 Crossmultiplication

In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can crossmultiply to simplify the equation or determine the value of a variable.
For an equation like the following:
 (note that "b" and "d" must be nonzero for these to be real fractions)
one can crossmultiply to get
Contents
Procedure
In practice, the method of crossmultiplying means that we multiply the numerator of each (or one) side by the denominator of the other side, effectively "crossing" the terms over.
The mathematical justification for the method is from the following longer mathematical procedure.
If we start with the basic equation:
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   we get:
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:
 .
and we can divide both sides of the equation by any of the elements  in this case we will use "d"  yielding:
Another variation of the same process multiply by 1 using alternate denominators
 divide out the common denominator
 ad = cb
These give the same results as crossmultiplication.Each step in these processes is based on a single, fundamental property of equations. Crossmultiplication 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):
we can use cross multiplication to determine that:
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
Crossmultiplying yields:
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:
are solved using cross multiplication, since the missing "b" term is implicitly equal to 1: e.g.:
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^{[1]} was a shorthand version for a particular form of cross multiplication, often taught to students by rote. This rule was known to Indian (Vedic) mathematicians in the 6th century BCE^{[citation needed]} and Chinese mathematicians prior to the 7th century CE^{[2]}, 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:
where the variable to be evaluated is in the righthand denominator, the Rule of Three states that:
For instance, if we rewrote the equation used as an example above like so (inverting the proportions and swapping sides):
the Rule of Three can be used to calculate x directly
In this context, a is referred to as the 'extreme' of the proportion, and b and c are called the 'means'.
References
 ^ This was sometimes also referred to as the Golden Rule  see Golden Rule, Brewer's Dictionary of Phrase and Fable though that usage is rare compared to other uses of Golden Rule
 ^ Shen, Kangren et al, tr. The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford: Oxford University Press, 1999.
Further reading
 'Dr Math', Rule of Three
 'Dr Math', Abraham Lincoln and the Rule of Three
 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 bookkeeping., 1827  facsimile of the relevant section
 Hersee J, Multiplication is vexation  an article tracing the history of the rule from 1781
 The Rule of Three as applied by Michael of Rhodes in the fifteenth century
 The Rule Of Three in Mother Goose
 Rudyard Kipling: You can work it out by Fractions or by simple Rule of Three, But the way of Tweedledum is not the way of Tweedledee.
Categories: Fractions
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Look at other dictionaries:
cross multiplication — noun see cross multiply … New Collegiate Dictionary
crossmultiplication — noun An instance, or the process, of cross multiplying … Wiktionary
cross multiplication — noun Etymology: cross (III) : the multiplying of a numerator of one fraction by the denominator of another (as in clearing an equation of fractions) … Useful english dictionary
crossmultiplication — … Useful english dictionary
crossmultiply — cross mul·ti·ply (krôsʹmŭlʹtə plī , krŏsʹ ) intr.v. cross mul·ti·plied, cross mul·ti·ply·ing, cross mul·ti·plies To multiply the numerator of one of a pair of fractions by the denominator of the other. cross multiplication n. * * * … Universalium
cross multiply — cross multiplication. Math. to remove fractions from an equation by multiplying each side by the common multiple of the denominators of the fractions of the opposite side. [1950 55] * * * … Universalium
cross multiply — intransitive verb Etymology: back formation from cross multiplication : to find the two products obtained by multiplying the numerator of each of two fractions by the denominator of the other * * * cross multiplication. Math. to remove fractions… … Useful english dictionary
cross multiply — intransitive verb Date: 1951 to clear an equation of fractions when each side consists of a fraction with a single denominator by multiplying the numerator of each side by the denominator of the other side and equating the two products obtained • … New Collegiate Dictionary
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