TITLE: Explaining why cross multiplication works using a pie
QUESTION [10 upvotes]: $\dfrac ab= \dfrac cd$ is true precisely when $ad = bc$.
As an example $\frac 26 = \frac 39$ is true because $2*9 = 3*6$.
$\frac 26$ can be thought of as dividing a pie into $6$ equal slices, then taking two of them.  
Similarly $\frac 39$ refers to dividing a pie into $9$ equal slices, then taking $3$ of them.
In this particular scenario I'm struggling a bit to understand if the expressions $2*9$ or $6*3$ represent any meaningful quantities, like total number of slices in both pies etc. I would appreciate any help. Thanks!

REPLY [2 votes]: One way to prove the cross-multiplication rule is to see that (assuming $x\neq0\neq x'$)
$$
\frac{x}{y} = \frac{x'}{y'} \Leftrightarrow \frac{xx'}{yx'} = \frac{xx'}{xy'} \Leftrightarrow yx' = xy'
$$
In "cake reasoning", this is equivalent to:

as starting point, assume you have one cake split in six parts, and two of them are taken (but not yet eaten), and one cake split in nine parts, three of them taken, as in the traditional cake analogy for fractions.
cut each slice -- both taken and not taken -- of the first cake in three parts (you are multiplying both terms of the left-hand fraction by $x'$)
cut each slice of the second cake in two parts (multiply the terms of the right-hand fraction by $x$)
you will have 6 pieces of cake taken from either cake, hence the amount of "taken" cake in the two cases will be equal if (and only if) the slices are the same size: the first cake will now have $6*3=18$ total slices, the second one will have $2*9$.