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Variable y is directly proportional to the variable x.
In mathematics, two variables are proportional if there is always a constant ratio between them. The constant is called the coefficient of proportionality or proportionality constant.
If one variable is always the product of the other variable and a constant, the two are said to be directly proportional. are directly proportional if the ratio is constant.
If the product of the two variables is always a constant, the two are said to be inversely proportional. are inversely proportional if the product xy is constant.
The statement "y is directly proportional to x" is written mathematically as "," or "," where c is the proportionality constant.
The statement "y is inversely proportional to x" is written mathematically as "." This is equivalent to "y is directly proportional to ."
Given two variablesx and y, y is directly proportional to x if there is a non-zero constant k such that
On a map of a sufficiently small area, drawn to scale distances, the distance between any two points on the map is directly proportional to the length of the projected line between the two locations that the points represent, with the constant of proportionality being the scale of the map.
The force, acting on a certain object across short distances due to gravity, is directly proportional to the object's sufficiently small mass; the constant of proportionality between the force and the mass is known as gravitational acceleration.
The net force acting on an object is proportional to the acceleration of that object. The constant of proportionality in this, Newton's Second Law, is the classical mass of the object.
is equivalent to
it follows that if y is directly proportional to x, with (nonzero) proportionality constant k, then x is also directly proportional to y, with proportionality constant .
The concept of inverse proportionality can be contrasted with direct proportionality. Consider two variables said to be "inversely proportional" to each other. If all other variables are held constant, the magnitude or absolute value of one inversely proportional variable decreases if the other variable increases, while their product (the constant of proportionality k) is always the same. As an example, the time taken for a journey is inversely proportional to the speed of travel.
Formally, two variables are inversely proportional (also called varying inversely, in inverse variation, in inverse proportion, in reciprocal proportion) if each of the variables is directly proportional to the multiplicative inverse (reciprocal) of the other, or equivalently if their product is a constant. It follows that the variable y is inversely proportional to the variable x if there exists a non-zero constant k such that
or equivalently Hence the constant is the product of x and y.
The graph of two variables varying inversely on the Cartesian coordinate plane is a rectangular hyperbola. The product of the x and y values of each point on the curve equals the constant of proportionality (k). Since neither x nor y can equal zero (because k is non-zero), the graph never crosses either axis.
The concepts of direct and inverse proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates; the two coordinates correspond to the constant of direct proportionality that specifies a point as being on a particular ray and the constant of inverse proportionality that specifies a point as being on a particular hyperbola.
Brian Burell: Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster, 1998, ISBN9780877796213, pp. 85-101
Lanius, Cynthia S.; Williams Susan E.: PROPORTIONALITY: A Unifying Theme for the Middle Grades. Mathematics Teaching in the Middle School 8.8 (2003), pp. 392-96 (JSTOR)
Seeley, Cathy; Schielack Jane F.: A Look at the Development of Ratios, Rates, and Proportionality. Mathematics Teaching in the Middle School, 13.3, 2007, pp. 140-42 (JSTOR)
Van Dooren, Wim; De Bock Dirk; Evers Marleen; Verschaffel Lieven : Students' Overuse of Proportionality on Missing-Value Problems: How Numbers May Change Solutions. Journal for Research in Mathematics Education, 40.2, 2009, pp. 187-211 (JSTOR)