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### Fractions and Common Denominators

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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?
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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)

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

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

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/
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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

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.

or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Fractions

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