Fractions and Common Denominators

Date: 06/04/2001 at 01:54:54
From: AT
Subject: Common Denominators

Why do you need a common denominator to add fractions, but not to 
multiply them? I have tried to answer this question by using the 
distributive property and the definition of division. I have always 
been taught that you need a common denominator to add fractions. I 
have never been told axiomatically why this has to be done. Also, 
why don't you need a common denominator to multiply fractions?

Date: 06/04/2001 at 12:47:52
From: Doctor Peterson
Subject: Re: Common Denominators

Hi, AT.

Let's approach fractions algebraically. A fraction is essentially a 
division problem stated but not worked out; rather than divide 3 by 4 
to get 0.75, we just write 3/4. This is much like algebra, where we 
write a/b and do not evaluate it, because we don't know the values. 
The reason we don't evaluate a fraction is that it has been found 
easier to manipulate the fraction as an expression (and then evaluate 
it at the end if necessary) rather than to evaluate it first, 
especially if we want an exact answer.

To multiply two fractions, we have to manipulate the expression

    (a/b) * (c/d)

until it is in a valid fractional form, one number "over" another. 
Let's do it:

    (a/b) * (c/d) = a * 1/b * c * 1/d
                  = a * c * 1/b * 1/d
                  = (ac) * 1/(bd)
                  = (ac) / (bd)

That was easy because multiplication and division work well together; 
division is a form of multiplication (by the reciprocal), and 
commutes.

Now let's add two fractions, manipulating

    (a/b) + (c/d)

until it looks like a fraction. We can't apply a commutative property, 
since addition and multiplication or division don't commute, but must 
use the distributive property (which relates addition and 
multiplication) instead. In particular, we have to choose a new 
denominator for the answer. It turns out that we can use (bd) as a 
denominator, and convert both fractions to that denominator by 
multiplying by 1:

    (a/b) + (c/d) = (a/b)(1) + (c/d)(1)
                  = (a/b)(d/d) + (c/d)(b/b)
                  = (ad)/(bd) + (bc)/(bd)

Now we can apply the distributive property to make this a single 
fraction:

                  = (ad) * 1/(bd) + (bc) * 1/(bd)
                  = (ad + bc)* 1/(bd)
                  = (ad+bc)/(bd)

(In working with actual fractions, we can save work by finding the 
LEAST common denominator; using variables, we don't have that issue to 
worry about.)

So why did we need a common denominator in the latter case, but not in 
the former? In both cases we needed a new denominator; but in 
multiplication, it arose by the commutative property, and is hardly 
noticeable as a common denominator. In doing our addition, we had to 
get the common denominator first so we could factor it out.

Here are a couple explanations we have given at a lower level, which 
may help:

   Common Denominators
   http://mathforum.org/dr.math/problems/curr2.8.96.html   

   When to Add or Multiply Denominators?
   http://mathforum.org/dr.math/problems/laurie.04.02.01.html   

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

Date: 06/04/2001 at 22:00:46
From: Doctor Ian
Subject: Re: Common Denominators 

Hi AT, 

If you have something like

  3/7 + 2/7 = ?

you can use the distributive property to simplify the expression on 
the left:

  3/7 + 2/7 = 3*(1/7) + 2*(1/7)
            = (3 + 2)*(1/7)
            = 5*(1/7)
            = 5/7

Without common denominators, you can't do this. That's why you need to 
have common denominators to add.

If you have something like

  3/4 * 5/6

it's really just a series of multiplications and divisions:

  ((3 / 4) * 5) / 6

which is why you don't need to have common denominators. 

Does this help? Write back if you'd like to talk about this some more, 
or if you have any other questions. 

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