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Proportionality (mathematics)

In mathematics, two quantities are proportional if they vary in such a way that one of them is a constantmultiple of the other.


The mathematical symbol '�' is used to indicate that two values are proportional. For example, A � B.

In Unicode this is symbol U+221D.

Direct proportionality

Given two variables x and y, y is '(directly) proportional to x (x and y vary directly, or x and y are in direct variation) if there is a non-zero constant k such that

y = kx.\,

The relation is often denoted

y \propto x

and the constant ratio

k = y/x\,

is called the proportionality constant or constant of proportionality.


  • If an object travels at a constant speed, then the distance traveled is proportional to the time spent traveling, with the speed being the constant of proportionality.
  • The circumference of a circle is proportional to its diameter, with the constant of proportionality equal to Ï€.
  • On a map drawn to scale, the distance between any two points on the map is proportional to the distance 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 due to gravity is proportional to the object's mass; the constant of proportionality between the mass and the force is known as gravitational acceleration.



y = kx\,

is equivalent to

x = \left(\frac{1}{k}\right)y,

it follows that if y is proportional to x, with (nonzero) proportionality constant k, then x is also proportional to y with proportionality constant 1/k.

If y is proportional to x, then the graph of y as a function of x will be a straight line passing through the origin with the slope of the line equal to the constant of proportionality: it corresponds to linear growth.

Inverse proportionality

As noted in the definition above, two proportional variables are sometimes said to be directly proportional. This is done so as to contrast direct proportionality with inverse proportionality.

Two variables are inversely proportional (or varying inversely, or in inverse variation, or in inverse proportion or reciprocal proportion) if one of the variables is directly proportional with 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

y = {k \over x}

The constant can be found by multiplying the original x variable and the original y variable.

Basically, the concept of inverse proportion means that as the absolute value or magnitude of one variable gets bigger, the absolute value or magnitude of another gets smaller, such that their product (the constant of proportionality) is always the same.

For example, the time taken for a journey is inversely proportional to the speed of travel; the time needed to dig a hole is (approximately) inversely proportional to the number of people digging.

The graph of two variables varying inversely on the Cartesian coordinate plane is a hyperbola. The product of the X and Y values of each point on the curve will equal the constant of proportionality (k). Since k can never equal zero, the graph will never cross either axis.

Hyperbolic coordinates

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 locates a point on a ray and the constant of inverse proportionality that locates a point on a hyperbola.

Exponential and logarithmic proportionality

A variable y is exponentially proportional to a variable x, if y is directly proportional to the exponential function of x, that is if there exist non-zero constants k and a

y = k a^x.\,

Likewise, a variable y is logarithmically proportional to a variable x, if y is directly proportional to the logarithm of x, that is if there exist non-zero constants k and a

y = k \log_a (x).\,

Experimental determination

To determine experimentally whether two physical quantities are directly proportional, one performs several measurements and plots the resulting data points in a Cartesian coordinate system. If the points lie on or close to a straight line that passes through the origin (0, 0), then the two variables are probably proportional, with the proportionality constant given by the line's slope.

Unrelated proportionality

Given two variables x and y, y

Proportional representation

Proportional representation (PR) is a goal of voting systems. While some systems that pursue this goal (such as closed party list) can address other proportionality issues (gender, religion, ethnicity), and these advantages are often used to promote such variants, it is not a feature of PR as such to ensure an even split of men vs. women, ethnic or religious representation that resembles the population, or any other goal. As it is used in practice in politics, the only proportionality being respected is a close match between the percentage of votes that groups of candidates obtain in elections in representative democracy, and the percentage of seats they receive (e.g., in legislative assemblies). Thus a more exact term is party-proportional representation , sometimes used by those who wish to highlight systems that emphasize party choice less, candidate or gender choice more, or who wish not to promote systems (such as closed party-listmixed-member proportional) that overly empower the parties, at the expense of voter choice of exactly which individuals go to the legislature as representatives. In contrast those who subordinate gender, ethnic, religious, regional or candidate choice to party choice (usually party members themselves) often use the term full representation. This terminology debate is considered central to the winning (or losing) ofelectoral reform referendums by some advocates who consider referenda to have been lost by it. See notes below.

Proportionality is Relative

When comparing two or more voting systems, most typically instant runoff, mixed member and single transferable vote, one may refer to one voting system as offering more proportional representation or more party-proportional representation than another one is comparing it to. Because historical comparisons of party popular vote and seats in a legislature are objective, this is more exact than simply referring to one system as being proportional and another not. Nor is party proportionality a goal of all electoral reforms, since the political party is often seen as a problem within the system rather than a solution. For example, sometimes the undeclared objective is to undemocratically prevent communist parties from winning elections, such as what happened in French 1958 election (where PCF won only 10 MPs albeit having been the second party in terms of proportional representation; in contrast, CNIP received less votes but won 132 MPs). Another example are independent candidates not affiliated with any party (and usually campaigning on a general mistrust of parties, which is at least their implied reasoning for not working within any of them).

