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Reverted edits by Emedoh (talk) to last version by Bovineboy2008

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	In [[mathematics]], specifically in [[elementary arithmetic]] and [[elementary algebra]], given an equation between two [[Fraction (mathematics)\|fraction]]s or [[rational expression]]s, one can '''cross-multiply''' to simplify the equation or determine the value of a variable.		The author is known by the title of Figures Lint. To gather cash is one of the issues I love most. For a whilst she's been in South Dakota. My day job is a meter reader.<br><br>My website :: [http://chatmast.com/index.php?do=/BookerSipessk/info/ home std test kit]

	~~For an equation like the following:~~

	~~:<math>\frac a b = \frac c d</math> (note that "b" and "d" must be non-zero for these to be [[real number\|real]] fractions)~~

	~~one can cross-multiply to get~~

	~~:<math>ad = bc \quad \mathrm {or} \quad a = \frac {bc} {d}.</math>~~

	~~== Procedure ==~~

	~~In practice, the method of ''cross-multiplying'' means that we multiply the numerator of each (or one) side by the denominator of the other side, effectively "crossing" the terms over.~~

	~~:<math>\frac a b \nwarrow \frac c d \quad \frac a b \nearrow \frac c d.</math>~~

	The ~~mathematical justification for the method~~ is ~~from the following longer mathematical procedure.~~

	~~If we start with the basic equation:~~

	~~:<math>\frac {a} {b} = \frac {c} {d}</math>~~

	~~We can multiply the terms on each side~~ by the ~~same number and the terms will remain equal. Therefore, if we multiply the fraction on each side by the product~~ of ~~the denominators of both sides - <math>(bd)\,\!</math> - we get:~~

	~~:<math>\frac {a} {b} \times {bd} = \frac {c} {d} \times {bd}</math>~~

	~~We can reduce the fractions to lowest terms by noting that the b's on the left hand side and the d's on the right hand side cancel, leaving:~~

	~~:<math>ad = bc \,</math>~~.

	~~and we can divide both sides~~ of the ~~equation by any of the elements - in this case we will use "d" - yielding:~~

	~~:<math>a = \frac {bc} {d}~~.~~</math>~~

	~~Another variation of the same process~~

	~~:<math>\frac {a} {b} = \frac {c} {d}</math>~~

	~~:<math>\frac {a} {b} \times \frac {d} {d} = \frac {c} {d} \times \frac {b} {b}</math>          multiply by 1 using alternate denominators~~

	~~:<math>\frac {ad} {bd} = \frac {cb} {db}</math>           divide out the common denominator~~

	~~:<math>{ad} = {cb}</math>~~

	~~These give the same results as cross-multiplication.~~

	Each step in these processes is based on a single, fundamental property of [[equation]]s. Cross-multiplication was devised as a shortcut, in particular as an easily understood procedure to teach students.

	~~== Use ==~~
	~~This is a common procedure in mathematics, used to reduce fractions or calculate a value for a given variable in a fraction. If we have an equation like (where ''x'' is a variable):~~

	~~:<math>\frac x b = \frac c d</math>~~

	~~we can use cross multiplication to determine that:~~

	~~:<math>dx = bc \quad \mathrm {or} \quad x = \frac {bc} {d}</math>~~

	For a ~~simple example, let~~'s ~~say that we want to know how far a car will get~~ in ~~7 hours, if we happen to know that its speed is constant and that it already travelled 90 miles in the last 3 hours~~. ~~Converting the word problem into ratios we get~~

	~~:<math>\frac {\mathrm {x}\ miles} {7\ hours} = \frac {90\ miles} {3\ hours}</math>~~

	~~Cross-multiplying yields:~~

	~~:<math>\begin{align}~~
	~~& \frac x {7} \times 21 = \frac {90} {3} \times 21 \\~~
	~~& x \times 3 = {90} \times 7 = 630 \\~~
	~~& x = 210\ \mathrm {miles} \\~~
	~~\end{align}</math>~~

	It is ~~important to keep track of the units, in this case 'miles' and 'hours', though they have been left out of the above equations for simplicity~~.

