Adding Fractions

Date: 01/06/2003 at 16:03:17
From: Sharon
Subject: Adding fractions

My problem is 3/4 + 1/6. I have figured out how to get the same 
denominators, but in the example 3/4 + 1/6 the denominator is 12 but 
it says to build up my numerator, so then it is written 9/12 + 2/12 = 
11/12. I don't understand how they get 9 and 2 for numerators.

Date: 01/06/2003 at 17:34:52
From: Doctor Ian
Subject: Re: Adding fractions

Hi Sharon,

Suppose you have a piece of cake, and you cut it into 4 pieces and
give one piece away. You now have 3/4 of the cake, right? 

              +-----------+
              |           |
              |   1/4     |
              |           |
  +-----------+-----------+  1/4 + 1/4 + 1/4 = 3/4
  |           |           |
  |   1/4     |   1/4     |
  |           |           |
  +-----------+-----------+
   
Now, suppose you cut each of these pieces into three smaller pieces:

              +-----------+
              |   .   .   |
              |   .   .   |
              |   .   .   |
  +-----------+-----------+
  |   .   .   |   .   .   |
  |   .   .   |   .   .   |
  |   .   .   |   .   .   |
  +-----------+-----------+

How big is each of these pieces?  Well, if we'd cut the cake into
pieces of this size to begin with, 

  ............+-----------+
  .   .   .   |   .   .   |
  .   .   .   |   .   .   |
  .   .   .   |   .   .   |
  +-----------+-----------+
  |   .   .   |   .   .   |
  |   .   .   |   .   .   |
  |   .   .   |   .   .   |
  +-----------+-----------+

we'd have had 12 pieces. (Count them to make sure.) So each of these
smaller pieces is 1/12 of the original cake, and when we have 3/4 of
the cake, that's the same as having 9/12 of the cake:

              +-----------+                 +-----------+
              |   .   .   |                 |           |
              | 1 . 2 . 3 |                 |     1     |
              |   .   .   |                 |           |
  +-----------+-----------+  =  +-----------+-----------+
  |   .   .   |   .   .   |     |           |           |
  | 4 . 5 . 6 | 7 . 8 . 9 |     |     2     |     3     |
  |   .   .   |   .   .   |     |           |           |
  +-----------+-----------+     +-----------+-----------+

             9/12                          3/4

Does that make sense?  

Now, how do we get from 3/4 to 9/12 without drawing a picture? Well,
remember, we divided something into 4 pieces, and kept three of them.
If we divided it into 3 times as many pieces, we'd keep three times
as many of those, right?  So 

   keep 3 pieces                       keep 3*3 pieces
  --------------- must be the same as -----------------
  out of 4 pieces                     out of 3*4 pieces

Does _that_ make sense?  The way we write it in symbols is 

  3   3 * 3    9 
  - = ----- = --
  4   3 * 4   12

Does this answer your question?  If so, you should be able to 'build
up' 1/6 in the same way that we 'built up' 3/4:

  1   ? * 1    ? 
  - = ----- = --
  6   ? * 6   12
  
If not, write back and let me know where I lost you. 

I hope this helps!

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