Lesson Video: Inverse Variation Mathematics

In this video, we will learn how to create formulas linking two quantities that vary directly and indirectly.

17:02

Video Transcript

In this video, we will learn how to identify inverse proportion and write equations describing inverse variation to solve problems. But before we discuss inverse variation, let’s recap what we mean by direct variation. And we’ll also look at some of the properties of variables that are directly proportional to one another.

We can say that two variables are in direct variation, or direct proportion, if their ratio is constant. As an example, if we work a job which pays an hourly wage, then the variables of hours worked and wages would be in direct variation. The more hours that we work, then the higher our wages will be. If we have two variables 𝑦 and π‘₯, then we use the proportionality symbol to describe the relationship between the variables. We can read this as a statement as 𝑦 varies directly with π‘₯ or 𝑦 is directly proportional to π‘₯.

Since we have the ratio between 𝑦 and π‘₯ is constant, then we can write that 𝑦 is equal to π‘šπ‘₯, where π‘š is not equal to zero. This value of π‘š is called the constant of variation or the constant of proportionality. Sometimes we may see the letter π‘˜ being used instead. And this time π‘˜ would be the constant of variation.

When it comes to drawing the graph of a direct variation, we have a straight line which passes through the origin. But of course this is not the only type of proportional relationship. Let’s take, for example, the relationship between the velocity of a car and the time taken to reach a destination. This can be given by the formula that the time taken is equal to the distance divided by the velocity. But we know that as the velocity would increase, then the time taken to reach the destination would decrease. This is an example of an inverse variation.

We can define this by saying that two variables 𝑦 and π‘₯ are in inverse variation if 𝑦 is in direct variation with the reciprocal of π‘₯. And we write it using the same proportionality symbol like this. When 𝑦 varies directly with one over π‘₯, that’s equivalent to saying 𝑦 varies inversely with π‘₯.

So when we have an inverse variation, for example, π‘Ž varies inversely with 𝑏, then we know that we will have a proportionality symbol along with a reciprocal. And just like in direct variation, we can write an inverse variation as an equation along with a constant of proportionality. We can say that 𝑦 is equal to π‘š over π‘₯, where π‘š is not equal to zero and π‘š is the constant of proportionality. Alternatively, the letter π‘˜ can be used instead to represent the constant of proportionality.

Let’s now look at some example questions. And in the first example, we’ll see how we can determine the constant of proportionality.

𝑦 varies inversely with π‘₯. Given that 𝑦 equals eight when π‘₯ equals seven, what is the constant of proportionality?

Let’s recall that two variables 𝑦 and π‘₯ are said to vary inversely if 𝑦 is directly proportional to or in a direct variation with the reciprocal of π‘₯. And we write that variation like this: 𝑦 is proportional to one over π‘₯. This means that there is a nonzero constant π‘š such that 𝑦 is equal to π‘š over π‘₯. π‘š is the constant of proportionality. However, the answer to this question is not simply π‘š or any other letter that we use to represent the constant of proportionality. Instead, what we need to do here is use the given information about these values of 𝑦 and π‘₯ to work out a numerical value for the constant of proportionality.

We substitute 𝑦 is equal to eight and π‘₯ is equal to seven into the equation. This gives us eight is equal to π‘š over seven. We can then multiply both sides of this equation by seven. And since eight times seven is equal to 56, we have that π‘š is equal to 56. And so we find that the constant of proportionality is 56.

Earlier in this video, we recalled that the graph of a direct variation is a linear graph which passes through the origin. But what would the graph of an inverse variation look like? Let’s take the two variables 𝑦 and π‘₯, where 𝑦 varies inversely with π‘₯. The graph of this will be an equation of the form 𝑦 equals π‘š over π‘₯. This will be a reciprocal graph, and it would look something like this. Notice that as the value of π‘₯ increases, the value of 𝑦 decreases. And similarly, as the value of π‘₯ decreases, the value of 𝑦 increases. We can use this information in the next question to determine which one of several different graphs represents an inverse variation.

Which of the following graphs represents inverse variation?

We can remember that when we have an inversely proportional relationship between two variables 𝑦 and π‘₯, we can write this as 𝑦 is proportional to one over π‘₯. This means that we can also say that there is some constant of proportionality π‘š such that 𝑦 is equal to π‘š over π‘₯. We can also recognize that as the value of π‘₯ increases, the value of 𝑦 must decrease.

If we look at the graphs of the functions that we’re given, the graphs of B, C, and D do not follow this pattern. In fact, the graph represented by C is actually the graph of a direct variation. But we can eliminate these three options. This leaves us with the answer of graph A. And we can indeed see that as π‘₯ increases, 𝑦 decreases. Similarly, as π‘₯ decreases, 𝑦 increases. We can also confirm the answer is graph A because this is the graph of a reciprocal function.

In the next example, we’ll see a very typical example of an inverse variation question where we first find the constant of proportionality and then use this to solve a problem.

