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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:
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--bd--we get
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
and we can divide both sides of the equation by any of the elements--in this case we will use d--getting
Another justification of cross-multiplication is as follows. Starting with the given equation
multiply by = 1 on the left and by = 1 on the right, getting
Cancel the common denominator bd = db, leaving
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
where x is a variable we are interested in solving for, we can use cross-multiplication to determine that
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
Note that even simple equations like
are solved using cross-multiplication, since the missing b term is implicitly equal to 1:
The rule of three was an 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 math education and still figures in the French national curriculum for secondary education.
For an equation of the form
where the variable to be evaluated is in the right-hand denominator, the rule of three states that
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, 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 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
and then using cross-multiplication to calculate x:
An anonymous manuscript dated 1570 said:
"Multiplication is vexation, / Division is as bad; / The Rule of three doth puzzle me, / And Practice drives me mad."
Double rule of three
An extension to the rule of three was the double rule of three, which involved finding an unknown value where five rather than three other values are known.
An example of such a problem might be If 6 builders can build 8 houses in 100 days, how many days would it take 10 builders to build 20 houses at the same rate?, and this can be set up as
^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.
^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 Education. Springer Science. p. 177. ISBN978-1-4614-9155-2.