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Multiplying Fractions, Dividing Common Factors: Why Does It Work?

Date: 06/10/2010 at 14:10:42
From: Jim
Subject: Multiplying Fractions

A student's former teacher taught her students that if the numerator of
one fraction and the denominator of the other fraction had a common
factor, then they could divide each by that factor before multiplying.

For example, say we have

   4/6 x 8/9
   
Since the 6 and 8 are multiples of 2, you could divide both by two,
creating the following result:

   4/3 x 4/9.
   
When multiplied, 4/6 x 8/9 has the same result as 4/3 x 4/9.

My question is ... why does this work? What is the underlying math
principle? My kids and I would love to know!

Thanks for your help.



Date: 06/10/2010 at 15:37:00
From: Doctor Ian
Subject: Re: Multiplying Fractions

Hi Jim,

Let's start with a particularly simple example, where one numerator is the
same as the other denominator:

  2   3
  - * -
  3   5

The first thing to notice is that when we follow the rule for multiplying
fractions, we get

  2   3   2 * 3
  - * - = -----
  3   5   3 * 5

Now, because multiplication is commutative, we can change the order of the
operands, so we could rewrite that as

  2   3   2 * 3   3 * 2 
  - * - = ----- = -----
  3   5   3 * 5   3 * 5

And we can "un-multiply" that to get

  2   3   2 * 3   3 * 2   3   2
  - * - = ----- = ----- = - * -
  3   5   3 * 5   3 * 5   3   5

But 3/3 is the same as 1, and multiplying by 1 is the same as doing
nothing:

  2   3   2 * 3   3 * 2   3   2       2   2
  - * - = ----- = ----- = - * - = 1 * - = -
  3   5   3 * 5   3 * 5   3   5       5   5

So here, the numerator and denominator just cancel out. Does this make
sense?

The important idea here is that when multiplying fractions, we can group
factors in the numerator and denominator any way we want. And if we have
something that appears in both the numerator and denominator, we can
"separate that out" to get n/n, or 1.

That's really what we're _doing_ when we cancel. We're separating out n/n
pairs, which "turn into 1," and thus disappear.

This really comes into its own when we break things into prime factors
before multiplying. For example,

   4   35
  -- * -- = ?
  15   72

We can break the numerators and denominators up into prime factors:

   4   35   2*2      5*7
  -- * -- = --- * ---------
  15   72   3*5   2*2*2*3*3

Those all just get multiplied together ...

   4   35      2*2*5*7
  -- * -- = -------------
  15   72   2*2*2*3*3*3*5

... and if we align them, we can spot pairs that can be canceled:

   4   35   2 * 2                 * 5 * 7
  -- * -- = ------------------------------
  15   72   2 * 2 * 2 * 3 * 3 * 3 * 5

We could go to the trouble of separating them ...

   4   35   2   2   5         7       
  -- * -- = - * - * - * -------------
  15   72   2   2   5   2 * 3 * 3 * 3

... but in practice, we don't bother. We would just cross them out, and
multiply what's left:

            x   x                   x
   4   35   2 * 2                 * 5 * 7       7       7
  -- * -- = ------------------------------ = ------- = --
  15   72   2 * 2 * 2 * 3 * 3 * 3 * 5        2*3*3*3   54
            x   x                   x

This is a lot easier than doing the multiplication, then reducing it by
finding common factors.

Still making sense? If so, we're almost done! Let's look at your example:

  4   8
  - * - = ?
  6   9

When we note that 6 and 8 have a common factor ...

   4    2*4
  --- * --- = ?
  2*3    9

... we recognize that we could separate those out, but we just toss the
common factor, and keep the rest:

   4     4
  --- * --- = ?
   3     9

It's the same result as writing everything out, and moving everything
around -- but a lot quicker.

Does this help? 

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



Date: 06/11/2010 at 08:08:27
From: Jim
Subject: Thank you (Multiplying Fractions)

After hearing me drone on and on about the importance of the commutative
property, my 7th graders really connected with this explanation. They are
now making up their own examples trying to "stump" each other. This is
also giving them invaluable practice working with factors.

Thanks so much!
Associated Topics:
Middle School Fractions

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