How is it sometimes helpful to use cross multiplication in order to complete proportions with a variable? How can it be helpful to do cross multiplication with proportions with variables such as ${2\over 4}={3\over x}$?  In this one, the value of x has to be found.  It can be found this way:  1. Do the cross multiplication by multiplying 2 by x and 3 by 4 to get: $$2x=12$$  2. Divide 2 by each side to get: $${2x\over 2}={12\over 2}$$ 3.  This is what you get: $$x=6$$  Now it's time to check the solution.  1. Substitute 6 for x: $${2\over 4}={3\over 6}$$  2. Find the cross multiplication by multiplying 2 by 6 and 3 by 4: $$12=12$$ The solution checks here.  Therefore, ${2\over 4}={3\over 6}$.  Is this helpful to you?  Also, I still have this question in the title to ask to you, so please answer it.
 A: One way to demonstrate that two fractions are equivalent is to reduce them to simplest form.  If they have the same simplest form, then they are equivalent.  For instance, since
$$\frac{9}{12} = \frac{3 \cdot 3}{3 \cdot 4} = \frac{3}{4}$$
and 
$$\frac{15}{20} = \frac{5 \cdot 3}{5 \cdot 4} = \frac{3}{4}$$
we may conclude that 
$$\frac{9}{12} = \frac{15}{20}$$
by substitution.
Observe that $9 \cdot 20 = 12 \cdot 15$.  Why should this be the case?  
Consider the cancellations we made to reduce each fraction to simplest form.  We obtain $9/12$ from $3/4$ by multiplying the numerator and denominator of $3/4$ by $3$.  We obtain $15/20$ by multiplying the numerator and denominator of $3/4$ by $5$.
In general, if $a, b$ are integers with $b \neq 0$, then any two fractions that are equivalent to $a/b$ can be expressed in the form $(ma)/(mb)$ and $(na)/(nb)$, where $m$ and $n$ are non-zero integers.  
$$\frac{ma}{mb} = \frac{na}{nb} = \frac{a}{b}$$
If we take the equation 
$$\frac{ma}{mb} = \frac{na}{nb}$$
and cross-multiply, we obtain
$$(ma)(nb) = mnab = (mb)(na)$$
Consequently, if we have two fractions that are equivalent such as $2/4$ and $3/x$, we can solve the equation 
$$\frac{2}{4} = \frac{3}{x}$$
by cross-multiplication.
In fact, if $a, b, c, d$ are integers with $b, d \neq 0$, mathematicians define $$\frac{a}{b} = \frac{c}{d}$$ if and only if $ad = bc$.
A: I think you're confusing cross-multiplication with cross product. A cross product is a binary operation on two vectors $u,v \in$R$^3$ that produces a perpendicular vector to both vectors $u,v$. "Cross multiplication" is what you were trying to do above. 
