Direct Variation. This describes the relationship between two variables. An example of direct variation is the total cost at the checkout counter in a store (C) and the total number of loaves of bread (L). The higher L is, the higher C will be. The lower L is, the lower C will be. You could also speak of this in reverse. The lower C is, then the lower L must be, etc. This is stated: "C varies directly as L," or C is directly proportional to L." This is expressed this way in mathematics:

In the expression above, k is represents a constant. We normally call this constant in this shopping context the unit price.

In general, direct variation between two generic variables y and x is expressed this way:

This expression is interpreted as "y varies directly as x," or "y is directly proportional to x."

Inverse Variation. This again describes the relationship between two variables. An example of inverse variation is the number of incorrect problems on a quiz (I) and the percentage of questions you got correct (P). The higher I gets, then the lower P gets. The lower I gets, then the higher P is. This is stated: "I varies inversely as P." This is expressed in mathematics in this manner:

In general, inverse variation between two generic variables y and x is expressed this way:

This expression is interpreted as "y varies inversely as x," or "y is inversely proportional to x."

Joint Variation. Joint variation is really just "double direct" variation. Here again is an example. Let's say that your grade on a test (G) varies jointly as the amount of time you studied (S) and the amount of time you have to take the test (T). This says that G varies jointly as S and T. In mathematics we would write this this way:

You can see that this is really G varying directly as S, and G varying directly as T. That's why I called it double direct variation.

Combined Variation. Several different variations can be included in a single problem. This is called combined variation. Here is an example problem:

	This is problem number 36 on page 234.
	We first need to translate this word problem into a mathematical variation statement. We notice the words "directly proportional" are the equivalent of "varies directly as."
	Here we substitute the values given in the problem (prior to the question). This allows us to find the constant. We see that k = 48.
	Now we can restate the variation with the known constant.
	Now we can substitute in the numbers given in the question part of the problem. q is the only unkown.
	Simplify.
	Solve for q.

Some Notes from Class

Variation @ Purple Math

Chapter 2 Review Notes