Fractions and Cross Multiplication

Date: 07/14/98 at 02:12:54
From: Lauri Novotny
Subject: Multiplying fractions

Hello Dr. Math,

A friend recently asked for my help in refreshing her basic math 
skills (she is an older student who will soon be going back to 
school). I agreed to help her, and we were quickly into fractions 
(+,-,x,/). When we started to do multiplication, my first response was 
to tell her that (a/b)(c/d) = ac/bd, but then there was something in 
my brain that said to cross multiply. I looked it up in a book and I 
know now that my first response was correct, but it has left me 
wondering when do you "cross multiply." I have tried to find the 
answer but without much success.  Can you help me?

Laura

Date: 07/15/98 at 12:17:50
From: Doctor Peterson
Subject: Re: Multiplying fractions

Hi, Laura. I always like to help someone who's helping someone else!

There are a couple of ways I have seen the idea of cross 
multiplication used. Only the first of them is really something I 
recommend.

1. To show that two fractions are equal

If I have an equation like:

    5   30
    - = --
    6   36

and want to check whether it is true, one way is to cross multiply:

    5 * 36 = 180
    30 * 6 = 180

Since these are equal, the fractions are equal. This is most useful in 
algebra, where you can quickly transform an equation like:

    x-2   x+3
    --- = ---
    x+4   x-2

to:

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

It's a better habit to think of this instead as multiplying both sides 
by the product of the denominators, but thinking of it as cross-
multiplying can be a convenient trick.

2. To add two fractions

Some people think of the numerator of a sum as cross-multiplying, 
since you add two crossed products:

     a     c    ad + bc
    --- + --- = -------
     b     d       bd

I don't like expressing it this way, because it leaves out the step of 
finding the least common denominator (since bd is only a common 
denominator, not necessarily the smallest), making the work often more 
complicated than it has to be.

It sounds as if you have what it takes to help your friend. If you 
haven't looked at our Dr. Math archives, you might want to check them 
out, since we have some good explanations of things like multiplying 
and adding fractions:

   http://mathforum.org/dr.math/tocs/fractions.elem.html   

- Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/