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### Common Denominators

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Date: 2/8/96 at 8:48:56
From: Anonymous
Subject: Addition of fractions

Why do fractions need common denominators in order to be
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Date: 4/1/96 at 2:11:33
From: Doctor Jodi
Subject: Re: Addition of fractions

Hi there!  Let me try to give you an example of why common
denominators are important.  Suppose I want to know how tall I am.
I don't have any string, but I DO have a yardstick and a collection
of short sticks.

I find out that I am 1 yardstick and 3 short rulers tall.  Let's say I
stop here.  Now, I say to myself, I am 4 sticks tall.  If each stick
is a meter stick, then that's 12 feet!  I have friends who are over 6
feet tall, and I've heard of basketball players being 7 or so feet tall,
but 12 feet tall?  That's Jack-in-the-Beanstalk-GIANT tall and I know
that if I were in that story I *wouldn't* be the giant.  And I'm sure
that I didn't eat any strange mushrooms for breakfast, so I couldn't be
changing size (I hope!) like Alice.

So I try again.  I know that each of the short sticks is LESS than 1
yard (That's why I'm calling them short!).  So if I want to measure
myself in yards, I'll find that I'm this tall:  1 +  first stick in yards
+ second stick in yards + third stick in yards.

Now I need to find out how long each of the sticks is in yards.

I measure the first stick and I find out that it is 1/6 of a yard.

I measure the second stick and find that it is 3/4 of a yard.

Finally, I measure the third stick and find that it is 15/36 of a yard.

So I'm 1 + 1/6 + 3/4 + 15/36 yards tall.  So, I say to myself, I'm
1 and 19/36 of a yard, right?  Nope, I've made another mistake.

3/4 = 27/36, as you can see by multiplying numerator and denominator by
9.  But I wound up thinking I was 19/36 tall by adding the numerators
and taking the highest denominator.  This method, or any other attempt
to add fractions with different denominators, will give the wrong

Why? It's hard to give a good explanation, but I *can* compare it to
trying to add the yard stick to the three smaller sticks without taking
their size into account.  Similarly, if you add fractions with different
denominators, you're adding "sticks" of different sizes.  First, you
need to put them in a common system of measurement so that you can
compare their sizes.  This is the common denominator.

Here's one way to see the difference in sizes of different fractions:

Take a piece of paper and fold it equally into four pieces.
Now take another piece of paper and fold it equally into two pieces.

You can repeat this with as many different sizes as you want.  If you
are careful, you can also make 3 folds, representing 1/3, etc.

You can do the same thing with pies.

Suppose you were sharing two pizzas among several friends and made the
mistake of adding eighths and quarters together...
Would you be content with one-eighth of a pizza while your friend was
getting one-quarter?  Probably not....

Well, I hope that this helps you understand the importance of common
deonominators.  Write back if you have any more questions.

-Doctor Jodi,  The Math Forum

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Associated Topics:
Elementary Fractions

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