1. Math compares things to one another.
   1. For example, if two things are exactly the same, they’re equal. If a number we’ll call x and another number we will call y were the same value, we would write y = x.
   2. However, if y was twice as big as x, we’d write y = 2x. If you double the value of x, they’d be equal.
   3. And if x was larger than y, we would write x > y.
2. Things that change vary.
   1. If y was three times larger than x, we would write, y = 3x. This is true because in order for the two numbers (x and y) to be equal, we have to multiply x by three because y is three times larger.

The topic, typically in an Algebra class, of Variation is an excellent way to develop the ability to read, write and manipulate equations.  This is exactly what mathematical literacy is!  So, let’s give this topic the focus it deserves.  Let’s dive into an example and see how we can make sense of it.

The number y varies directly with x.  When y is 10, x is 8.  What is x when y is 16?

In order to understand this, we need to define varies directly.  Varies means changes, directly means by multiplication (more on this later), by the constant of variation, which might be our first oxymoron of the day!  The constant of variation is what we multiply by to make the two values equal.  In the example earlier, where y was three times as large as x, the constant of variation was 3.  We do not know the constant of variation here, and need to find it.  We will use the letter k to represent the constant of variation.

Since we are told that y varies directly with x, we can write the equation below. 

y = k·x

This equation describes the relationship between our numbers x and y.  Now, we know that when y = 10 that x = 8.  Let’s plug those in and see if we can figure out what the constant of variation is.

10 = k·8

All we need to do is use Algebra (inverse operations) to solve for k.  When we do this, we need to reduce.  This gives us 10/8 = 5/4, or 1.25.  Either way works fine.

Let’s be 100% sure that we understand what we have found.  The constant of variation, k, makes the values of y and x equal.  Therefore, the following is true.

y = 1.25x

In other words, no matter what y is, it will be 1.25 times greater than x.  This is true for 10 and 8. 

PRO – TIP:  Do not forget to write the equation above.  This is what we use to answer the remaining questions because this is how y and x are related.

Let’s take a look at our original question again.

The number y varies directly with x.  When y is 10, x is 8.  What is x when y is 16?

Even though we weren’t asked to find the value of constant of variation, or asked to write an equation relating y and x, we need to in order to answer the next question.  In doing these things what we have done is made sense of the clues provided.  Those clues are: (1) y varies directly with x, and (2) when y is 10, x is 8.  Now, let’s use what we know to find the value of x when y is 16.

y = 1.25x

16 = 1.25x

Divide both sides by 1.25.  Before doing so, should x be a larger or smaller number than y? 

12.8 = x

Let’s recap what has happened.  For direct variation we begin with y = kx. Then, we plug in the values provided to find the value of k.  Next, we write the equation with the value found for k.  After that, we plug in what is provided and answer the question.  Things will get a little trickier, but this is the basic idea and it will not change.

A quick word on direct variation.  Things that vary directly grow in the same direction.  So if y varies directly with x, then as y grows, so does x.  This could be something like the more sunlight the more plant growth (within reason, of course…not always true in places like Arizona).  Another example is, as your student skills improve so do your grades.

There’s another type of variation to know. It’s called indirect, or inverse, or inversely proportional.  That means that as one value grows, the other shrinks.  For example, the more money you spend, the less money you have.  Or, the more you study, the less time you have for playing video games.

It is typically the case that the type of variation will be stated.  Sometimes, the situation will be described and you have to determine the type of variation.  Let’s see an example of how indirect variation works.

The number y varies indirectly with x.  When y is 10, x is 8.  What is x when y is 9?

Let’s think about this for just a second.  Since they’re inversely proportional, or indirectly varied, as one grows the other shrinks.  If y starts at 10 and is now 9, it has gotten smaller.  That means that x will be more than where it started, at 8.

Let’s start with our situation, writing an equation using k, the constant of variation.

This is how inversely proportional works.  There are other ways to write this, like yx = k.  The style is irrelevant.

Now, plug in our values of x and y to find the constant of variation.

Writing our original equation, with the value of k, we get the following.

We were told that y = 9, and asked to find the value of x.  This involves a little Algebra, but isn’t too difficult, if you’re careful.

If we multiply both sides by x, we get this.

9x = 80

It is true there are short cuts here, but knowing this level of Algebra is beneficial.  Now we just divide by 9 and we’re done.

x = 8.89
(Rounded to three significant figures.)

This is good because we expected x to be a larger value because y was smaller.

Let’s summarize what we have so far. 

1. Identify if the relationship is direct or inverse.
   1. Write y = kx, or y = k/x accordingly
2. Plug in the initial values.
3. Solve for
4. Write the equation using your value for k.
5. Plug in the information provided and use inverse operations to answer the questions.

That’s really it.  Nothing more … except some details.  Here is an example.

Given that y varies directly with the square of x, and that when y = 9, x = 4, find y when x is 2.

The key phrase here is, with the square of x.  All we need to do is account for that fact and everything else is the same.  Here’s what it will look like.

You can download a PowerPoint with this link:  Click Here.  Here are a couple of screen shots of the lesson.

Practice Problems

1. Given that y is inversely proportion with the square root of x, and that when y = 20, x = 4, find the value of x when y is 18.
   
   
   
   
   
2. The number of children at a playground decreases whenever swarms of killer bees are around. Is this an example of direct proportion, inversely proportional, or neither?  Why?
   
   
   
   
   
3. Does the table below show inverse or direct variation?

4. Write an equation that fits the table of values above in the form of y = kx,  or yx =  
   
   
   
   
   
   
5. Fill in the missing values in the table above.
   
   
6. The force of attraction (F) between two planets is inversely proportional to the square of the distance between them (d). The units to measure force are newtons.  When F = 10²⁰ newtons, d = 10⁹ 
   1. Write an equation relating F and d.
      
      
      
      
      
   2. Find the force between two planets that are 10¹² meters apart.