππ₯. π is called the constant of proportionality. We can substitute π¦=8 and π₯=7 into this equation to get 8=π7. Multiplying the equation through by 7 gives π=8Γ7=5 6. It is worth noting that we could have found π directly by noting that π¦=ππ₯ can be rearranged to give π₯π¦=π. In other words, the product of the variables is constant and equal to π. Hence, we can always find the constant of proportionality by multiplying the corresponding variables: π=8Γ7=56. 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. Letβs 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 InverseDecide if π₯ varies directly or inversely with π¦ and use this to find the value of π¦ when π₯=3. AnswerWe recall that π₯ varies directly with π¦ if their ratio remains constant; however, π₯ varies indirectly with π¦ if their product remains constant. Therefore, we can determine whether π₯ and π¦ follow either of these relations by calculating the quotient and product of each pair of the π₯- and π¦ -values and checking if these remain constant. We can add these values to the table.
We see that the ratio between the corresponding π₯- and π¦-values varies; however, their product remains constant at 140. Hence, π₯ varies inversely with π¦, and π₯π¦=140. We can use this equation to determine the value of π¦ when π₯=3 by substituting π₯=3 into the equation. This gives 3π¦=140. Dividing the equation though by 3 gives π¦=1403. Finally, we can write this as a mixed fraction: π¦=4623. Hence, π₯ varies inversely with π¦, and when π₯=3, π¦=4623. 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 π¦β1π₯ and π₯ο§ and π¦ο§ and π₯ο¨ and π¦ο¨ are corresponding values in the relationship, we must have π₯π¦=π₯π¦.ο§ ο§ο¨ο¨ We can rearrange this equation to get π¦π¦=π₯π₯.ο§ο¨ο¨ο§ In other words, π¦ο§, π¦ο¨, π₯ο¨, and π₯ο§ 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 π¦=ππ₯; this is called the reciprocal graph and it has the following shape. We can see as the value of π₯ increases, the value of π¦ decreases; similarly, as the value of π₯ decreases, the value of π¦ increases. Letβs use this to determine which of several different graphs represents inverse variation. Example 3: Identifying the Graph of an Inverse VariationWhich of the graphs shown represents inverse variation? AnswerWe begin by recalling that, in an inversely proportional relationship, the product of the variables remains constant, so π₯π¦=π, for some constant π. Hence, as the value of π₯ increases, the value of π¦ must decrease. We can see in the diagram that graphs B, C, and D do not follow this pattern. As the values of π₯ increase, we can see that the π¦-values are also increasing, so none of these graphs can represent inverse variation. In graph A, we can see that as π₯ increases, π¦ decreases. Similarly, as π₯ decreases, π¦ 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 π¦=ππ₯, which we can rearrange to get π₯π¦=π. Hence, only graph A represents inverse variation. Letβs 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 VariationA group of scouts receives a donation of $1β β000 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.
AnswerPart 1 We recall that two variables π and π are said to vary inversely if π is directly proportional to the reciprocal of π. In other words, πβ1π. This means that there is some constant πβ 0 such that π=ππ, where πβ 0. π is called the constant of proportionality. Therefore, to find an equation for π in terms of π, we need to determine the value of π. To do this, letβs determine a pair of values for π and π. 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 π=1, we have π=1000. Substituting these values into the proportionality equation gives 1000=π1=π. So, π=1 000, and we can substitute this into the proportionality equation to get π=1000π. Part 2 If 25 scouts from the group are going to the jamboree, then we have π=25, and the amount each scout receives is the corresponding value of π. Since we have an equation for π in terms of π, we can substitute π=25 into the equation to find the corresponding value of π to get π=100025= 40. 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 UnknownFor a rectangle of fixed area, the length π varies inversely with its width π€. Given that π=22cm when π€=16cm, determine the value of π when π€=44.cm AnswerThere are two ways we can answer this question. First, we recall that two variables π and π€ are said to vary inversely if π is directly proportional to the reciprocal of π€. In other words, πβ1π€. This means that there is some constant πβ 0 such that π=ππ€. We can find the value of π by substituting π=22 and π€=16 into the equation to get 22=π16. Multiplying the equation through by 16 gives us π=22Γ 16=352.cmο¨ This is the area of the rectangle. We can substitute this value into the proportionality equation to get π= 352π€. We now substitute π€=44 into this equation to get π=35244=8cm. 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 πο¨, we have 22Γ16=πΓ44.ο¨ Dividing the equation through by 44 we get π=22Γ1644=8.ο¨cm 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 VariationThe number of hours π 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? AnswerWe 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 π‘, we must have 23Γ35=π,1 15Γπ‘=π. For some constant π , equating the left-hand side of each equation gives us 23Γ35=1 15Γπ‘. Dividing the equation through by 115 gives π‘=23Γ35115= 7.h Hence, it would take 115 workers 7 hours to carry out the task. Letβs finish by recapping some of the important points from this explainer. Key Points
What is it called when the ratio of two variables is constant?Proportional relationships are relationships between two variables where their ratios are equivalent. Another way to think about them is that, in a proportional relationship, one variable is always a constant value times the other. That constant is know as the "constant of proportionality".
When the ratio of variations in the related variables is constant it is called positive correlation?What Is Positive Correlation? A positive correlation is a relationship between two variables that move in tandemβthat is, in the same direction. A positive correlation exists when one variable decreases as the other variable decreases, or one variable increases while the other increases.
What is a constant of variation?Constant of Variation
The ratio between two variables in a direct variation or the product of two variables in an inverse variation. In the direct variation equations = k and y = kx, and the inverse variation equations xy = k and y = , k is the constant of variation.
What is the constant ratio called?The constant of proportionality is the ratio that relates two given values in what is known as a proportional relationship. Other names for the constant of proportionality include the constant ratio, constant rate, unit rate, constant of variation, or even the rate of change.
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