Why do cross products work?

Answer 1

There was a time when this rule was considered a wonderful piece of magic, called the "rule of three"; students just learned to do it without expecting to understand why it worked. That makes me sad! With algebra, it's almost obvious, and certainly not something special. I'll try to express this in a way that students who don't know algebra (or don't realize how much of it they have already seen) can follow. Remember that if you have two equal quantities and multiply them by the same amount, the products will again be equal. So if we multiply the fractions a/b and c/d by b, the results are equal: a c --- * b = --- * b b d which can be written as bc a = ---- d Now we can multiply both fractions by d: bc ad = ---- * d d which, of course, means ad = bc I personally prefer not to cross multiply, but just to multiply by whichever denominator helps. In your example, 3/15 = n/30, I would just multiply both sides by 30 and get n = 30*3/15 = 30/15*3 = 2*3 = 6.

I would like to add that cross products are used to find the unknown of a proportion (or ratio) that already exists by the nature of the equality.

For example a recipe calls for 2 cups of flour to make 4 pancakes but I want 12.

I can say 2 is to 4 what x is to 12 and set up the ratio:

2/4 :: x/12

2 x 12 = 4x (cross multiply)

24 = 4x

24/4 = x (and divide)

x = 6 cups of flour

this method is also used (transparently) in chemistry and engineering, and is crucial in solving algebraic equations.