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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will be able to identify problems where you can use the formula for inverse variation to solve them. Learn what is involved and how easy they are to solve.

We have all kinds of problems in math. One of them is called **inverse variation**. What does it mean? It means that as one thing gets larger, another gets smaller by a fixed factor. This is the type of problem that we will be considering in this video.

What exactly is inverse variation? We actually see it going around us all the time. For example, think of when you get together with your friends and family for some pizza. Say you have just one large pizza for everybody. If there are just four of you altogether, then everybody can have a quarter of the pizza. But what if there were eight of you? Well, your slice of pizza would suddenly get smaller. Each person would only get half of a quarter of a pizza, an eighth of a pizza. This is inverse variation. As one number gets larger, the other number gets smaller.

And, this being algebra and all, we do have formulas for inverse variation. It is **y = k/x**, where *k* is your *constant of variation*, the number that tells you how fast or slow things change with each other. Your *y* and *x* are the two things that are changing with each other. So, for our pizza example, our y can be the size of our pizza slice and our x can be the number of people.

Usually, when you get problems of inverse variation, you are given your two things that are changing with each other. Your job is to find your constant of variation and then use that constant to find new information. For example, going back to our pizza party, our problem could say that the number of people at a party compared to how much of the pizza each person gets varies inversely. If there are four people and each person gets one-fourth of the pizza, how much of the pizza would each person get if there were eight people?

To solve this, we need to look for our first data point, which is our first pair of numbers of the things that are changing with each other. In this case, it is that when there are four people, then each person gets a fourth of the pizza. This is our first data point. We will use this number to find our constant of variation. To do this we will label our number of people as 'x' and our pizza slice as 'y.' Then we will plug these numbers into our equation to find k. Doing this, we get *1/4 = k/4*. Solving this for k, we find that *k = 1*. Since k is a constant, now that we have found k, we know that for all data points in this scenario, k will always equal 1. We will use this information to find our missing data point.

The problem wants to know how much of the pizza each person will get if there are eight people. So we will plug in 8 for x and then solve for y. So we have *y = 1/8*. So each person will get 1/8 of the pizza and 1/8 is our answer!

Let's try another problem. Eight people rent a cabin for the weekend for 70 dollars each. How much will each person pay if there are ten people if there is an inverse variation relationship between the number of people and cost per person? We first label our x and y. It really doesn't matter how we label as long as we stick to our x and y labels throughout the problem. I will label my number of people as 'x' and the cost per person as 'y'. So my first step is to find my constant of variation. My first data point is eight people for 70 dollars each. I plug in my x and y into my formula to find k: I have *70 = k/8*. Solving for k, I get *k = 70 x 8*, which equals *560*.

Now that I have k, I will use this to find my missing data. I have 10 people and I want to find the cost per person, so my new equation is *y = 560/10*. Because I know the total cost of the cabin is 560 dollars, I solve this for y to get 56, so each person pays 56 dollars. I am done and 56 dollars is my answer!

So, what have we learned? We've learned that an inverse variation means that as one thing gets larger, another gets smaller by a fixed factor. To solve these kinds of problems, we use the formula for inverse variation problems, which is y = k/x, where k is our constant of variation and x and y are our two things that change with each other. Our problem will usually give us one data point for x and y. We use this to find our k. The problem will then ask us to find the missing data for another point. We will use the k we found from our first point to find the other point. We use our formula twice: once to find our k and twice to find our missing data. We label our two items with x and y. It doesn't matter which we labeled with x and which we labeled with y as long as we stick to this label throughout our problem.

Once you are finished reviewing this lesson you should be able to recall the formula for and calculate an inverse variation.

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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

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