Cross-multiplication

In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable.

Given an equation like:

$\frac a b = \frac c d$

(where b and d are not zero), one can cross-multiply to get:

$ad = bc \qquad \mathrm{or} \qquad a = \frac {bc} d.$

In Euclidean geometry the same calculation can be achieved by considering the ratios as those of similar triangles.

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.

$\frac a b \nwarrow \frac c d \quad \frac a b \nearrow \frac c d.$

The mathematical justification for the method is from the following longer mathematical procedure. If we start with the basic equation:

$\frac a b = \frac c d$

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—bd—we get:

$\frac a b \times bd = \frac c d \times bd.$

We can reduce the fractions to lowest terms by noting that the two occurrences of $b$ on the left-hand side cancel, as do the two occurrences of d on the right-hand side, leaving:

$ad = bc$

and we can divide both sides of the equation by any of the elements—in this case we will use d—getting:

$a = \frac {bc} d.$

Another justification of cross-multiplication is as follows. Starting with the given equation:

$\frac a b = \frac c d$

multiply by d/d = 1 on the left and by b/b = 1 on the right, getting:

$\frac a b \times \frac d d = \frac c d \times \frac b b$

and so:

$\frac {ad} {bd} = \frac {cb} {db}.$

Cancel the common denominator bd = db, leaving:

$ad = cb.$

Each step in these procedures is based on a single, fundamental property of equations. Cross-multiplication is a shortcut, an easily understandable procedure that can be taught to 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 this, where x is a variable we are interested in solving for:

$\frac x b = \frac c d$

we can use cross multiplication to determine that:

$x = \frac {bc} d.$

For 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

$\frac x {7\ \mathrm{hours}} = \frac {90\ \mathrm{miles}} {3\ \mathrm{hours}}.$

Cross-multiplying yields:

$x = \frac {7\ \mathrm{hours} \times 90\ \mathrm{miles}} {3\ \mathrm{hours}}$

and so:

$x = 210\ \mathrm{miles}.$

Note that even simple equations like this:

$a = \frac {x} {d}$

are solved using cross multiplication, since the missing b term is implicitly equal to 1:

$\frac a 1 = \frac x d.$

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] is a shorthand version for a particular form of cross-multiplication, that may be taught to students by rote. It figures in the French national curriculum for secondary education.^[2]

For an equation of the form:

$\frac a b = \frac c x$

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

$x = \frac {bc} a.$

In this context, a is referred to as the extreme of the proportion, and b and c are called the means.

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 and Chinese mathematicians prior to the 7th century CE,^[3] 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 example, Cocker's Arithmetick introduces its discussion of the Rule of Three^[4] with the problem, "If 4 Yards of Cloth cost 12 Shillings, what will 6 Yards cost at that Rate?" The Rule of Three gives the answer to this problem directly; whereas in modern arithmetic, we would solve it by introducing a variable x to stand for the cost of 6 yards of cloth, writing down the equation:

$\frac {4\ \mathrm{yards}} {12\ \mathrm{shillings}} = \frac {6\ \mathrm{yards}} { x}$

and then using cross-multiplication to calculate x:

$x = \frac {12\ \mathrm{shillings} \times 6\ \mathrm{yards}} {4\ \mathrm{yards}} = 18\ \mathrm{shillings}.$

References

1. This was sometimes also referred to as the Golden Rule, though that usage is rare compared to other uses of Golden Rule. See E. Cobham Brewer (1898). "Golden Rule". Brewer's Dictionary of Phrase and Fable. Philadelphia: Henry Altemus.
2. "Socle de connaissances, pilier 3". French ministry of education. 30 December 2012. Retrieved 24 September 2015.
3. Shen Kangshen; John N. Crossley; Anthony W.-C. Lun (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford: Oxford University Press.
4. Edward Cocker (1702). Cocker's Arithmetick. London: John Hawkins. p. 103.

Further reading

• Brian Burell: Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster, 1998, ISBN 9780877796213, pp. 85-101
• '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 book-keeping., 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 Tweedle-dum is not the way of Tweedle-dee.