Ideological differences often color debate about the terminology. Persons more trusting of parties tend to fuse (or confuse) concerns about all forms of proportionality into that of party-proportionality, as if assuming that parties (or just their own party) are trustworthy entities to address gender or ethnic or religious inequity. Persons less trusting of parties tend to highlight the shift of influence from the voter to the party to argue against any more party-proportional representation system. These debates can be difficult to understand especially in the context of misinformation by one side or another promoting or detracting from a particular system. All terminology in this field should be understood as changing.

Versus plurality systems

PR goals are often contrasted to the results of plurality voting systems, such as those commonly used in the United States and (much of) the United Kingdom, where disproportional seat distribution results from the division of voters into multiple electoral districts, especially "winner takes all" plurality ("first-past-the-post" or FPTP) districts.

In countries employing plurality systems (notably the UK, US, Canada and India) most alternative systems tend to be described as forms or types of proportional representation but this terminology is inexact (as explained above) and has sometimes (in the British Columbia electoral reform referendum, 2009 and Ontario electoral reform referendum, 2007) resulted in expert advocates using a different set of criteria to decide what to present to the public than the public uses, resulting in referendums that suffer an overwhelming defeat. In particular the focus on party-proportionality was in Ontario, BC (and a similar referendum in PEI) exploited by the "No" campaigns which were able to emphasize of shift of power to parties from the public. In the Ontario case this was particularly effective, as closed-party-list representation had not yet been ruled out.

Party-proportionality and its problems

Proportional systems almost always use political parties as the measure of representation (thus in practice these systems are party-proportional). For example, a party that receives 15% of the votes under such a system receives 15% of the seats for its candidates.

Some systems (notably Israel or Italy) are criticized for being too party-proportional, often leaving the balance of power in the hands of a small party with idiosyncratic beliefs, or fragmenting the "left" or "right" into too many small parties incapable of campaigning or holding a government together.

Even more basic goals that party-proportional systems can help address, such as gender or ethnic equity, are criticized on grounds that the public must be free to choose a legislature that does not look like the public, and that regional representation is most important.

Voting systems that achieve more party-proportional representation

The majority of debate in English about voting systems is about whether to move to more proportionality not less. This is because the establis

Avogadro constant

In chemistry and physics, the Avogadro constant (symbols: L, N A) is defined as the ratio of the number of entities (usually atom s or molecule s) N in a sample to the amount of substance n (unit mole) through the relationship N A = N/ n. Thus it is the proportionality factor that

Constant term

In mathematics, a constant term is a term in an algebraic expression that does not contain any variables, and therefore has constant value. For example, in the quadratic polynomial

x^2 + 2x + 3,\

the 3 is a constant term.

After like terms are combined, an algebraic expression will have at most one constant term. Thus, it is common to speak of the quadratic polynomial


where x is the variable, and has a constant term of c. If c = 0, then the constant term will not actually appear when the quadratic is written.

Any polynomial written in standard form has a unique constant term, which can be considered the coefficient of x0. In particular, the constant term is always the lowest degree term of the polynomial. This also applies to multivariate polynomials. For example, the polynomial


has a constant term of −4, which can be considered the coefficient of x0y0. For any polynomial, the constant term can be obtained by substituting in 0 for all of the variables.

The concept can be extended to power series and other types of series: in the power series

a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots,

a0 is the constant term. In general a constant term is one that does not involve any variables at all. However in expressions that involve terms with other types of factors than constants and powers of variables, the notion of constant term cannot be used in this sense, since that would lead to calling "4" the constant term of (x-3)^2+4, whereas substituting 0 for x in this polynomial makes it evaluate to 13.

From Yahoo Answers


Answers:sure.... y = -k x example think of a line y = mx + b where m is - and b is zero.....

Question:This is for a pendulum lab in grade 12 physics This is for a pendulum lab in grade 12 physics. My y-axis is labeled Frequency (Hz) and my x-axis is labeled Length (cm). My teacher said that the value of 'k' should be around 0.5... So for example, for one of the co-ordinates: Length = 21.9 and Frequency = 0.8889... Help?

Answers:Y=KX you need the values of Y and X and then solve for K

Question:Determine the value of the proportionality constant and then write the proportionality equation describing the relationship between: (a) frequency and length (b) period and length

Answers:Do you mean a simple pendulum? Denote f=frequency, L=length, g=acceleration due to gravity, T=period f=1/T (a) f= 1/(2pi) * sqrt (g/L) (b) T= 2pi * sqrt (L/g)


Answers:F = G m1 m2 / r^2 G is the gravitational constant. So on the surface of the earth (G m1 / r^2) gives the small g = 9.81 m/s/s with m1 the mass of the earth and r the radius of the earth.

From Youtube

Finding the Proportionality Constant :This is a short video about finding the proportionality constant

Beg Algebra: Direct Proportion :www.mindbites.com In order to explain direct proportionality, Professor Burger uses a real-world example of a spring and Hooke's Law. Hooke's law states that the distance a spring stretches varies directly to the force applied. If force, f, is directly proportional to distance, d, then d~f or d=kf. This equation allows us to find the constant, k, of how much the spring stretches when force is applied. After we have found this number, we can determine the distance the spring will stretch with varying forces applied. A lesson on inverse proportions can be found here: www.mindbites.comTaught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Beginning Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at www.thinkwell.com The full course covers linear equations, inequalities, polynomials, rational expressions, relations and functions, roots and radicals, quadratic equations and systems of equations. Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College. He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the ...