Inverse Proportion: This Goes Up, That Goes Down

If you’ve ever sat in your car as an ambulance or fire truck passes you with their lights and siren on, you have probably noticed that the farther away the vehicle got from you, the quieter the siren eventually became.

This is a basic, real world illustration of the algebraic concept of inverse proportion. In a more technical and mathematical explanation, inverse proportionality refers to the relationship between two variables. If one of these variables increases, causing the other variable to decrease, their relationship may be described as inversely proportional to each other. Conversely, a direct proportion describes the relationship between two variables where one increases, the other increases along with it. This direct proportionality also holds true if the variables decrease, as long as they move together in the same direction. 

Inverse and direct proportions are found in algebra and are quite simple in theory and explain simple concepts that we are all familiar with, just maybe not on a technical level. Below we will explain certain aspects of the inverse proportion relationship, giving examples and showing equations that will hopefully better illustrate them. If you’re feeling a little shaky with your algebra skills, our course on beginning algebra will explain the basic principles of this subject.

Reciprocals

A more technical way to explain a relationship that is inversely proportional is to describe it in terms of reciprocals. A reciprocal of a number is that number divided by one. For example, the reciprocal of 10 is 1/10. Every number has a reciprocal except for 0, because 1/0 is undefined. If you multiply a number by its reciprocal, you get 1 (2 x 1/2 = 1). Now, back to the inverse proportions. If one variable is directly proportional to the reciprocal, also referred to as the multiplicative inverse, of another variable, they are inversely proportional.

Example: Let’s take two variables, X and Y, which are inversely proportional to each other. As you now know, this means X is directly proportional to the reciprocal of Y, and this relationship may be expressed as the following equation, where α is the Greek letter alpha, indicating direct proportionality.

This relationship can also be explained in plain language with an everyday situation we all might potentially experience. Imagine you are riding in a train at a constant speed and you are traveling a distance of 100 miles. If you travel for one hour and reach your destination after that hour, your average speed was 100 miles per hour (mph). If the trip took you two hours to reach your 100 mile destination, then your average speed was 50 mph (100 miles/2 hours). By doubling the number of hours traveled, you are decreasing the average speed by half. The variables of time and speed changed by reciprocal factors (time changed by 2, speed by 1/2), thus making them inversely proportional.

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Equation

The equation for an inverse proportion is as follows, where the variable y is inversely proportional to the variable x, as long as there exists a constant, k, which is a non-zero constant.

The constant (k) can be found by simply multiplying the original X and Y variables together. When graphed, the products of the X and Y values at each point along the curved line will equal the constant (k), and because this number can never be 0, it will never reach either axis, where the values are 0.

On the other side of the coin, the equation for a direct proportion is y=kx. k is still the constant, and x and y are still the variables. Not only is the equation different, but we will illustrate in the next section that, when graphed, it looks different, as well.

Illustration of an Inverse Proportion

To explain how an inverse proportion works, as well as what it looks like when graphed, we will describe below how this relationship looks in graph form, as well as the equations from which it springs. The smooth, curved line used on a graph in this type of problem is referred to as a hyperbola.

To illustrate the concept of inverse proportion, the following real world scenario will shows how it works. In the following example, we want to compare how fast your average speed is while driving, compared to how quickly you arrive at your destination, with the following table showing the times and speeds you measured on these trips.

Avg. Speed (x)	Mins. to Destination (y)	X times Y
40	7.5	300
45	6.7	300
50	6	300
55	5.5	300

In this example, the X variable (average speed traveled) goes up, while the Y variable (time it took to arrive, in minutes) goes down. Notice how the product of the X and Y variables always equals the same thing, which is our constant, k. This was explained in the Equation section above. When plotted onto a graph, the table from above appears as a smoothly curving line, starting on the left side of the upper right quadrant, gently sloping down to the bottom right of the quadrant, like a hill, and never once touching the X and Y axises, because the values of those locations are 0, and as you know, the constant is never 0.

Compared to a graph of a direct proportion, it looks quite different. This line is absolutely straight, going from the bottom left or right of the graph, and ascending to the opposite corner. If you follow this straight line and at any point analyze the corresponding values, you will find that as the X variable increases, so does the Y variable – hence, a direct proportion.

So that’s inverse proportions for you – simple in theory, but potentially confounding when written out and analyzed, or if you’re a math whiz, it only gets easier to understand when there are variables and graphs involved. If you’ve been following along with our courses, and are ready for the most difficult one, our most advanced algebra course will complete your online algebra education, then these concepts should be pretty easy for you to understand. And if the advanced algebra course is still child’s play, our course on matrix algebra should really blow your hair back. Wherever your algebraic expertise lies, we hope we were able to illuminate the concept of inverse proportions for you today, and maybe now you’ll look at a simple trip to the grocery store as a graph with a hyperbola.