Multiplying Fractions

Date: 02/06/2001 at 23:22:56
From: Alysha Houston
Subject: Multiplying fractions

I need help on how to solve this:


   8 * 3
   -   -
   9   5


My teacher said something about cross-cancelling, but I didn't 
understand. Can you please explain? Thanks.

Date: 02/07/2001 at 08:47:02
From: Doctor Rick
Subject: Re: Multiplying fractions

Hi, Alysha.

You know the basic method for multiplying fractions, right? You 
multiply the numerators, and that's the numerator of the product. 
Multiply the denominators, and that's the denominator of the product.

Let's do that, but DON'T do the multiplications yet - just write them 
where they go.

  8   3   8*3
  - * - = ---
  9   5   9*5

Now, to reduce the fraction to lowest terms, we want to look for a 
common factor in the numerator and denominator. By not multiplying 
first, we make this job easier: we have already done some factoring of 
the numerator and denominator. I'll finish the factoring, but keep the 
factors together to show where they came from:

  8*3   (2*2*2)*3
  --- = ---------
  9*5    (3*3)*5

Now you can see a common factor: 3. We can "pull them out" of the 
fraction:

  8*3   (2*2*2)   3
  --- = ------- * -
  9*5    (3)*5    3

But 3/3 = 1, so all that's left is

  8*3   (2*2*2)    8
  --- = ------- = --
  9*5    (3)*5    15

That's the answer. What we call "cancelling" is really "pulling out" 
the same number in the numerator and denominator, making a factor of 
1, as I did above when I pulled out a factor of 3/3.

Now we can talk about "cross-cancelling." Notice that the 3 in the 
numerator came from the 3 of 8*3, while the 3 in the denominator came 
from the 9 of 9*5. In other words, the 3 in the numerator was from the 
numerator of the second fraction, 3/5, while the 3 in the denominator 
was from the denominator of the first fraction, 8/9.

You don't need to write nearly as much as I did. You can just look at 
the problem

  8   3
  - * - = ?
  9   5

and visualize it like this:

  8*3
  --- = ?
  9*5

Then look for common factors in one number of the numerator and the 
other number of the denominator. We see that 3 is a factor of both 3 
(in the numerator) and 9 (in the denominator). Divide each of these 
numbers by 3 (on paper, you'd cross out the 3 and write a 1 above it, 
and cross out the 9 and write a 3 below it):

  8*1
  --- = ?
  3*5

You can look again for other common factors. Do the 8 and 5 have a 
common factor? No. Therefore we're done, and the product is

  8*1    8
  --- = --
  3*5   15

Do you see why it's called cross-cancellation? You look for common 
factors in these pairs:

  8    3
   \  /
 __ \/ __
    /\
   /  \
  9    5

It's possible you might find common factors vertically, too - between 
the 8 and 9, or between the 3 and 5. But if you did, it would mean 
that the fractions you started with weren't in lowest terms. If the 
fractions are in lowest terms, then you only need to look for common 
factors in the "cross-terms."

You can find more help in our Dr. Math Archives. Go to the Search Dr. 
Math page at http://mathforum.org/mathgrepform.html   , and try 
searching for the words

  multiply fractions cross cancel

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