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### 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

-- = --
bd   bd

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

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...

-- = 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/
```
Associated Topics:
Elementary Fractions
Elementary Multiplication
Middle School Fractions

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