Understanding the relationships between two variables is the goal for most of science. Whether you have a specific scientific question in mind such as: What happens to the global temperature if the amount of carbon dioxide in the atmosphere increases, or how does the strength of gravity vary when you move further away from the source, or you are more interested in an abstract mathematical setting, finding out the difference between direct and inverse relationships is essential if you want to describe these relationships. In short, direct relationships increase or decrease together, but inverse relationships move in opposite directions.

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In a direct relationship, an increase in one quantity leads to a corresponding decrease in the other. This has the mathematical formula of *y* = *kx*, where *k* is a constant. For a circle, circumference = pi × diameter, which is a direct relationship with pi as a constant. A bigger diameter means a bigger circumference.

In an inverse relationship, an increase in one quantity leads to a corresponding decrease in the other. Mathematically, this is expressed as *y* = *k*/*x*. For a journey, travel time = distance ÷ speed, which is an inverse relationship with the distance traveled as a constant. Faster travel means a shorter journey time.

## The Background: How Does y Vary with x?

Scientists and mathematicians dealing with direct and inverse relationships are answering the general question, how does *y* vary with *x*? Here, *x* and *y* stand in for two variables that could be basically anything. For example, how does the height that a ball bounces (*y*) depend on how high it’s dropped from (*x*)? By convention, *x* is the independent variable and *y* is the dependent variable. So the value of *y* depends on the value of *x*, not the other way around, and the mathematician has some control over *x* (for example, she can choose the height from which to drop the ball). When there is a direct or inverse relationship, *x* and *y* are proportional to each other in some way.

## Direct Relationships

A direct relationship is proportional in the sense that when one variable increases, so does the other. Using the example from the last section, the higher from which you drop a ball, the higher it bounces back up. A circle with a bigger diameter will have a bigger circumference. If you increase the independent variable (*x*, such as the diameter of the circle or the height of the ball drop), the dependent variable increases too and vice-versa.

A direct relationship is linear. The circumference of a circle is *C* = π_*D_*, where *C* means circumference and *D* means diameter. Pi is always the same, so if you double the value of *D*, the value of *C* doubles too. If you plotted a graph of this relationship, it would equate to a straight line with zero circumference at *D* = 0, 3.14 at *D* = 1 and 31.4 at *D* = 10. The gradient of the graph tells you the value of the constant.

## Inverse Relationships

Inverse relationships work differently. If you increase *x*, the value of *y* decreases. For example, if you move more quickly to your destination, your journey time will decrease. In this example, *x* is your speed and *y* is the journey time. Doubling your speed halves the journey time, and increasing the speed by ten times makes the journey time ten times shorter.

Mathematically, this type of relationship has the form: *y* = *k* / *x*, where *k* is some constant (filling the same role as pi in the direct relationship example). Inverse relationships aren’t straight lines, though. As you start to increase *x*, *y* decreases really quickly, but as you continue increasing *x* the rate of decrease of *y* gets slower.

For example, if *x* is the length of one pair of sides of a rectangle, *y* is the length of the other pair of sides, and *k* is the area, the formula *k* = *xy* is valid, so *y* = *k* ÷ *x*. In this case, *y* is inversely related to *x*. For an area *k* = 12, this gives *y* = 12 ÷ *x*. For *x* = 3, this shows *y* = 4. For *x* = 6, then *y* = 2. For *x* = 12, then *y* = 1. At first an increase of 3 in *x* decreases *y* by 2, but then an increase of 6 in *x* only decreases *y* by 1. This is why inverse relationships are declining curves that get shallower the further you move along them.

## Direct vs. Inverse Relationships: The Difference

In direct relationships, an increase in *x* leads to a correspondingly sized increase in *y*, and a decrease has the opposite effect. This makes a straight-line graph. In inverse relationships, increasing *x* leads to a corresponding decrease in *y*, and a decrease in *x* leads to an increase in *y*. This makes a curving graph where the decline is rapid at first but gets slower for larger values of *x*.