	~~note that even simple equations like this:~~

	:<~~math~~>~~a = \frac {x} {d}~~<~~/math~~>

	~~are solved using cross multiplication, since the missing "b" term is implicitly equal to 1~~: ~~e.g.~~:

	~~:<math>\frac a 1 = \frac x d.</math>~~

	~~Any equation containing fractions or rational expressions can be simplified by multiplying both sides by the [[least common denominator]]. This step is called "clearing fractions".~~

	~~== Rule of Three ==~~

	~~The Rule of Three<ref>This was sometimes also referred to as the Golden Rule - see~~ [http://~~www.bartleby~~.com/81/7351.html Golden Rule], [[Brewer's Dictionary of Phrase and Fable]] though that usage is rare compared to other uses of [[Golden Rule]]</ref> was a shorthand version for a particular form of cross multiplication, often taught to students by rote. This rule was already known to Hebrews by the 15th century BCE as it is a special case of the Kal va-chomer (קל וחומר). It was also known by Indian (Vedic) mathematicians in the 6th century BCE{{Citation needed\|date=July 2008}} and Chinese mathematicians prior to the 7th century CE,<ref>Shen, Kangren et al, tr. The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford: Oxford University Press, 1999.</ref> though it was not used in Europe until much later. ~~The Rule of Three gained notoriety for being particularly difficult to explain: see [[Cocker's Arithmetick]] for an example of how the premier textbook in the 17th century approached the subject.~~

	~~For an equation of the form:~~

	~~:<math>\frac a b = \frac c x</math>~~

	~~where the variable to be evaluated is in the right-hand denominator, the Rule of Three states that:~~

	~~:<math>x = \frac {bc} {a}.</math>~~

	~~For instance, if we re-wrote the equation used as an example above like so (inverting the proportions and swapping sides):~~

	~~:<math>\frac {3\ \mathrm {hours}} {90\ \mathrm {miles}} = \frac {7\ \mathrm {hours}} {x\ \mathrm {miles}} \quad</math>~~

	~~the Rule of Three can be used to calculate <math>x</math> directly~~

	~~:<math>x = \frac {90\ \mathrm {miles} \times 7\ \mathrm {hours} } {3\ \mathrm {hours}} = 210\ \mathrm {miles}</math>~~

	~~In this context, <math>a</math> is referred to as the 'extreme' of the proportion, and <math>b</math> and <math>c</math> are called the 'means'.~~

	~~==References==~~
	~~<references />~~

	== ~~Further reading ==~~
	* [http://mathforum.org/library/drmath/view/60822.html 'Dr Math', ''Rule of Three'']
	* [http://mathforum.org/library/drmath/view/62685.html 'Dr Math', ''Abraham Lincoln and the Rule of Three'']
	* [http://~~www.pballew.net~~/arithm18.html ''Pike's System of arithmetick abridged: designed to facilitate the study of the science of numbers, comprehending the most perspicuous and accurate rules, illustrated by useful examples: to which are added appropriate questions, for the examination of scholars, and a short system of book-keeping.'', 1827] - facsimile of the relevant section
	* [http://faculty.ed.uiuc.edu/westbury/paradigm/Hersee.html Hersee J, ''Multiplication is vexation''] - an article tracing the history of the rule from 1781
	* [http://brunelleschi.imss.fi.it/michaelofrhodes/math_toolkit_three.html The Rule of Three as applied by Michael of Rhodes in the fifteenth century]
	* [http://www.rhymes.org.uk/a61-multiplication.htm The Rule Of Three in Mother Goose]
	* [http://www.literaturepage.com/read/thejunglebook-127.html Rudyard Kipling: You can work it out by Fractions or by simple Rule of Three, But the way of Tweedle-dum is not the way of Tweedle-dee.]

	~~[[Category:Fractions]~~]

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