Cross-Multiplication

Date: 08/04/2002 at 01:51:27
From: Cammy
Subject: How do you cross multply?

Please, I know it's simple but I still have trouble.

I can't figure out a way to type it out, so can you just show me the 
basics so I can understand?

Thanks, Cammy

Date: 08/04/2002 at 16:14:12
From: Doctor Ian
Subject: Re: How do you cross multply?

Hi Cammy,

When you cross-multiply, you're really multiplying each side by n/n, 
where n is the denominator of the other side:

         a   c
         - = - 
         b   d


     a   d   c   b
     - * - = - * - 
     b   d   d   b

Note that this doesn't change anything, because multiplying anything 
by 1 doesn't change its value.  So now we have

     ad   bc
     -- = -- 
     bd   bd

Since you're dividing both sides by the same thing, you can ignore 
it, to get

     ad = bc

Once you get used to the idea, you can skip all the middle steps and 
just multiply the things that are crosswise from each other:

  a       c
    .    .
     .  .
  -   =  -
     .  .        
    .     .
  b        d

which is where we get the name 'cross-multiplying'. 

It's always good to understand something in more than one way, so 
here's another way to look at it. When you have any two things that 
are equal, 

   this = that

then if neither of them is zero, it will be true that

   this
   ---- = 1
   that

right?  Let's look at that for

         a   c
         - = - 
         b   d


      a/b
     ----- = 1
      c/d

 
     a   d
     - * - = 1                Because to divide by a fraction,
     b   c                    you invert and multiply...

        ad
        -- = 1
        bc

        ad = bc               Here, we're just noting that 
                              if this/that = 1, then 
                              this = that...

If you just learn to cross-multiply without understanding _why_ it 
works, then it can seem like magic. But after you do a couple of dozen 
of these the long way, your brain will automatically start looking 
ahead a few steps, and you'll start taking the shortcut naturally, 
because you're _ready_ to. And at that point, you'll never really be 
able to forget how to cross-multiply, because you'll always be able to 
figure it out again if you need to. 

In addition, you might want to take a look at this:

   Flipping and Switching Fractions
   http://mathforum.org/library/drmath/view/58193.html 

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/