Rule of Three (mathematics)

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.

For an equation like the following:

\frac a b = \frac c d (note that "b" and "d" must be non-zero for these to be real fractions)

one can cross-multiply to get

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

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 b's on the left hand side and the d's on the right hand side cancel, leaving:

ad = bc \,.

and we can divide both sides of the equation by any of the elements - in this case we will use "d" - yielding:

a = \frac {bc} {d}.

Another variation of the same process

\frac {a} {b} = \frac {c} {d}
\frac {a} {b} \times \frac {d} {d} = \frac {c} {d} \times \frac {b} {b}          multiply by 1 using alternate denominators
\frac {ad} {bd} = \frac {cb} {db}           divide out the common denominator
{ad} = {cb}

These give the same results as cross-multiplication.

Each step in these processes is based on a single, fundamental property of equations. Cross-multiplication 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):

\frac x b = \frac c d

we can use cross multiplication to determine that:

dx = bc \quad \mathrm {or} \quad x = \frac {bc} {d}

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

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

Cross-multiplying yields:

\begin{align}

& \frac x {7} \times 21 = \frac {90} {3} \times 21 \\ & x \times 3 = {90} \times 7 = 630 \\ & x = 210\ \mathrm {miles} \\ \end{align}

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:

a = \frac {x} {d}

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

\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] 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 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:

\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}.

For instance, if we re-wrote the equation used as an example above like so (inverting the proportions and swapping sides):

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

the Rule of Three can be used to calculate x directly

x = \frac {90\ \mathrm {miles} \times 7\ \mathrm {hours} } {3\ \mathrm {hours}} = 210\ \mathrm {miles}

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

References

Further reading

  • , 1827 - facsimile of the relevant section
  • - 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.