Cross-multiplication

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:

${\displaystyle {\frac {a}{b}}\nwarrow {\frac {c}{d}},\quad {\frac {a}{b}}\nearrow {\frac {c}{d}}.}$

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

${\displaystyle {\frac {a}{b}}={\frac {c}{d}},}$

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

${\displaystyle {\frac {a}{b}}\times bd={\frac {c}{d}}\times bd.}$

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

${\displaystyle ad=bc,}$

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

${\displaystyle a={\frac {bc}{d}}.}$

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

${\displaystyle {\frac {a}{b}}={\frac {c}{d}},}$

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

${\displaystyle {\frac {a}{b}}\times {\frac {d}{d}}={\frac {c}{d}}\times {\frac {b}{b}},}$

and so

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

Cancel the common denominator bd = db, leaving

${\displaystyle ad=cb.}$

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.

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

${\displaystyle {\frac {x}{b}}={\frac {c}{d}},}$

where x is a variable we are interested in solving for, we can use cross-multiplication to determine that

${\displaystyle x={\frac {bc}{d}}.}$

For example, suppose we want to know how far a car will travel in 7 hours, if we 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

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

Cross-multiplying yields

${\displaystyle x={\frac {7\ {\text{hours}}\times 90\ {\text{miles}}}{3\ {\text{hours}}}},}$

and so

${\displaystyle x=210\ {\text{miles}}.}$

Note that even simple equations like

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

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

${\displaystyle {\frac {a}{1}}={\frac {x}{d}}.}$

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.

The rule of three[1] was a historical shorthand version for a particular form of cross-multiplication that could be taught to students by rote. It was considered the height of Colonial maths education[2] and still figures in the French national curriculum for secondary education,[3] and in the primary education curriculum of Spain.[4]

For an equation of the form

${\displaystyle {\frac {a}{b}}={\frac {c}{x}},}$

where the variable to be evaluated is in the right-hand denominator, the rule of three states that

${\displaystyle x={\frac {bc}{a}}.}$

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 Chinese mathematicians prior to the 2nd century CE,[5] though it was not used in Europe until much later.

The rule of three gained notoriety for being particularly difficult to explain. Cocker's Arithmetick, the premier textbook in the 17th century, introduces its discussion of the rule of three[6] 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

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

and then using cross-multiplication to calculate x:

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

An anonymous manuscript dated 1570[7] said: "Multiplication is vexation, / Division is as bad; / The Rule of three doth puzzle me, / And Practice drives me mad."

Charles Darwin refers to his use of the rule of three in estimating the number of species in a newly discerned genus.[8] In a letter to William Darwin Fox in 1855, Charles Darwin declared “I have no faith in anything short of actual measurement and the Rule of Three.”[9] Karl Pearson adopted this declaration as the motto of his newly founded journal Biometrika.[10]

• Cross-ratio
• Odds ratio
• Turn (angle)

• 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
• 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.

• Media related to Cross-multiplication at Wikimedia Commons