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Why Does Cross Multiplication Work?

Date: 05/09/2002 at 18:27:43
From: Kimberly Geisler
Subject: Why does cross multiplication work?

I am a sixth grade teacher in PA. We just started working on 
proportions in math and a few students want to know why we cross 
multiply to solve a proportion like: 

  3/15 = n/30 

where we cross multiply 3 x 30= 90 then this means that 15n = 90. 
To solve this, I divide 90 by 15 and get 6 = n.  

Why does the rule a/b = c/d also mean ad = bc?  Why can we solve 
proportions by cross multiplying?  And I apologize if you have 
answered this before.  Thank you so much!


Date: 05/09/2002 at 22:55:09
From: Doctor Peterson
Subject: Re: Why does cross multiplication work?

Hi, Kimberly. 

You may well find something in our archives by going to the 
search page and entering "cross multiply"; I won't bother to look.

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.

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


Date: 05/11/2002 at 16:53:36
From: Kimberly Geisler
Subject: Why does cross multiplication work?

Dear Dr. Peterson,

I would really like to thank you for responding to my question.  
It is really going to be a big help on Monday when I go into 
school. 
 
I have used this web site for other math activities and problems. 
It is nice to know that there is such a reliable resource for 
teachers and students of mathematics.

Much thanks and appreciation,
Kimberly Geisler
Associated Topics:
Middle School Algebra
Middle School Division
Middle School Fractions
Middle School Ratio and Proportion

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