Additional Resources
- (MathHints.Com) Direct, Inverse, Joint, and Combined Variation
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mathhints.com/beginning-algebra/direct-inverse-and-joint-variation/
Section 2.5 Variation Relationships
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In some situations, it makes sense to describe the relationship between variables using words rather than equations. This can often bring additional focus and attention to the core structure of the relationship and can avoid getting lost in the equations. We are going to focus on two specific phrasings that are used to discuss these relationships.
Direct or Proportional Variation.
The idea of two variables being "in proportion" to each other is that they form a constant ratio. This is known as a direct variation. A simple example is purchasing items (without a bulk discount). If a burger is $4, then 2 burgers is $8, 3 burgers is $12, and so on. The ratio of number of burgers to the cost remains the same. There are two mathematical phrasings that capture this type of relationship. The first is "\(A\) is proportional to \(B\)" and the other is "\(A\) varies directly with (or as) \(B\text{.}\)" As an equation, this is often written as \(A \propto B\text{,}\) which means that \(A = kB\) for some constant \(k\text{.}\) This constant is known as the constant of proportionality.
When two variables are directly proportional to each other, it establishes a linear relationship between them. This means that if one increases, so does the other, and always in an equal proportion of the amount of increase. This type of relationship appears throughout physics, though it usually requires assuming that certain other variables are being held constant. Here are a few examples (and don’t worry if you don’t understand them right now):
- Distance is proportional to the time traveled (at a constant speed): \(d \propto t\text{.}\)
- Voltage is proportional to the current (with a constant resistance): \(V \propto I\text{.}\)
- Force is proportional to the accerlation (with a constant mass): \(F \propto a\text{.}\)
The reason for the \(\propto\) notation is that we don’t always care about the value of the constant of proportionality. It emphasizes the variables in the relationship without needing to introduce another symbol that would need to be defined or discussed. However, when it comes to working through problems, we almost always see that constant of proportionality show up.
Inverse Variation.
Direct variations have the property that both variables increase or decrease together. Inverse variations have the property that the two variables move in opposite directions. So as one variable increases, the other decreases. The common phrasing for this is "\(A\) is inversely proportional to \(B\text{,}\)" and is often written as \(A \propto \frac{1}{B}\text{.}\) Here are a few examples:
- Speed is inversely proportional to the time traveled (for a constant distance): \(s \propto \frac{1}{t}\text{.}\)
- Current is inversely proportional to the resistance (with a constant voltage): \(I \propto \frac{1}{R}\text{.}\)
- Acceleration is inversely proportional to the mass (with a constant force): \(a \propto \frac{1}{m}\text{.}\)
This phrasing can be used to loosely indicate that the two variables move in opposite directions from each other. For example, one might say something like "fun is inversely proportional to the amount of work that needs to be done" to mean that there’s more fun when there’s less work and that there’s less fun when there is more work. However, in the context of physics, it can only mean the precise mathematical relationship above.
Joint and Combined Variations.
Joint and combined variations are variations with more than two variables. Technically, a joint variation is a variation built from two direct variations. So the phrase "\(A\) varies jointly with \(B\) and \(C\)" means that \(A \propto BC\text{.}\) (You will also see "\(A\) is proportional to \(B\) and \(C\)".) This is different from a combined variation, which is a combination of a direct variation and an inverse variation. Fortunately, the relationships are expressed explicitly in the language. For example, we would say "\(A\) varies directly with \(B\) and inversely with \(C\)" to mean that \(A \propto \frac{B}{C}\text{.}\)
With these types of variations, there is only one constant of proportionality. In other words, \(A \propto BC\) means \(A = kBC\text{,}\) as opposed to something like \(A = k_B B \cdot k_C C\text{.}\) The reason for this that there is no benefit to introducing extra constants. In the end, the two constants can just be multiplied together into one, and we wouldn’t be able to determine two separate constants without having to introduce another equation.
Variations with Variable Expressions.
The most general type of relationships that we can have with variables allow us to not just have a variable, but a variable expression. This should be seen as just a standard symbolic substitution. For example, the phrase "\(A\) is inversely proportional to the square of \(B\)" takes the phrase "the square of \(B\)" and replaces it with \(B^2\text{,}\) and then the relationship can be written as \(A \propto \frac{1}{B^2}\text{.}\) This can be done in all sorts of ways. Here are some examples:
- The distance an object falls when dropped is proportional to the gravitational acceleration and the square of the time: \(s \propto a t^2\text{.}\)
- The gravitational force between two objects is proportional to the product of the masses of the objects and inversely proportional to the square of the distance between them: \(F \propto \frac{m_1 m_2}{r^2}\text{.}\)