A group of scouts receives a donation of 1000 dollars 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 π‘š, the amount each scout receives, in terms of 𝑛, 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?

Let’s note that this question is all about inverse variation. We are told that the amount that each scout receives varies inversely with the number of scouts going. And that makes sense that it is an inverse variation. Let’s imagine that just one scout is going. They would have all of this 1000 dollars to themselves. But if more and more scouts decide that they are going to go to the jamboree, then the less money that each individual scout would have towards the travel cost.

We are given here the letters to assign for each variable. π‘š is the amount each scout receives, and 𝑛 is the number of scouts who are going. If we have two variables π‘š and 𝑛 which vary inversely, then we can write that π‘š is directly proportional to the reciprocal of 𝑛. We can then immediately say that there is some constant of proportionality π‘˜, which is not equal to zero, such that π‘š is equal to π‘˜ over 𝑛. π‘˜ is called the constant of proportionality. And so, in order to find an equation for π‘š in terms of 𝑛, we need to work out this value of π‘˜.

Sometimes in questions like this, we are given a known pair of values for 𝑛 and π‘š. We are not given this here, but we can work some out. Remember, we’ve already noted that if there’s just one student, then they will get all of the 1000 dollars. One student means that 𝑛 is equal to one. The amount of money π‘š would be 1000. We can then substitute these values into the proportionality equation. This gives us 1000 is equal to π‘˜ over one. So π‘˜ is equal to 1000. We can then substitute this value of π‘˜ into this proportionality equation. And so we get the answer for the first part of this question: π‘š is equal to 1000 over 𝑛.

In order to answer the second part of this question, we use the equation that we’ve worked out in part one to help us. We know that the relationship is that the amount of money that any scout receives, which is π‘š, is equal to 1000 over the number of scouts. If 25 scouts are going, then that means that 𝑛 is equal to 25. And we need to work out the value of π‘š, the amount of money that each scout would receive. Substituting 𝑛 equals 25 into this equation, we have π‘š is equal to 1000 over 25, so π‘š is equal to 40. So the answer is that if 25 scouts are going to the jamboree, then each scout would receive 40 dollars.

Let’s take a look at one final example.

The number of hours 𝑛 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?

In this problem, we are investigating the relationship between the number of hours it takes workers to do a task and the number of workers doing that task. We are told that these two variables are related with an inverse variation. It can be tempting to think that as the number of workers increases, then the time taken also increases. But this would be incorrect, but it’s actually the reverse. As the number of workers increases, then the time taken will decrease. This is why this is an inverse variation. We can describe the time taken or the number of hours using the variable 𝑛. And let’s denote the number of workers with the variable 𝑀.

Recall that if we have two variables 𝑛 and 𝑀 which are in an inverse variation, then we can write this as 𝑛 is directly proportional to one over 𝑀. We can then say that there is some constant of proportionality π‘˜ such that 𝑛 is equal to π‘˜ over 𝑀. And in order to find what the value of π‘˜ will be, we’re given a piece of information that it takes 23 workers 35 hours to do the task. So we can substitute 𝑛 is equal to 35 and 𝑀 is equal to 23 into this equation. This gives us that 35 is equal to π‘˜ over 23. When we multiply both sides of this equation by 23, we have 35 times 23 is equal to π‘˜. We know then that π‘˜ is equal to 805. We then substitute this value of π‘˜ into the equation of proportionality. We have that 𝑛 is equal to 805 over 𝑀.

We now have an equation that relates the number of hours 𝑛 with the number of workers 𝑀. We could use this equation to work out the time needed for any number of workers. And we can use it here to work out the time needed for 115 workers. The value of 𝑀 is the number of workers, and that’s equal to 115. And we need to work out 𝑛, the number of hours. Substituting this into the equation, we have 𝑛 is equal to 805 over 115, which simplifies to seven. And so the answer is that the time taken for 115 workers to carry out the task is seven hours.

But there is an alternative way in which we could’ve worked out this inverse variation problem. And this is by using the property that if two variables vary inversely with each other, then their product remains constant. In this context, we’d be saying that the number of hours 𝑛 multiplied by the number of workers 𝑀 is equal to some constant π‘˜. In the first part of this problem, we were told that it takes 23 workers 35 hours to complete the task. And that product will be the same as 115 workers multiplied by 𝑛, the number of hours. To solve for 𝑛, we would divide through by 115. And when we do that, we get a value 𝑛 is equal to seven. And so we have confirmed the earlier answer that it takes seven hours for 115 workers.

We can now summarize the key points of this video. Two variables 𝑦 and π‘₯ are said to be in an inverse variation, or inverse proportion, if 𝑦 is proportional to one over π‘₯. This also means that their product remains constant. If 𝑦 and π‘₯ are in an inverse variation, this is equivalent to 𝑦 equals π‘š over π‘₯ for some constant π‘š, which is not equal to zero. We call π‘š the constant of proportionality. Finally, we also saw that the graph of variables in an inverse variation is a reciprocal graph.

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