# Cross multiplication

The cross multiplication (also known as cross multiplication and multiplication over cross ) is a method of transforming an equation in which both sides are represented by a fraction or a fraction term in such a way that there are no more fractions or fraction terms afterwards. The method is often used for ratio equations and rule of three problems .

## description

A fractional equation of the form is given

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

with and . If both sides of this equation are now multiplied by, the result is after shortening the fractions ${\ displaystyle b \ neq 0}$${\ displaystyle d \ neq 0}$${\ displaystyle bd}$

${\ displaystyle ad = bc}$.

With cross multiplication, the denominator on the right side is multiplied by the numerator on the left and the denominator on the left side with the numerator of the right. In the clear illustration there is a cross, hence the name:

## example

The equation is given:

${\ displaystyle {\ frac {x} {2x-2}} = {\ frac {2x + 3} {4x}}}$

We are looking for the value of x , which, however, must be neither 1 nor 0 because of the division by 0. The fractions cannot be shortened. Cross multiplication results in:

4 x 2 = (2 x - 2) (2 x + 3)

The brackets on the right side are removed by multiplying them out twice :

4 x 2 = 4 x 2 + 2 x - 6

Now you subtract (4x 2 + 2x) or (4x 2 - 6):

−2 x = −6 or 6 = 2 x

Dividing by −2 or by 2 gives:

x = 3

As a sample, 3 could be plugged into the first equation. It follows:

${\ displaystyle {\ frac {3} {4}} = {\ frac {3} {4}}}$

So the solution is valid.