Common Denominators

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

Why do fractions need common denominators in order to be 
added? 

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 
answer.

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