Cross-multiplication

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In an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable. That is, 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 proper fractions)

one can cross multiply to get

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

[edit] Procedure

Cross multiplication works as follows. If each side of an equation is multiplied by the same number, the two sides will remain equal (symmetry). Therefore, if we multiply the fraction on each side by the product of the denominators of each side $(bd)\,\!$ we get the following:

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

We reduce to lowest terms by noting that the b's on the left hand side and the d's on the right hand side cancel.

$ad = bc \,$

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

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

We use the term "cross multiplying" because it appears as though we have multiplied the numerator of each side by the denominator of the other side, "crossing" the terms over to the other side.

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

[edit] 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 'kinds' of things (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".

[edit] Historical use

The Rule of Three (sometimes called the Golden Rule)^[1] was a shorthand version for a particular form of cross multiplication, often taught to students by rote. for an equation of the form:

$\frac a b = \frac c x$