explain what happens to the cross products when the terms of a proportion are cross multiplied.

In mathematics, specifically in uncomplicated arithmetic and elementary algebra, given an equation betwixt 2 fractions or rational expressions, one can cross-multiply to simplify the equation or make up one's mind the value of a variable.

The method is also occasionally known as the "cross your heart" method because lines resembling a eye outline can exist drawn to remember which things to multiply together.

Given an equation like

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

where b and d are non zip, ane tin can cross-multiply to get

${\displaystyle ad=bc\quad {\text{or}}\quad a={\frac {bc}{d}}.}$

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

Procedure [edit]

In practice, the method of cross-multiplying ways that nosotros multiply the numerator of each (or ane) side by the denominator of the other side, finer 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 process. If we beginning with the basic equation

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

nosotros can multiply the terms on each side past 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 go

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

We tin reduce the fractions to lowest terms past noting that the ii occurrences of b on the left-hand side cancel, every bit do the two occurrences of d on the correct-paw side, leaving

${\displaystyle ad=bc,}$

and we tin can carve up both sides of the equation by any of the elements—in this case nosotros 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 = one on the left and by b / b = i 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

${\displaystyle advertising=cb.}$

Each step in these procedures is based on a single, fundamental belongings of equations. Cross-multiplication is a shortcut, an easily understandable procedure that tin can exist taught to students.

Use [edit]

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

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

where 10 is a variable we are interested in solving for, nosotros can use cross-multiplication to make up one's mind that

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

For case, suppose we want to know how far a car volition travel in 7 hours, if we know that its speed is constant and that information technology already travelled 90 miles in the last iii hours. Converting the word problem into ratios, nosotros go

${\displaystyle {\frac {ten}{7\ {\text{hours}}}}={\frac {90\ {\text{miles}}}{iii\ {\text{hours}}}}.}$

Cross-multiplying yields

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

and then

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

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:

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

Any equation containing fractions or rational expressions tin can be simplified by multiplying both sides by the to the lowest degree common denominator. This step is called clearing fractions.

Rule of three [edit]

The dominion of three ^[i] was a historical autograph version for a particular form of cross-multiplication that could be taught to students past rote. It was considered the peak of Colonial math education^[two] and even so figures in the French national curriculum for secondary education.^[3]

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 dominion of three states that

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

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 second century CE,^[four] though it was non used in Europe until much later.

The rule of three gained notoriety^{[ citation needed ]} for being particularly difficult to explain. Cocker'due south Arithmetick, the premier textbook in the 17th century, introduces its discussion of the rule of 3^[5] with the problem "If iv yards of cloth cost 12 shillings, what will half-dozen yards cost at that rate?" The rule of 3 gives the answer to this problem directly; whereas in modern arithmetic, we would solve it by introducing a variable 10 to stand for the cost of 6 yards of material, writing down the equation

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

and so using cross-multiplication to calculate x:

${\displaystyle 10={\frac {12\ {\text{shillings}}\times vi\ {\text{yards}}}{four\ {\text{yards}}}}=eighteen\ {\text{shillings}}.}$

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

Double rule of 3 [edit]

An extension to the rule of three was the double rule of three, which involved finding an unknown value where v rather than three other values are known.

An case of such a problem might be If 6 builders tin can build 8 houses in 100 days, how many days would it take 10 builders to build xx houses at the same rate?, and this can be set up as

${\displaystyle {\frac {\frac {8\ {\text{houses}}}{100\ {\text{days}}}}{half-dozen\ {\text{builders}}}}={\frac {\frac {20\ {\text{houses}}}{x}}{10\ {\text{builders}}}},}$

which, with cross-multiplication twice, gives

${\displaystyle x={\frac {20\ {\text{houses}}\times 100\ {\text{days}}\times 6\ {\text{builders}}}{viii\ {\text{houses}}\times 10\ {\text{builders}}}}=150\ {\text{days}}.}$

Lewis Carroll'southward "The Mad Gardener's Song" includes the lines "He thought he saw a Garden-Door / That opened with a key: / He looked again, and found information technology was / A double Rule of Iii".^[seven]

Run across also [edit]

• Cross-ratio
• Odds ratio

References [edit]

1. ^ This was sometimes also referred to equally the golden dominion, though that usage is rare compared to other uses of golden dominion. See E. Cobham Brewer (1898). "Gold Rule". Brewer's Dictionary of Phrase and Fable. Philadelphia: Henry Altemus.
2. ^ Ubiratan D'Ambrósio; Joseph W. Dauben; Karen Hunger Parshall (2014). "Mathematics Education in America in the Premodern Period". In Alexander Karp; Gert Schubring (eds.). Handbook on the History of Mathematics Didactics. Springer Science. p. 177. ISBN978-1-4614-9155-2.
3. ^ "Socle de connaissances, pilier iii". French ministry of instruction. 30 December 2012. Retrieved 24 September 2015.
4. ^ Shen Kangshen; John North. Crossley; Anthony Westward.-C. Lun (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford: Oxford University Printing.
5. ^ Edward Cocker (1702). Cocker's Arithmetick. London: John Hawkins. p. 103.
6. ^ Concise Oxford Dictionary of Quotations, 1964.
7. ^ Sylvie and Bruno, Chapter 12.

Farther reading [edit]

• 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 Iii
• 'Dr Math', Abraham Lincoln and the Dominion of 3
• Pike's Organization of arithmetick abridged: designed to facilitate the study of the science of numbers, comprehending the nearly 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 Dominion of Three as applied by Michael of Rhodes in the fifteenth century
• The Rule Of Three in Mother Goose
• Rudyard Kipling: You lot can work it out by Fractions or by simple Rule of Three, But the fashion of Tweedle-dum is not the manner of Tweedle-dee.

External links [edit]

• Media related to Cross-multiplication at Wikimedia Commons

southernpothead1969.blogspot.com

Source: https://en.wikipedia.org/wiki/Cross-multiplication