11.1 Direct and inverse variations

Direct variation

As a simple example, consider the variable c which is the number of cars in a parking lot, and the variable t which is the number of tires in the parking lot. Assuming each car has four tires, we might see numbers like this.

These two columns stand in a very particular relationship to each other which is referred to as direct variation .

Definition of “direct variation”

When the left-hand column goes up, the right-hand column goes up. This is characteristic of direct variation, but it does not prove a direct variation. The function $y=x+1$ has the characteristic that whenever $x$ goes up, $y$ also goes up; however, it does not fulfill the definition of direct variation.

The equation for this particular function is, of course, $t\left(c\right)=4c$ . In general, direct variation always takes the form $y=kx$ , where $k$ is some constant—a number, not a function of $x$ . This number is referred to as the constant of variation .

Note that, in real life, these relationships are not always exact! For instance, suppose $m$ is the number of men in the room, and $w$ is the weight of all the men in the room. The data might appear something like this:

Not all men weigh the same. So this is not exactly a direct variation. However, looking at these numbers, you would have a very good reason to suspect that the relationship is more or less direct variation.

How can you confirm this? Recall that if this is direct variation, then it follows the equation $w=km$ , or $w/m=k$ . So for direct variation, we would expect the ratio w/m to be approximately the same in every case. If you compute this ratio for every pair of numbers in the above table, you will see that it does indeed come out approximately the same in each case. (Try it!) So this is a good candidate for direct variation.

Inverse variation

Suppose 5 cars all travel 120 miles. These cars get different mileage. How much gas does each one use? Let m be the miles per gallon that a car gets, and g be the number of gallons of gas it uses. Then the table might look something like this.

These variables display an inverse relationship .

Definition of “inverse variation”

Note that as the first column gets bigger , the second column gets smaller . This is suggestive of an inverse relationship, but it is not a guarantee. $y=10–x$ would also have that property, and it is not inverse variation.

The equation for this particular function is $g=120/m$ . In general, inverse variation can always be expressed as $y=k/x$ , where $k$ is once again the constant of variation .

If $y=k/x$ , then of course $xy=k$ . So inverse variation has the characteristic that when you multiply the two variables, you get a constant. In this example, you will always get 120. With real life data, you may not always get exactly the same answer; but if you always get approximately the same answer, that is a good indication of an inverse relationship.

More complex examples

In the year 1600, Johannes Kepler sat down with the data that his teacher, Tycho Brahe, had collected after decades of carefully observing the planets. Among Brahe’s data was the period of each planet’s orbit (how many years it takes to go around the sun), and the semimajor axis of the orbit (which is sort of like a radius, but not quite—more on this in “Ellipses”). Today, these figures look something like this.

What can we make of this data? As a goes up, $T$ clearly also goes up. But they are not directly proportional. For instance, looking at the numbers for Uranus and Neptune, we see that 165 is almost exactly twice 84; but 450 is much less than twice 287. Is there a consistent pattern? Kepler went down in history for figuring out that the square of the period is directly proportional to the cube of the semimajor axis: in numbers, $T^2=ka3$ . You can confirm this for yourself, using the numbers above. (What is $k$ ?)

So we see that the concepts of “directly proportional” and “inversely proportional” can be applied to situations more complex than $y=kx$ or $y=k/x$ .

The situation becomes more interesting still when multiple independent variables are involved. For instance, Isaac Newton was able to explain Kepler’s results by proposing that every body in the world exerts a gravitational field that obeys the following two laws.

• When the mass of the body doubles, the strength of the gravitational field doubles
• When the distance from the body doubles, the strength of the field drops in a fourth

Science texts express these laws more concisely: the field is directly proportional to the mass, and inversely proportional to the square of the radius. It may seem as if these two statements require two different equations. But instead, they are two different clues to finding the one equation that allows you to find the gravitational field F at a distance r from a given mass m. That one equation is $F=\frac{Gm}{r^2}$ where $G$ , the constant of proportionality, is one of the universal constants of nature. This does not come from combining the two equations $F=km$ and $F=\frac{k}{r^2}$ as a composite function or anything else. Rather, it is one equation that expresses both relationships properly: doubling the mass doubles the field, and doubling the radius drops the field in a fourth.