Changes

<center>For lessons on the related [[topic]] of '''''[[Balance]]''''', follow [https://nordan.daynal.org/wiki/index.php?title=Category:Balance '''''this link'''''].</center>

<center>For lessons on the related [[topic]] of '''''[[Balance]]''''', follow [https://nordan.daynal.org/wiki/index.php?title=Category:Balance '''''this link'''''].</center>

===Examples===

===Examples===

* If an object travels at a constant [http://en.wikipedia.org/wiki/Speed speed], then the distance traveled is proportional to the time spent traveling, with the speed being the constant of proportionality.

* If an object travels at a constant [https://en.wikipedia.org/wiki/Speed speed], then the distance traveled is proportional to the time spent traveling, with the speed being the constant of proportionality.

* The [http://en.wikipedia.org/wiki/Circumference circumference] of a [[circle]] is proportional to its [http://en.wikipedia.org/wiki/Diameter diameter], with the constant of proportionality equal to π.

* The [https://en.wikipedia.org/wiki/Circumference circumference] of a [[circle]] is proportional to its [https://en.wikipedia.org/wiki/Diameter diameter], with the constant of proportionality equal to π.

* On a map drawn to [http://en.wikipedia.org/wiki/Scale 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.

* On a map drawn to [https://en.wikipedia.org/wiki/Scale 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 [http://en.wikipedia.org/wiki/Mass mass]; the constant of proportionality between the mass and the force is known as gravitational acceleration.

* The [[force]] acting on a certain object due to [[gravity]] is proportional to the object's [https://en.wikipedia.org/wiki/Mass mass]; the constant of proportionality between the mass and the force is known as gravitational acceleration.

===Properties===

===Properties===

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.

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 [http://en.wikipedia.org/wiki/Slope slope] of the line equal to the constant of proportionality: it corresponds to [[linear]] [[growth]].

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 [https://en.wikipedia.org/wiki/Slope slope] of the line equal to the constant of proportionality: it corresponds to [[linear]] [[growth]].

==Inverse proportionality==

==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.

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.

[[File:Proportion3.jpg|center]]

[[File:Proportion3.jpg|center]]

The constant can be found by multiplying the [[original]] x variable and the original y [http://en.wikipedia.org/wiki/Variable variable].

The constant can be found by multiplying the [[original]] x variable and the original y [https://en.wikipedia.org/wiki/Variable variable].

Basically, the [[concept]] of inverse proportion [[Meaning|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.

Basically, the [[concept]] of inverse proportion [[Meaning|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 [http://en.wikipedia.org/wiki/ speed] of travel; the time needed to dig a hole is (approximately) inversely proportional to the number of people digging.

For example, the time taken for a journey is inversely proportional to the [https://en.wikipedia.org/wiki/ 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 [http://en.wikipedia.org/wiki/Cartesiancoordinate Cartesian coordinate] plane is a [http://en.wikipedia.org/wiki/Hyperbola 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.

The graph of two variables varying inversely on the [https://en.wikipedia.org/wiki/Cartesiancoordinate Cartesian coordinate] plane is a [https://en.wikipedia.org/wiki/Hyperbola 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.

==Experimental determination==

==Experimental determination==

To determine [[experiment]]ally whether two [[physical]] quantities are directly proportional, one [[performs]] several [[measurements]] and plots the resulting [[data]] points in a [http://en.wikipedia.org/wiki/Cartesiancoordinate 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.

To determine [[experiment]]ally whether two [[physical]] quantities are directly proportional, one [[performs]] several [[measurements]] and plots the resulting [[data]] points in a [https://en.wikipedia.org/wiki/Cartesiancoordinate 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.

[[Category: Mathematics]]

[[Category: Mathematics]]